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href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="category_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#motivation'>Motivation</a></li> <ul> <li><a href='#as_a_compromise_notion'>As a compromise notion</a></li> <li><a href='#as_a_motivation_for_categories'>As a motivation for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</a></li> </ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#prederivators'>Prederivators</a></li> <li><a href='#derivators'>Derivators</a></li> <li><a href='#SemiDerivators'>Semiderivators</a></li> <li><a href='#historical_remarks'>Historical remarks</a></li> </ul> <li><a href='#computing_homotopy_kan_extensions'>Computing homotopy Kan extensions</a></li> <ul> <li><a href='#equivalences'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalences</a></li> <li><a href='#ExactSquares'>Exact squares</a></li> </ul> <li><a href='#2CategoryOfDerivators'>The 2-category of derivators</a></li> <ul> <li><a href='#free_cocompletions'>Free cocompletions</a></li> </ul> <li><a href='#constructions_of_derivators'>Constructions of derivators</a></li> <ul> <li><a href='#from_homotopy_theory'>From homotopy theory</a></li> <ul> <li><a href='#theorem_dwyer_hirschhorn_kan_and_smith'>Theorem (Dwyer, Hirschhorn, Kan, and Smith)</a></li> <li><a href='#theorem_cisinski_2003'>Theorem (Cisinski, 2003)</a></li> </ul> <li><a href='#FromInfinityCategories'>From <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</a></li> <li><a href='#exotic_examples'>Exotic examples</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#PresentableDerivatorsAndCombinatorialModelCategories'>Presentable derivators and combinatorial model categories</a></li> </ul> <li><a href='#related_pages'>Related pages</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#relation_to_categories'>Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>An <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> can be flattened into a <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ho(C)</annotation></semantics></math>, called its <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category</a>, by forgetting higher morphisms. A <em>derivator</em> is a refinement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ho(C)</annotation></semantics></math>, in the sense that it retains enough information about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> for many purposes, like computing <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+limits">homotopy limits</a>. Roughly speaking, the idea is to retain the data of the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy categories</a> of all categories of <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, together with the induced <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between them. Since derivators can be studied using only ordinary <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> theory, they are often practical in situations when one requires more information than the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category</a> retains, but not the whole <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>Very similar structures were invented independently by <a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a> (who introduced the name “derivator”), <a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a> (who called his version “homotopy theories”), and Franke (who considered only the <a class="existingWikiWord" href="/nlab/show/stable+derivator">stable</a> case and called his version a “system of triangulated diagram categories”). The definition given below combines elements from the work of all three.</p> <h2 id="motivation">Motivation</h2> <p>The notion of derivator can be motivated in several ways.</p> <h3 id="as_a_compromise_notion">As a compromise notion</h3> <p>Suppose we start from the perspective that what we really study in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> are <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a>. The <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a> (the 1-category obtained by setting all equivalent 1-morphisms equal) is a fairly coarse invariant, but for some purposes it is sufficient. On the other hand, sometimes we need more than merely the homotopy category, but doing everything with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories can be technically daunting. Frequently, all we need from an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category that is not present in its homotopy category is to know that we have well-behaved constructions of <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>s and <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a>s.</p> <p>A <em>derivator</em> is thus a compromise notion, containing more information than a homotopy category, but being easier to work with than an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category. It consists of a homotopy category together with extra structure that enables one to compute with homotopy limits and colimits. Any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category with sufficiently many limits and colimits has an underlying derivator, and working with these derivators suffices for a surprising number of things we may want to do with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories. However, derivators are an essentially 1-categorical notion, so we can study them using ordinary <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> theory. Thus derivators provide a “truncated” version of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>, which gives us the language to characterize higher category theory using only usual <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, without any emphasis on any particular model (in fact, without assuming we even know any such model).</p> <p>For instance, it turns out that we can also express many convenient <a class="existingWikiWord" href="/nlab/show/universal+properties">universal properties</a> in terms of derivators. A striking example is the theory of <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated categories</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, its (bounded) <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D^b(A)</annotation></semantics></math> is not defined by a universal property. A natural statement would be that, given a triangulated category, the category of additive functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">A\to T</annotation></semantics></math> which send <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>s to distinguished triangles is equivalent to the category of triangulated functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">D^b(A)\to T</annotation></semantics></math>, but this statement is false, and in fact, does not even make sense (unless <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/semisimple+category">semi-simple</a>). But, in practice, everything behaves as if the above statement were meaningful and true. The ‘reason’ why this does not work is that the <a class="existingWikiWord" href="/nlab/show/mapping+cone">cone</a> of a map in a triangulated category is not defined by a universal property. On the other hand, the cone of a morphism of complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\to Y</annotation></semantics></math> is canonically defined: this is the <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> of the diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>←</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">0\leftarrow X \to Y</annotation></semantics></math>. In the world of derivators, we can remedy this situation and recover a suitable universal property.</p> <h3 id="as_a_motivation_for_categories">As a motivation for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</h3> <p>On the other hand, we may ask: Why do we take for granted that the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/topological+space">spaces</a> provides a good notion (among others) of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>? And why should we expect everything to be <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in higher groupoids anyway? A priori this may seem arbitrary (although it certainly works very well). Instead we might ask: what is the mathematical structure over which everything is canonically enriched? How can we even correctly formulate such a question?</p> <p>The notion of <em>derivator</em> provides a way to correctly formulate such a question, and the answer turns out to be mostly what we expect. Every type of “category theory” (at least, category theory without an <em>a priori</em> given <a class="existingWikiWord" href="/nlab/show/enriched+category">enrichment</a>) that we might want to do is automatically and uniquely enriched in <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a>, i.e. in the homotopy category of <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> or <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>. (For ordinary 1-category theory, this enrichment is trivial, i.e. factors through sets considered as discrete homotopy types.) A derivator is simply a structure with the characteristics of ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>: <a class="existingWikiWord" href="/nlab/show/category">categories</a>, <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s, <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s, <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a>s, <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">Grothendieck fibration</a>s. We can then show that any derivator acquires such an enrichment, so that homotopy types are a canonical enriching place for “category theories.”</p> <h2 id="definition">Definition</h2> <h3 id="prederivators">Prederivators</h3> <p>Let <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> denote the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of <a class="existingWikiWord" href="/nlab/show/large+category">large</a> <a class="existingWikiWord" href="/nlab/show/categories">categories</a> (not necessarily even <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a>), and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math> be some 2-category of <a class="existingWikiWord" href="/nlab/show/small+categories">small categories</a>, thought of as <strong><a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>s</strong>. One common choice is the 2-category of <em>all</em> small categories (which generates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>), but we might also choose the 2-category of finite categories. A <strong>prederivator</strong> with domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math> is a strict <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Dia</mi> <mi>op</mi></msup><mover><mo>→</mo><mi>D</mi></mover><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> Dia^{op} \overset{D}{\to} Cat </annotation></semantics></math></div> <p>As usual, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">(-)^{op}</annotation></semantics></math> denotes the 1-cell dual of a 2-category. Thus, a prederivator is a “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>-valued <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a>” on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math>. Prederivators form a 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PDer</mi></mrow><annotation encoding="application/x-tex">PDer</annotation></semantics></math> whose morphisms are <a class="existingWikiWord" href="/nlab/show/pseudonatural+transformations">pseudonatural transformations</a> and whose 2-cells are <a class="existingWikiWord" href="/nlab/show/modification">modification</a>s.</p> <p>Another common convention is to use the double dual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Dia</mi> <mi>coop</mi></msup></mrow><annotation encoding="application/x-tex">Dia^{coop}</annotation></semantics></math> which reverses both 1-cells and 2-cells, although confusingly sometimes in the literature this double dual is still denoted “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Dia</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Dia^{op}</annotation></semantics></math>”. The motivation for the latter choice seems to be that then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(A)</annotation></semantics></math> is the category of “<a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>” instead of the category of “<a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.” We have chosen the convention above since the main purpose of a derivator is a calculus of homotopy limits and colimits, and it is more usual to take limits and colimits of covariant functors. However, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Dia</mi> <mi>coop</mi></msup></mrow><annotation encoding="application/x-tex">Dia^{coop}</annotation></semantics></math> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Dia</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Dia^{op}</annotation></semantics></math> via the 2-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">(-)^{op}</annotation></semantics></math> (as long as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math> is closed under opposite categories in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>), there is really very little difference.</p> <p>There are two main motivating examples. Firstly, any category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> defines a “representable” prederivator by the assignation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↦</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A\mapsto Hom(A,C)</annotation></semantics></math>, sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Dia</mi></mrow><annotation encoding="application/x-tex">A\in Dia</annotation></semantics></math> to the category of functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A\to C</annotation></semantics></math>. This defines an embedding of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PDer</mi></mrow><annotation encoding="application/x-tex">PDer</annotation></semantics></math>.</p> <p>Secondly, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is any <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>, there is a prederivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> which sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Dia</mi></mrow><annotation encoding="application/x-tex">A\in Dia</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/localization">localization</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)(A)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(A,C)</annotation></semantics></math> relative to the <a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">objectwise weak equivalences</a> (as we allowed categories in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> to have large <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>s, these localization exist). In general, this is a non-representable prederivator, although of course if the weak equivalences are just the <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>s, it reduces to the representable case. Note that to construct it, we don’t need anything besides ordinary <a class="existingWikiWord" href="/nlab/show/2-category">(2-)</a><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>.</p> <h3 id="derivators">Derivators</h3> <p>A <strong>derivator</strong> is a prederivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> which satisfies a list of axioms. These axioms are of two sorts.</p> <p>The first set of axioms says that there exist well-behaved homotopy limits and colimits, and more generally <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a>s. Specifically, we require the following.</p> <ul> <li> <p><strong>(Der3)</strong> For any functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">u : X\to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><mo>:</mo><mi>D</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u^*:D(Y)\to D(X)</annotation></semantics></math> admits a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">u_!</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">u_*</annotation></semantics></math>. These can be understood as <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>u</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>u</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>u</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mover><munder><mo>→</mo><mrow><msub><mi>u</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>u</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>→</mo><mrow><msub><mi>u</mi> <mo>!</mo></msub></mrow></mover></mover><mi>D</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (u_! \dashv u^* \dashv u_*) : D(X) \stackrel{ \overset{u_!}{\to} } { \stackrel{ \overset{u^*}{\leftarrow} }{ \underset{u_*}{\to} } } D(Y) </annotation></semantics></math></div></li> <li> <p><strong>(Der4)</strong> For any <a class="existingWikiWord" href="/nlab/show/comma+square">comma square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>f</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>v</mi></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>→</mo><mi>u</mi></munder></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{A & \overset{g}{\to} & B \\ ^f\downarrow &\swArrow& \downarrow^v\\ C& \underset{u}{\to} & E} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Beck-Chevalley+transformation">Beck-Chevalley transformation</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><msup><mi>g</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>u</mi> <mo>*</mo></msup><msub><mi>v</mi> <mo>!</mo></msub><mspace width="1em"></mspace><mtext>and</mtext><mspace width="1em"></mspace><msup><mi>v</mi> <mo>*</mo></msup><msub><mi>u</mi> <mo>*</mo></msub><mo>→</mo><msub><mi>g</mi> <mo>*</mo></msub><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> f_! g^* \to u^* v_! \quad \text{and} \quad v^* u_* \to g_* f^* </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>s. Intuitively, this says that the Kan extensions in question are <em><a href="http://ncatlab.org/nlab/show/Kan+extension#Pointwise">pointwise</a></em>. In the presence of the second set of axioms, it suffices to require this when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a> (for the first case) and when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is so (for the second).</p> </li> </ul> <p>The second set of axioms are “<a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>” conditions. Of course, we cannot assert that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is exactly a sheaf (in the appropriate <a class="existingWikiWord" href="/nlab/show/stack">2-categorical sense</a>), since the terminal category in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math> is 2-categorically dense and so any sheaf on it is representable (and represented by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(1)</annotation></semantics></math>), whereas we want to also allow non-representable derivators. But we do need some sheaf-like properties in order to do <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. All of the following axioms can be understood as asserting that for some <a class="existingWikiWord" href="/nlab/show/cover">covering family</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Y</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Y_i \to X\}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math>, the canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Desc</mi><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>Y</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(X) \to Desc(D,\{Y_i\})</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(X)</annotation></semantics></math> to the category of <a class="existingWikiWord" href="/nlab/show/descent">descent</a> data for the covering, while not an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a>, has some weaker good properties. They can also be understood as 2-categorical <a class="existingWikiWord" href="/nlab/show/sketch">sketch</a> conditions.</p> <p>The standard axioms are:</p> <ul> <li> <p><strong>(Der1)</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Dia</mi> <mi>coop</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">D\colon Dia^{coop} \to Cat</annotation></semantics></math> takes <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>s to products. Sometimes we require this only for finite coproducts. In particular, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">D(\emptyset) = 1</annotation></semantics></math>.</p> </li> <li> <p><strong>(Der2)</strong> For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Dia</mi></mrow><annotation encoding="application/x-tex">X\in Dia</annotation></semantics></math>, consider the family of functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mn>1</mn><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x\colon 1\to X</annotation></semantics></math> determined by the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Then the induced functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>x</mi></munder><mi>D</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> D(X) \overset{(x^*)}{\to} \prod_x D(1)</annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/conservative+functor">conservative</a> (though not generally faithful). This in turn implies that the same is true for any jointly <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a> family of functors.</p> </li> <li> <p><strong>(Der5)</strong> For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Dia</mi></mrow><annotation encoding="application/x-tex">X\in Dia</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/interval+category">interval category</a>, then the induced functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> D(X\times I) \to Hom(I,D(X)) </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a> and <a class="existingWikiWord" href="/nlab/show/full+functor">full</a> (though again, it is not generally faithful). Since this functor is also conservative by (Der2), it is then a <a class="existingWikiWord" href="/nlab/show/weakly+smothering+functor">weakly smothering functor</a>.</p> </li> </ul> <p>There is substantial variation in (Der5). Sometimes it is convenient to assume this property when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is any (perhaps finite) <a class="existingWikiWord" href="/nlab/show/free+category">free category</a>. Some references do not include (Der5) at all in the definition, instead calling a derivator <strong>strong</strong> if it satisfies (Der5).</p> <div class="query"> <p><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>: It’s clear to me that these are desirable requirements, which are moreover satisfied by all derivators of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math>, but I would really like a conceptual explanation for why these axioms are <em>sufficient</em>.</p> </div> <p>It is easy to see that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has pointwise left and right Kan extensions along all functors in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math>, then its representable prederivator is a derivator. Somewhat more difficult to prove is that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> (or more generally, has well-behaved homotopy Kan extensions), then the prederivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> is also a derivator. Thus the derivator encodes the notions of <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> and of <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>. Note again that this way of seeing homotopy (co)limits does not use anything besides usual (2-)category theory.</p> <h3 id="SemiDerivators">Semiderivators</h3> <p>It may sometimes be useful to consider prederivators which are like derivators, but which “do not have <em>all</em> limits and colimits.” Let us say that a <strong>semiderivator</strong> is a prederivator satisfying (Der1), (Der2), and (Der5).</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is a semiderivator, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>D</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\in D(B)</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-shaped diagram, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">v\colon B\to C</annotation></semantics></math> is a functor in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math>, then a <strong>pointwise left extension</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> is an object “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mo>!</mo></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">v_! x</annotation></semantics></math>” in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(C)</annotation></semantics></math>, together with a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>→</mo><msup><mi>v</mi> <mo>*</mo></msup><msub><mi>v</mi> <mo>!</mo></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">x \to v^* v_! x</annotation></semantics></math> which is <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> in the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">/</mo><msup><mi>v</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x / v^*)</annotation></semantics></math> (this says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mo>!</mo></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">v_! x</annotation></semantics></math> is a “local” or “partial” value of the left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">v_!</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, although the entire functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">v_!</annotation></semantics></math> may not exist) which additionally satisfies the <a href="/nlab/show/Beck-Chevalley+condition#Local">local Beck-Chevalley condition</a> for any comma square as above. We have a dual notion of <strong>pointwise right extension</strong>.</p> <p>We say a semiderivator is <strong>complete</strong> (resp. <strong>cocomplete</strong>) if it admits all pointwise right (resp. left) extension. Clearly a semiderivator is a derivator just when it is both complete and cocomplete.</p> <h3 id="historical_remarks">Historical remarks</h3> <ul> <li> <p>Grothendieck’s definition of a <em>derivator</em> included only axioms (Der1), (Der2), (Der3), and (Der4).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a>‘s definition of a <em>homotopy theory</em> included axioms (Der1), (Der2), (Der3), a weaker form of (Der4), and (Der5) for finite free categories. His <em>pointed homotopy theories</em> add the axiom of a <a class="existingWikiWord" href="/nlab/show/pointed+derivator">pointed derivator</a>, and his <em>stable homotopy theories</em> include a weaker version of the axiom now used for a <a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a>.</p> </li> <li> <p>Franke’s definition of a <em>system of triangulated diagram categories</em> was irreducibly <a class="existingWikiWord" href="/nlab/show/pointed+derivator">pointed</a>, taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math> to consist of categories enriched over pointed sets. (See <a class="existingWikiWord" href="/nlab/show/pointed+derivator">pointed derivator</a> for the relationship of this approach to that of adding a “pointedness” axiom to an unpointed notion of derivator.) In this context, he assumed axioms analogous to (Der1), (Der2), (Der3), (Der4), and (Der5) for the interval category, plus the <a class="existingWikiWord" href="/nlab/show/stable+derivator">stability</a> axiom. He also observed that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math> consists entirely of posets, then (Der4) follows from the other axioms.</p> </li> </ul> <p>Axioms (Der1)–(Der4) are clearly the easiest to motivate and the most obviously necessary. Axiom (Der5) is used in order to do things with morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(*)</annotation></semantics></math>, by first lifting them to objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(I)</annotation></semantics></math>. In particular, it is necessary to conclude that a stable derivator gives a <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>.</p> <h2 id="computing_homotopy_kan_extensions">Computing homotopy Kan extensions</h2> <p>We now describe an “omnibus” theorem which is the main thing enabling us to compute with <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a>s in a derivator.</p> <h3 id="equivalences"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalences</h3> <p>For any functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">u\colon I\to J</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math>, we say it is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-<strong>equivalence</strong> if the induced transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>I</mi></msub><msub><mo stretchy="false">)</mo> <mo>!</mo></msub><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>I</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>→</mo><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>J</mi></msub><msub><mo stretchy="false">)</mo> <mo>!</mo></msub><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>J</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(\pi_I)_! (\pi_I)^* \to (\pi_J)_!(\pi_J)^*</annotation></semantics></math> is an isomorphism, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>I</mi></msub><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\pi_I\colon I\to *</annotation></semantics></math> is the projection to the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a>, and similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>J</mi></msub></mrow><annotation encoding="application/x-tex">\pi_J</annotation></semantics></math>. This means that homotopy colimits of <em>constant</em> diagrams of shapes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> are equivalent. By the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, this is equivalent to saying that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>J</mi></msub><msub><mo stretchy="false">)</mo> <mo>!</mo></msub><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>J</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>D</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>I</mi></msub><msub><mo stretchy="false">)</mo> <mo>!</mo></msub><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>I</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> D(*)((\pi_J)_!(\pi_J)^* X , Y) \to D(*)((\pi_I)_!(\pi_I)^* X , Y)</annotation></semantics></math></div> <p>is an isomorphism for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>D</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X,Y\in D(*)</annotation></semantics></math>, and by adjunction this is equivalent to saying that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>J</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>X</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>J</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>D</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>I</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>X</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>I</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> D(J)((\pi_J)^* X , (\pi_J)^*Y) \to D(I)((\pi_I)^* X , (\pi_I)^*Y)</annotation></semantics></math></div> <p>is an isomorphism—i.e. that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">u^*</annotation></semantics></math> is fully faithful when restricted to the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mi>J</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">\pi_J^*</annotation></semantics></math>. In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">u^*</annotation></semantics></math> is fully faithful, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence.</p> <p>In a representable derivator (i.e. in ordinary category theory), the colimit of a constant diagram of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/copower">copower</a> with the set of connected components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>. Thus, in a representable derivator, any functor that induces an isomorphism on sets of connected components will be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence, and the converse is true as long as the category in question is not a <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a>.</p> <p>By contrast, in the derivator coming from a model category or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category, the colimit of a constant diagram of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a copower with the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> regarded as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid, so in this case any functor that induces a homotopy equivalence of nerves (a stronger condition) will be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence. In fact, one can show:</p> <div class="num_theorem" id="NerveEquiv"> <h6 id="theorem">Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a>, Cisinski)</strong> A functor which induces a homotopy equivalence of nerves is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence for any derivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> </div> <p>This was proven by <a href="#Heller">Heller</a> using the canonical <a class="existingWikiWord" href="/nlab/show/enriched+derivator">enrichment</a> of any derivator over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids. It also follows from Cisinski’s theorem that the nerve equivalences are the smallest <a class="existingWikiWord" href="/nlab/show/basic+localizer">basic localizer</a>, once we verify that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalences are in fact a basic localizer—as we will now proceed to do.</p> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>Any functor in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math> with a fully faithful left or right adjoint is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⊣</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f\dashv g</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mo>*</mo></msup><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">g^* \dashv f^*</annotation></semantics></math>, and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is fully faithful, then the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mi>g</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">1 \to g f</annotation></semantics></math> is an isomorphism, and thus so is the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><msup><mi>g</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">1\to f^* g^*</annotation></semantics></math>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">g^*</annotation></semantics></math> is also fully faithful and so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence. The other case is dual.</p> </div> <div class="num_lemma"> <h6 id="lemma_2">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">W_D</annotation></semantics></math> denote the class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">W_D</annotation></semantics></math> is saturated, in the sense that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">u\colon I\to J</annotation></semantics></math> is a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math> which becomes an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi><mo stretchy="false">[</mo><msubsup><mi>W</mi> <mi>D</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Dia[W_D^{-1}]</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Fix some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>D</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X,Y\in D(*)</annotation></semantics></math> and consider the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo lspace="verythinmathspace">:</mo><mi>Dia</mi><mo>→</mo><msup><mi>Set</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Phi\colon Dia \to Set^{op}</annotation></semantics></math> which sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>π</mi> <mi>I</mi> <mo>*</mo></msubsup><mi>X</mi><mo>,</mo><msubsup><mi>π</mi> <mi>I</mi> <mo>*</mo></msubsup><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(I)(\pi_I^*X, \pi_I^*Y)</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> inverts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalences, it factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi><mo stretchy="false">[</mo><msubsup><mi>W</mi> <mi>D</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Dia[W_D^{-1}]</annotation></semantics></math>. But if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> becomes an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi><mo stretchy="false">[</mo><msubsup><mi>W</mi> <mi>D</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Dia[W_D^{-1}]</annotation></semantics></math>, then it must be inverted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>, but that is the definition of being a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence (as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> vary).</p> </div> <div class="num_lemma"> <h6 id="lemma_3">Lemma</h6> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, the class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalences is a <a class="existingWikiWord" href="/nlab/show/basic+localizer">basic localizer</a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Saturation gives 2-out-of-3 property and closure under retracts. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has a terminal object, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A\to 1</annotation></semantics></math> has a fully faithful right adjoint and hence is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence. Finally, given a triangle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>u</mi></mover></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo></mo><mi>v</mi></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mi>w</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>C</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ A & & \overset{u}{\to} & & B\\ & _v \searrow & & \swarrow_w \\ & & C, } </annotation></semantics></math></div> <p>to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence, it suffices to show that the transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mo>!</mo></msub><msubsup><mi>π</mi> <mi>A</mi> <mo>*</mo></msubsup><mo>→</mo><msub><mi>w</mi> <mo>!</mo></msub><msubsup><mi>π</mi> <mi>B</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">v_! \pi_A^* \to w_! \pi_B^*</annotation></semantics></math> is an isomorphism. By (Der2) it suffices to check this for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c\in C</annotation></semantics></math>. We can then form the comma objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>v</mi><mo stretchy="false">/</mo><mi>c</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>f</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>v</mi></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>→</mo><mrow></mrow></munder></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow><mspace width="1em"></mspace><mtext>and</mtext><mspace width="1em"></mspace><mrow><mtable><mtr><mtd><mi>w</mi><mo stretchy="false">/</mo><mi>c</mi></mtd> <mtd><mover><mo>→</mo><mi>k</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>h</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>w</mi></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>→</mo><mrow></mrow></munder></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{v/c & \overset{g}{\to} & A\\ ^f\downarrow && \downarrow^v\\ *& \underset{}{\to} & C} \quad\text{and}\quad \array{w/c & \overset{k}{\to} & B\\ ^h\downarrow && \downarrow^w\\ * & \underset{}{\to} & C} </annotation></semantics></math></div> <p>and the transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>c</mi> <mo>*</mo></msup><msub><mi>v</mi> <mo>!</mo></msub><msubsup><mi>π</mi> <mi>A</mi> <mo>*</mo></msubsup><mo>→</mo><msup><mi>c</mi> <mo>*</mo></msup><msub><mi>w</mi> <mo>!</mo></msub><msubsup><mi>π</mi> <mi>B</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">c^* v_! \pi_A^* \to c^* w_! \pi_B^*</annotation></semantics></math> factors as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>c</mi> <mo>*</mo></msup><msub><mi>v</mi> <mo>!</mo></msub><msubsup><mi>π</mi> <mi>A</mi> <mo>*</mo></msubsup><mo>≅</mo><msub><mi>f</mi> <mo>!</mo></msub><msup><mi>g</mi> <mo>*</mo></msup><msubsup><mi>π</mi> <mi>A</mi> <mo>*</mo></msubsup><mo>≅</mo><msub><mi>f</mi> <mo>!</mo></msub><msubsup><mi>π</mi> <mrow><mi>v</mi><mo stretchy="false">/</mo><mi>c</mi></mrow> <mo>*</mo></msubsup><mo>→</mo><msub><mi>h</mi> <mo>!</mo></msub><msubsup><mi>π</mi> <mrow><mi>w</mi><mo stretchy="false">/</mo><mi>c</mi></mrow> <mo>*</mo></msubsup><mo>≅</mo><msup><mi>c</mi> <mo>*</mo></msup><msub><mi>w</mi> <mo>!</mo></msub><msubsup><mi>π</mi> <mi>B</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex"> c^* v_! \pi_A^* \cong f_! g^* \pi_A^* \cong f_! \pi_{v/c}^* \to h_! \pi_{w/c}^* \cong c^* w_! \pi_B^*</annotation></semantics></math></div> <p>using the analogous map for the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo stretchy="false">/</mo><mi>c</mi><mo>→</mo><mi>w</mi><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">v/c \to w/c</annotation></semantics></math>. Therefore, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo stretchy="false">/</mo><mi>c</mi><mo>→</mo><mi>w</mi><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">v/c \to w/c</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence, this composite is an isomorphism, and if this holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>, then by (Der2), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence.</p> </div> <p>Therefore, since the nerve equivalences are the smallest basic localizer, every nerve equivalence is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence for any derivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> <h3 id="ExactSquares">Exact squares</h3> <p>Now suppose given any square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>L</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>I</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>q</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>μ</mi></msup></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>u</mi></msup></mtd></mtr> <mtr><mtd><mi>J</mi></mtd> <mtd><munder><mo>→</mo><mi>v</mi></munder></mtd> <mtd><mi>K</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{L & \overset{p}{\to} & I\\ ^q\downarrow & \Downarrow^\mu & \downarrow^u\\ J & \underset{v}{\to} & K} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dia</mi></mrow><annotation encoding="application/x-tex">Dia</annotation></semantics></math> which commutes up to a specified 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>. Given a derivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, we say that this square is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-<strong>exact</strong> if the <a class="existingWikiWord" href="/nlab/show/Beck-Chevalley+transformation">Beck-Chevalley transformation</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>q</mi> <mo>!</mo></msub><msup><mi>p</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>v</mi> <mo>*</mo></msup><msub><mi>u</mi> <mo>!</mo></msub><mspace width="1em"></mspace><mtext>and</mtext><mspace width="1em"></mspace><msup><mi>u</mi> <mo>*</mo></msup><msub><mi>v</mi> <mo>*</mo></msub><mo>→</mo><msub><mi>p</mi> <mo>*</mo></msub><msup><mi>q</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> q_! p^* \to v^* u_! \quad \text{and} \quad u^* v_* \to p_* q^* </annotation></semantics></math></div> <p>are isomorphisms. (In fact, if one of these is an isomorphism, so is the other, since they are <a class="existingWikiWord" href="/nlab/show/mate">mates</a>.) Thus, the derivator axioms say that all comma squares are exact.</p> <p>Like the notion of <a class="existingWikiWord" href="/nlab/show/exact+square">exact square</a> in ordinary category theory, this is a “functional” definition, but we can also give a more explicit characterization, using more or less the same argument. Given such a square as above and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i\in I</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j\in J</annotation></semantics></math>, we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i/L/j)</annotation></semantics></math> for the category whose objects are triples</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℓ</mi><mo>∈</mo><mi>L</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>i</mi><mover><mo>→</mo><mi>α</mi></mover><mi>p</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>q</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>β</mi></mover><mi>j</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">\big(\ell\in L,\, i\overset{\alpha}{\to} p(\ell),\, q(\ell)\overset{\beta}{\to} j\big).</annotation></semantics></math></div> <p>The morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i/L/j)</annotation></semantics></math> are morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mi>ℓ</mi><mo>→</mo><mi>ℓ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\gamma\colon \ell \to \ell'</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> which make the evident triangles commute. Now there is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>,</mo><mi>v</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r\colon (i/L/j) \to K(u(i),v(j))</annotation></semantics></math> (the latter considered as a discrete category), which sends the above triple to the composite</p> <div class="maruku-equation" id="eq:commacomp"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mover><mi>u</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mi>μ</mi></mover><mi>v</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mover><mi>v</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> u(i) \overset{u(\alpha)}{\to} u(p(\ell)) \overset{\mu}{\to} v(q(\ell)) \overset{v(\beta)}{\to} v(j) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>A square as above is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-exact if and only if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i\in I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j\in J</annotation></semantics></math>, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>,</mo><mi>v</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i/L/j) \to K(u(i),v(j))</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>It is easy to verify that the horizontal or vertical pasting composite of exact squares is exact; hence if the given square is exact, then so is the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>L</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>I</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>q</mi></msup></mtd> <mtd><mo>⇓</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>u</mi></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>→</mo><mi>j</mi></munder></mtd> <mtd><mi>J</mi></mtd> <mtd><munder><mo>→</mo><mi>v</mi></munder></mtd> <mtd><mi>K</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{(L/j) & \overset{}{\to} & L & \overset{p}{\to} & I \\ \downarrow & \Downarrow & \downarrow^q & \Downarrow & \downarrow^u\\ * & \underset{j}{\to} & J & \underset{v}{\to} & K} </annotation></semantics></math></div> <p>for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j\in J</annotation></semantics></math>, where the left-hand square is a comma square. Conversely, if all of these squares are exact, then so is the given one, by (Der2). We play the same game by composing with comma squares on the top to conclude that the given square is exact if and only if all the induced squares</p> <div class="maruku-equation" id="eq:ijsquare"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>g</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>→</mo><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>K</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{(i/L/j) & \overset{f}{\to} & *\\ ^g\downarrow & \Downarrow & \downarrow^{\mathrlap{u(i)}}\\ * & \underset{v(j)}{\to} & K} </annotation></semantics></math></div> <p>are exact. But the square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>,</mo><mi>v</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>s</mi></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>t</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>→</mo><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>K</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{K(u(i),v(j)) & \overset{s}{\to} & *\\ ^t\downarrow & \Downarrow & \downarrow^{\mathrlap{u(i)}}\\ * & \underset{v(j)}{\to} & K} </annotation></semantics></math></div> <p>is exact, since it is a comma square, and by the universal property of a comma square, the square <a class="maruku-eqref" href="#eq:ijsquare">(2)</a> factors uniquely through this one by a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>,</mo><mi>v</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r\colon (i/L/j) \to K(u(i),v(j))</annotation></semantics></math>, which is precisely the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> defined above. Specifically, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>s</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">f = s r</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><mi>t</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">g = t r</annotation></semantics></math>. Therefore, the Beck-Chevalley transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mo>!</mo></msub><msup><mi>f</mi> <mo>*</mo></msup><mo>→</mo><mo stretchy="false">(</mo><mi>v</mi><mi>j</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>u</mi><mi>i</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">g_! f^* \to (v j)^* (u i)^*</annotation></semantics></math> is equal to the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mo>!</mo></msub><msup><mi>f</mi> <mo>*</mo></msup><mo>=</mo><msub><mi>t</mi> <mo>!</mo></msub><msub><mi>r</mi> <mo>!</mo></msub><msup><mi>r</mi> <mo>*</mo></msup><msup><mi>s</mi> <mo>*</mo></msup><mo>→</mo><msub><mi>t</mi> <mo>!</mo></msub><msup><mi>s</mi> <mo>*</mo></msup><mo>→</mo><mo stretchy="false">(</mo><mi>v</mi><mi>j</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>u</mi><mi>i</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> g_! f^* = t_! r_! r^* s^* \to t_! s^* \to (v j)^* (u i)^* </annotation></semantics></math></div> <p>where the second map is the Beck-Chevalley transformation for the comma square, which is an isomorphism. Thus, <a class="maruku-eqref" href="#eq:ijsquare">(2)</a> is exact just when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mo>!</mo></msub><msub><mi>r</mi> <mo>!</mo></msub><msup><mi>r</mi> <mo>*</mo></msup><msup><mi>s</mi> <mo>*</mo></msup><mo>→</mo><msub><mi>t</mi> <mo>!</mo></msub><msup><mi>s</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">t_! r_! r^* s^* \to t_! s^*</annotation></semantics></math> is an isomorphism. But this says exactly that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalence.</p> </div> <p>A square is said to be <a class="existingWikiWord" href="/nlab/show/homotopy+exact+square">homotopy exact</a> if it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-exact for all derivators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> (or, equivalently, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories, or for all model categories, or simplicially enriched categories).</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>A square is homotopy exact if and only if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i\in I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j\in J</annotation></semantics></math>, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>,</mo><mi>v</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i/L/j) \to K(u(i),v(j))</annotation></semantics></math> induces a weak homotopy equivalence of nerves.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>“If” follows directly from Theorem <a class="maruku-ref" href="#NerveEquiv"></a> and the previous theorem. Conversely, we can take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> to be the derivator of spaces (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids), where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-equivalences are precisely the nerve equivalences.</p> </div> <div class="num_cor"> <h6 id="corollary_2">Corollary</h6> <p>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i\in I</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j\in J</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>φ</mi><mo lspace="verythinmathspace">:</mo><mi>u</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>→</mo><mi>v</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\varphi\colon u(i) \to v(j)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><msub><mo stretchy="false">)</mo> <mi>φ</mi></msub></mrow><annotation encoding="application/x-tex">(i/L/j)_\varphi</annotation></semantics></math> denote the subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i/L/j)</annotation></semantics></math> consisting of those triples for which the composite <a class="maruku-eqref" href="#eq:commacomp">(1)</a> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math>. Then the square is homotopy exact if and only if each category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">/</mo><mi>L</mi><mo stretchy="false">/</mo><mi>j</mi><msub><mo stretchy="false">)</mo> <mi>φ</mi></msub></mrow><annotation encoding="application/x-tex">(i/L/j)_\varphi</annotation></semantics></math> has a contractible nerve.</p> </div> <p>This gives a convenient way to compute many homotopy limits and colimits, which works in any derivator, and <em>a fortiori</em> in any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category or model category. Example applications can be found at <a class="existingWikiWord" href="/nlab/show/homotopy+exact+square">homotopy exact square</a>.</p> <h2 id="2CategoryOfDerivators">The 2-category of derivators</h2> <p>We can consider now the 2-category of derivators. Actually, there are several different ways to define such a 2-category, depending on whether we require morphisms to preserve homotopy colimits, limits, both, or neither. Let us write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Der</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">Der_!</annotation></semantics></math> for the 2-category whose morphisms preserve colimits. Thus:</p> <ul> <li> <p>its <a class="existingWikiWord" href="/nlab/show/object">object</a>s are derivators,</p> </li> <li> <p>its 1-<a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s are <a class="existingWikiWord" href="/nlab/show/pseudonatural+transformations">pseudonatural transformations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>D</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">F:D\to D'</annotation></semantics></math> which commute with the functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">u_!</annotation></semantics></math> (i.e. the canonical comparison maps for these are isomorphisms),</p> </li> <li> <p>its 2-cells are the <a class="existingWikiWord" href="/nlab/show/modifications">modifications</a> between these.</p> </li> </ul> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mi>D</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Der</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mi>D</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_!(D,D') = Der_!(D,D')</annotation></semantics></math> for hom-categories in this 2-category.</p> <h3 id="free_cocompletions">Free cocompletions</h3> <p>Now, given a small category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, there is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functor, defined by evaluating at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Der</mi> <mo>!</mo></msub><mo>→</mo><mi>Cat</mi><mspace width="1em"></mspace><mo>,</mo><mspace width="2em"></mspace><mi>D</mi><mo>↦</mo><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">Der_!\to Cat \quad , \qquad D\mapsto D(X).</annotation></semantics></math></div> <p>Note that, for any (pre)derivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(X)</annotation></semantics></math> is canonically equivalent to the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(X,D)</annotation></semantics></math> of morphisms of <em>prederivators</em> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> (considering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as a representable prederivator). Of course, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> will usually not itself be a derivator, but nevertheless this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functor is <em>representable</em> in the 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Der</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">Der_!</annotation></semantics></math>.</p> <p>Thus, there exists a derivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{X}</annotation></semantics></math>, endowed with a morphism of prederivators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mover><mi>X</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">h:X\to \widehat{X}</annotation></semantics></math> (called the Yoneda embedding), such that, for any derivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, composing with h defines an equivalence of categories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo>^</mo></mover><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">Hom_!(\widehat{X},D)\simeq Hom(X,D)=D(X).</annotation></semantics></math></div> <p>In other words, the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mover><mi>X</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">h:X\to \widehat{X}</annotation></semantics></math> is the “free completion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by homotopy colimits” in the sense of derivators.</p> <p>Furthermore, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{X}</annotation></semantics></math> can be described rather explicitly: it is the derivator associated to the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model category of simplicial presheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. In particular, the usual <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">homotopy theory of simplicial sets</a> gives rise to the derivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>*</mo><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{*}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math> stands for the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> category. Note that in order to conclude this, we didn’t take for granted that <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>s should be important: its universal property is formulated purely with ordinary category theory.</p> <p>From there, you can see that any derivator is canonically enriched in the derivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>*</mo><mo>^</mo></mover><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>SSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat{*}\simeq Ho(SSet)</annotation></semantics></math>: as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math> acts uniquely on any prederivator, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>*</mo><mo>^</mo></mover><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>SSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat{*}\simeq Ho(SSet)</annotation></semantics></math> acts uniquely on any derivator (as far as we ask compatibility with homotopy colimits). Thus the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a> might be reformulated vaguely as: is there an algebraic model of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>*</mo><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{*}</annotation></semantics></math>? Then one may guess that some notion of higher groupoid might do the job.</p> <h2 id="constructions_of_derivators">Constructions of derivators</h2> <p>Viewing a derivator as a partway point between an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category and its homotopy category, and recalling that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories are often presented by <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a> and in particular <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>, we can construct derivators in two general ways.</p> <h3 id="from_homotopy_theory">From homotopy theory</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,W)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>, then the representable prederivator defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Cat</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">D_C(X) = Cat(X,C) = C^X</annotation></semantics></math> comes equipped with weak equivalences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">W^X</annotation></semantics></math> on each category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">C^X</annotation></semantics></math>. We define the <em>homotopy prederivator</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> by inverting these weak equivalences in each diagram category:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>X</mi></msup><mo>,</mo><msup><mi>W</mi> <mi>X</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mi>X</mi></msup><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msup><mi>W</mi> <mi>X</mi></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Ho(C)(X) = Ho(C^X,W^X) = C^X[(W^X)^{-1}]</annotation></semantics></math></div> <div class="num_theorem"> <h6 id="theorem_dwyer_hirschhorn_kan_and_smith">Theorem (Dwyer, Hirschhorn, Kan, and Smith)</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> with a functorial <span class="newWikiWord">three-arrow calculus<a href="/nlab/new/three-arrow+calculus">?</a></span>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> is a prederivator satisfying axioms (Der1), (Der2), and (Der5).</p> </div> <p>Actually, it is enough to assume that, for every small category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">C^X</annotation></semantics></math> admits a (not necessarily functorial) three-arrow calculus, but in practice this only happens when we have a functorial three-arrow calculus for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <div class="num_theorem"> <h6 id="theorem_cisinski_2003">Theorem (Cisinski, 2003)</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> (complete and cocomplete, but possibly without functorial factorisations), then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> is a derivator (with the strong version of Der5).</p> </div> <p>Axioms (Der1) is easy to check, and when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has functorial factorisations, we have a functorial three-arrow calculus. Axiom (Der3) is also easy, since <a class="existingWikiWord" href="/nlab/show/homotopy+limits">homotopy limits</a> and colimits in categories with weak equivalences are <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>s of the usual limits and colimits, and so they supply left and right adjoints to derived pullback functors. (This shows moreover that homotopy limits and Kan extensions in a model category coincide with the notions of homotopy limit and Kan extension in its homotopy derivator, so that by working in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> we really are studying the things we want to study.) Axiom (Der4) requires a bit of work; there is a proof for combinatorial model categories using the <a class="existingWikiWord" href="/nlab/show/injective+model+structure">injective model structure</a> in <a href="#Groth">(Groth)</a>. Axiom Der 5 is discussed in Theorem 9.8.5 in <a href="#RadBan06">(Radulescu Banu)</a>. The theorem above remains true for suitable (co)fibration categories for which a suitable version of the injective and projective structures are always available; see Theorem 9.5.5 in <a href="#RadBan06">(Radulescu Banu)</a>.</p> <p>Note that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is any complete and cocomplete 1-category, we can equip it with its <a class="existingWikiWord" href="/nlab/show/trivial+model+structure">trivial model structure</a> in which the only weak equivalences are isomorphisms. Then the above derivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> is the same as the representable prederivator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_C(X)</annotation></semantics></math>, which can easily be proven to be a derivator directly.</p> <p>In this connection, it is interesting to point out that a given homotopy category can admit multiple “enrichments” to a derivator. For instance, the homotopy category of the model category of chain complexes over a field is equivalent to the category of graded modules over that field, which is itself complete and cocomplete. Thus we have two distinct derivators, which have equivalent underlying homotopy categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(*)</annotation></semantics></math>.</p> <h3 id="FromInfinityCategories">From <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>, it has a <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> obtained by identifying equivalent <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a>. If our <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories are modeled by <a class="existingWikiWord" href="/nlab/show/quasicategories">quasicategories</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(-)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> of the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>, often denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tau_1</annotation></semantics></math>.</p> <p>Since categories are in particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories, for any category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> we have a <a class="existingWikiWord" href="/nlab/show/functor+%28%E2%88%9E%2C1%29-category">functor (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">C^X</annotation></semantics></math>, and thus a homotopy category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>X</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C^X)</annotation></semantics></math>. We define the <em>homotopy prederivator</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>X</mi></msup><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">Ho(C)(X) = Ho(C^X).</annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/limit+in+an+%28%E2%88%9E%2C1%29-category">limits</a> and colimits in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categorical sense, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> is a derivator (this is an interpretation of the theory of pointwise Kan extensions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories). A proof of this is sketched in <a href="#GPS14">GPS, Example 2.5</a>. The comparison of the derivator associated to a cofibration category (hence also to any model category) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and of the derivator associated to the Dwyer-Kan localization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is discussed by <a href="#Lenz">Lenz</a>.</p> <h3 id="exotic_examples">Exotic examples</h3> <p>There are also examples of derivators not satisfying (Der5) that do not arise from homotopy theory or higher category theory in this way. For instance, see Remark 5.4 and Example 5.5 in <a href="#LN16">Lagkas-Nikolos</a>.</p> <h2 id="properties">Properties</h2> <h3 id="PresentableDerivatorsAndCombinatorialModelCategories">Presentable derivators and combinatorial model categories</h3> <p>We discuss precise versions of the idea that derivators indeed constitute a model of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>.</p> <div class="num_theorem" id="RenaudinTheorem"> <h6 id="theorem_3">Theorem</h6> <p>The sub-<a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of “<a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a>” derivators with left adjoints as morphisms between them, among the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of all derivators (see <a href="#2CategoryOfDerivators">above</a>), is <a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/2-localization">2-localization</a> of the 2-category of <a class="existingWikiWord" href="/nlab/show/left+proper+model+category">left proper</a><sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup> <a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a> at the <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a>.</p> </div> <p>This is shown in (<a href="#Renaudin06">Renaudin 2006</a>). See at <a class="existingWikiWord" href="/nlab/show/Ho%28CombModCat%29">Ho(CombModCat)</a> for more.</p> <p>Notice that <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-categories">locally presentable (∞,1)-categories</a> are precisely those <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> that arise, up to <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a> as <a class="existingWikiWord" href="/nlab/show/simplicial+localizations">simplicial localizations</a> of <a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a>. Hence this theorem suggests that there is, at least, an <a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a> between the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of presentable derivators and the <a class="existingWikiWord" href="/nlab/show/homotopy+2-category">homotopy 2-category</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a> <a class="existingWikiWord" href="/nlab/show/Pr%28%E2%88%9E%2C1%29Cat">Pr(∞,1)Cat</a>. However, an actual proof of this seems to be missing.</p> <p>As a corollary, we obtain a canonical comparison <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>PrDer</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>Pr</mi><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho(PrDer) \longrightarrow Ho(Pr(\infty,1)Cat) </annotation></semantics></math></div> <p>from the homotopy category presentable derivators with left adjoint morphisms, to the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/Pr%28%E2%88%9E%2C1%29Cat">Pr(∞,1)Cat</a>.</p> <p>This is induced by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/2-localization+of+a+2-category">2-localization</a> from the fact that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup></mrow><annotation encoding="application/x-tex">L^H</annotation></semantics></math> that forms <a class="existingWikiWord" href="/nlab/show/simplicial+localizations">simplicial localizations</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>CombModCat</mi><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>Pr</mi><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L^H \;\colon\; CombModCat \longrightarrow Ho(Pr(\infty,1)Cat) </annotation></semantics></math></div> <p>sends <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a> to <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a> (see thereresentable infinity-category#PresentedByCombinatorialSimplicialModelCategories)).</p> <h2 id="related_pages">Related pages</h2> <p>Special kinds of derivators:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/pointed+derivator">pointed derivator</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></li> <li><a class="existingWikiWord" href="/nlab/show/monoidal+derivator">monoidal derivator</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+derivator">enriched derivator</a></li> <li><span class="newWikiWord">finitely complete prederivator<a href="/nlab/new/finitely+complete+prederivator">?</a></span></li> <li><span class="newWikiWord">regular derivator<a href="/nlab/new/regular+derivator">?</a></span></li> </ul> <p>The calculus of homotopy Kan extensions used in derivators:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+exact+square">homotopy exact square</a></li> </ul> <p>Special limits and structures in derivators:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/pullback+in+a+derivator">pullback in a derivator</a></li> <li><a class="existingWikiWord" href="/nlab/show/monomorphism+in+a+derivator">monomorphism in a derivator</a></li> <li><a class="existingWikiWord" href="/nlab/show/coend+in+a+derivator">coend in a derivator</a></li> <li><a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system+in+a+derivator">orthogonal factorization system in a derivator</a></li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The term <em>derivator</em> is originally due to <a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a>, introduced in <em><a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">Pursuing Stacks</a></em> . The first fifteen chapters of a 2000 page manuscript of Grothendieck (in French) about derivators can be found at:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, <em><a href="http://people.math.jussieu.fr/~maltsin/groth/Derivateurs.html">Les Dérivateurs</a></em></li> </ul> <p>Independently, there is a version due to <a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a> (who called them “homotopy theories”):</p> <ul id="Heller"> <li><a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a>, <em>Homotopy theories</em> , Memoirs of the American Mathematical Society, Vol. 71, No 383 (1988).</li> </ul> <ul> <li>Alex Heller, <em>Stable homotopy theories and stabilization</em> , <a href="http://www.ams.org/mathscinet-getitem?mr=1431157">MR</a> – see <a class="existingWikiWord" href="/nlab/show/pointed+derivator">pointed derivator</a> and <a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></li> </ul> <p>Apparently also independent is the development by Franke, who takes an enriched approach to the pointed case and also assumes stability:</p> <ul> <li>Jens Franke, <em>Uniqueness theorems for certain triangulated categories with an Adams spectral sequence</em>, <a href="http://www.math.uiuc.edu/K-theory/0139/">K-theory archive</a></li> </ul> <p>Georges Maltsiniotis has written an introduction to the topic (in French):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Georges+Maltsiniotis">Georges Maltsiniotis</a>, <em>Introduction à la théorie des dérivateurs, d’après Grothendieck</em> , Preprint (2001) <a href="http://www.math.jussieu.fr/~maltsin/ps/m.ps">ps</a></li> </ul> <p>He also gave a <a href="http://congreso.us.es/htag09/php/index.php?carga=courses">course</a> (in English) in Seville, Sep 2010, and part 3 is on derivators:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Georges+Maltsiniotis">Georges Maltsiniotis</a></p> <ul> <li> <p><a href="http://people.math.jussieu.fr/~maltsin/Seville/Lecture_I_Localizers.pdf">Lecture I (Localizers)</a></p> <p><a href="http://people.math.jussieu.fr/~maltsin/Seville/Lecture_II_Model_Categories.pdf">Lecture II (Model Categories)</a></p> <p><a href="http://people.math.jussieu.fr/~maltsin/Seville/Lecture_III_Derivators.pdf">Lecture III (Derivators)</a></p> <p><a href="http://people.math.jussieu.fr/~maltsin/Seville/Summary_on_Derivators.pdf">Summary on Derivators</a></p> </li> </ul> </li> </ul> <p>Part of the above material is adapted from</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em><a href="http://golem.ph.utexas.edu/category/2010/03/a_perspective_on_higher_catego.html#c032227">blog comment on derivators</a></em></li> </ul> <p>Cisinski has also written a number of papers on the subject (in French), which can be found at <a href="http://www.mathematik.uni-regensburg.de/cisinski/publikationen.html">his homepage</a>.</p> <ul> <li> <p>— Images directes cohomologiques dans les catégories de modèles. <em>Ann. Math. Blaise Pascal</em>, 10(2):195–244, 2003.</p> </li> <li> <p>— Catégories dérivables, Bull. Soc. Math. France, Tome 138 (2010) no. 3, pp. 317-393. <a href="http://www.numdam.org/item/BSMF_2010__138_3_317_0/">NUMDAM</a></p> </li> </ul> <p>Derivators were also recently used by <a class="existingWikiWord" href="/nlab/show/Gon%C3%A7alo+Tabuada">Gonçalo Tabuada</a> in a universal characterization of higher <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gon%C3%A7alo+Tabuada">Gonçalo Tabuada</a>, “Higher K-theory via universal invariants” <a href="http://arxiv.org/abs/0706.2420">arXiv</a>.</li> </ul> <p>An introduction to some of the theory of pointed and stable derivators, in English, can be found in the paper:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a> and <a class="existingWikiWord" href="/nlab/show/Amnon+Neeman">Amnon Neeman</a>, <em>Additivity for derivator K-theory</em> , <a href="http://www.ams.org/mathscinet-getitem?mr=2382732">MR</a></li> </ul> <p>An introductory discussion aimed towards <a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a>s is also in</p> <ul> <li id="Groth"><a class="existingWikiWord" href="/nlab/show/Moritz+Groth">Moritz Groth</a>, <em>Monoidal derivators and additive derivators</em>, <a href="http://arxiv.org/abs/1203.5071">arxiv/1203.5071</a>; <em>Derivators, pointed derivators, and stable derivators</em> (<a href="http://www.math.uni-bonn.de/~mgroth/groth_derivators.pdf">pdf</a>)</li> </ul> <p>Other references include:</p> <ul> <li id="RadBan06"> <p>Andrei Radulescu Banu, Cofibrations in Homotopy Theory, <a href="https://arxiv.org/abs/math/0610009">arXiv:math/0610009</a>, 2006</p> </li> <li id="Lenz"> <p>Tobias Lenz, Homotopy (Pre-)Derivators of Cofibration Categories and Quasi-Categories, Algebr. Geom. Topol. 18 No. 6 (2018), pp. 3601–3646. <a href="https://arxiv.org/abs/1712.07845">arXiv:1712.07845</a></p> </li> <li id="IN16"> <p>Ioannis Lagkas-Nikolos, <em>Levelwise modules over separable monads on stable derivators</em>, <a href="https://arxiv.org/abs/1608.06340">arXiv:1608.06340</a>, 2016</p> </li> <li id="Coley20"> <p>Coley, Ian, <em>The theory of half derivators</em>, <a href="https://arxiv.org/abs/2010.12057">arxiv:2010.12057</a>, 2020</p> </li> <li id="Richardson20"> <p>Richardson, James, <em>Enriched derivators</em>, <a href="https://arxiv.org/abs/2010.07740">arXiv:2010.07740</a>, 2020</p> </li> <li id="GPS14"> <p><a class="existingWikiWord" href="/nlab/show/Moritz+Groth">Moritz Groth</a>, <a class="existingWikiWord" href="/nlab/show/Kate+Ponto">Kate Ponto</a>, and <a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Mayer-Vietoris sequences in stable derivators</em>. Homology, Homotopy and Applications 16 (1) 2014, <a href="http://arxiv.org/abs/1306.2072">arxiv:1306.2072</a></p> </li> </ul> <h3 id="relation_to_categories">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories</h3> <p>An brief informal discussion of derivators as a 2-categorical tool for studying <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,1)</annotation> </semantics> </math>-categories</a> is contained in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Squeezing Higher Categories out of Lower Categories</em> (<a href="http://golem.ph.utexas.edu/category/2010/05/squeezing_higher_categories_ou.html">blog</a>)</li> </ul> <p>In the paper</p> <ul> <li id="Renaudin06"><a class="existingWikiWord" href="/nlab/show/Olivier+Renaudin">Olivier Renaudin</a>, <em>Plongement de certaines théories homotopiques de Quillen dans les dérivateurs</em>, Journal of Pure and Applied Algebra Volume 213, Issue 10, October 2009, Pages 1916-1935 <p>(<a href="https://arxiv.org/abs/math/0603339">arXiv:math/0603339</a>, <a href="https://doi.org/10.1016/j.jpaa.2009.02.014">doi:10.1016/j.jpaa.2009.02.014</a>)</p> </li> </ul> <p>it is proven that the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of “<a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a>” derivators is equivalent to the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> of the 2-category of <a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a> at the <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a> (see also at <em><a class="existingWikiWord" href="/nlab/show/Ho%28CombModCat%29">Ho(CombModCat)</a></em>). Thus in some sense derivators capture “all the information” about a combinatorial model category, hence also about a <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a>.</p> <p>Along somewhat similar lines, in</p> <ul> <li>Kevin Arlin, <em>On the ∞-categorical Whitehead theorem and the embedding of quasicategories in prederivators</em> (<a href="https://arxiv.org/abs/1612.06980">arxiv:1612.06980</a>)</li> </ul> <p>it is shown that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories can be literally embedded in “prederivators” as long as we allow <em>strict morphisms</em> between the latter (which is arguably somewhat against the spirit of derivators, but still interesting), and that at the 2-categorical level the embedding is conservative but generally fails to be full on large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories. This is developed further in</p> <ul> <li>Daniel Fuentes-Keuthan, Magdalena Kedziorek, Martina Rovelli, <em>A model structure on prederivators for (∞,1)-categories</em>, <a href="https://arxiv.org/abs/1810.06496">arxiv</a>, 2018</li> </ul> <p>On lifting derivators from the <a class="existingWikiWord" href="/nlab/show/homotopy+categories+of+%28infinity%2C1%29-categories">homotopy categories of (infinity,1)-categories</a> to the <a class="existingWikiWord" href="/nlab/show/homotopy+2-categories">homotopy 2-categories</a> of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C2%29-categories">(infinity,2)-categories</a>:</p> <ul> <li id="DiVittorio23"><a class="existingWikiWord" href="/nlab/show/Nicola+Di+Vittorio">Nicola Di Vittorio</a>, <em>2-derivators</em> (2023). [<a href="https://arxiv.org/abs/2309.05216">arXiv:2309.05216</a>]</li> </ul> <div class="footnotes"><hr /><ol><li id="fn:1"> <p>The condition of left properness does not appear in the arXiv version of <a href="#Renaudin06">Renaudin 2006</a>, but is added in the published version. By <a class="existingWikiWord" href="/nlab/show/Dugger%27s+theorem">Dugger's theorem</a> (see <a href="combinatorial+model+category#EveryCombinatorialModelCatQEquivalentToLeftProper">here</a>) every combinatorial model category is Quillen equivalent to a left proper one, but it is not immediate that every <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> of Quillen equivalences between left proper combinatorial model categories may be taken to pass through only left proper ones. <a href="#fnref:1" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on September 12, 2023 at 17:59:33. See the <a href="/nlab/history/derivator" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/derivator" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/887/#Item_126">Discuss</a><span class="backintime"><a href="/nlab/revision/derivator/73" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/derivator" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/derivator" accesskey="S" class="navlink" id="history" rel="nofollow">History (73 revisions)</a> <a href="/nlab/show/derivator/cite" style="color: black">Cite</a> <a href="/nlab/print/derivator" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/derivator" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>