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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> <h4 id="enriched_category_theory">Enriched category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>, <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, <a class="existingWikiWord" href="/nlab/show/double+category">double category</a>, <a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual double category</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+adjoint+functor">enriched adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+product+category">enriched product category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a></p> </li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>, <a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> </ul> <h2 id="extra_stuff_structure_property">Extra stuff, structure, property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/copowering">copowering</a> (<a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/powering">powering</a> (<a class="existingWikiWord" href="/nlab/show/cotensoring">cotensoring</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+category">monoidal enriched category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+enriched+category">cartesian closed enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+enriched+category">locally cartesian closed enriched category</a></p> </li> </ul> </li> </ul> <h3 id="homotopical_enrichment">Homotopical enrichment</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+homotopical+category">enriched homotopical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">model structure on homotopical presheaves</a></p> </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='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#the_infinity1category_of_dgcategories'>The (infinity,1)-category of dg-categories</a></li> <li><a href='#relation_to_stable_categories'>Relation to stable <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> <li><a href='#aspects_of_dgcategories'>Aspects of dg-categories</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#overviews'>Overviews</a></li> <li><a href='#homotopy_theory_of_dgcategories'>Homotopy theory of dg-categories</a></li> <li><a href='#derived_noncommutative_geometry'>Derived noncommutative geometry</a></li> <li><a href='#other_aspects'>Other aspects</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Differential graded categories</em> or <em>dg-categories</em> are linear analogues of <a class="existingWikiWord" href="/nlab/show/spectral+categories">spectral categories</a>. In other words they are <a class="existingWikiWord" href="/nlab/show/linear+%28infinity%2C1%29-category">linear</a> <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-categories">stable (infinity,1)-categories</a>. It is common and useful to view them as <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+categories">enhanced triangulated categories</a>.</p> <h2 id="definition">Definition</h2> <p>A <em>dg-category</em> over a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <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> is an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> <a class="existingWikiWord" href="/nlab/show/enriched+%28infinity%2C1%29-category">enriched</a> in the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+chain+complexes">(infinity,1)-category of chain complexes</a> of <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>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a>. Equivalently, it is an ordinary category <a class="existingWikiWord" href="/nlab/show/enriched+category">strictly enriched</a> in <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> (see <a href="#Haugseng13">Haugseng 13</a>).</p> <p>Hence a dg-category is a category with <a class="existingWikiWord" href="/nlab/show/mapping+complexes">mapping complexes</a> of morphisms between any two objects. By taking the <a class="existingWikiWord" href="/nlab/show/homologies">homologies</a> of these <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> in degree zero, one gets an ordinary category, called the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+dg-category">homotopy category of a dg-category</a>. Notice that a dg-category with a single object is the same thing as a <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>.</p> <h2 id="properties">Properties</h2> <h3 id="the_infinity1category_of_dgcategories">The (infinity,1)-category of dg-categories</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Dwyer-Kan+model+structure+on+dg-categories">Dwyer-Kan model structure on dg-categories</a> presents the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+dg-categories">(infinity,1)-category of dg-categories</a>.</p> <h3 id="relation_to_stable_categories">Relation to stable <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>By the <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a>, the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> of dg-categories is equivalent to the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-categories">(infinity,1)-categories</a> <a class="existingWikiWord" href="/nlab/show/enriched+%28infinity%2C1%29-category">enriched</a> in the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28infinity%2C1%29-category">symmetric monoidal (infinity,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/module+spectra">modules</a> over the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Mac+Lane+spectrum">Eilenberg-Mac Lane spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">H k</annotation></semantics></math>. The latter is equivalent, at least morally, to the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> of <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>-<a class="existingWikiWord" href="/nlab/show/linear+%28infinity%2C1%29-category">linear</a> <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-categories">stable (infinity,1)-categories</a>.</p> <p>More precisely, it is shown in <a href="#Cohn13">Cohn 13</a> that the <a class="existingWikiWord" href="/nlab/show/Morita+model+structure+on+dg-categories">Morita model structure on dg-categories</a> presents the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/idempotent+complete+%28infinity%2C1%29-category">idempotent complete</a> <a class="existingWikiWord" href="/nlab/show/linear+%28infinity%2C1%29-category">linear</a> <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-categories">stable (infinity,1)-categories</a>.</p> <h2 id="aspects_of_dgcategories">Aspects of dg-categories</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+dg-category">homotopy category of a dg-category</a>.</li> <li><a class="existingWikiWord" href="/nlab/show/equivalence+of+dg-categories">equivalence of dg-categories</a></li> <li><a class="existingWikiWord" href="/nlab/show/dg-modules">dg-modules</a>, <a class="existingWikiWord" href="/nlab/show/perfect+dg-modules">perfect dg-modules</a></li> <li><a class="existingWikiWord" href="/nlab/show/derived+dg-category">derived dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/dg-Yoneda+embedding">dg-Yoneda embedding</a></li> <li><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/dg-localization">dg-localization</a>, <span class="newWikiWord">dg-quotient<a href="/nlab/new/dg-quotient">?</a></span></li> <li><a class="existingWikiWord" href="/nlab/show/dg-nerve">dg-nerve</a></li> <li><a class="existingWikiWord" href="/nlab/show/Waldhausen+K-theory+of+a+dg-category">Waldhausen K-theory of a dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/semi-topological+K-theory+of+a+dg-category">semi-topological K-theory of a dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/derived+moduli+stack+of+objects+in+a+dg-category">derived moduli stack of objects in a dg-category</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-infinity+category">A-infinity category</a></li> <li><a class="existingWikiWord" href="/nlab/show/spectral+category">spectral category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a></li> </ul> <h2 id="references">References</h2> <p>Historically, dg-categories were introduced in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/G.+M.+Kelly">G. M. Kelly</a>, <em>Chain maps inducing zero homology maps</em>, Proc. Cambridge Philos. Soc. <strong>61</strong> (1965), 847–854, doi:<a href="https://doi.org/10.1017/S0305004100039207">10.1017/S0305004100039207</a></li> </ul> <p>whilst their modern development can be traced to</p> <ul> <li><span class="newWikiWord">A. I. Bondal<a href="/nlab/new/A.+I.+Bondal">?</a></span>, <a class="existingWikiWord" href="/nlab/show/Mikhail+Kapranov">Mikhail Kapranov</a>, <em>Enhanced triangulated categories</em>, Матем. Сборник, Том 181 (1990), No.5, 669–683 (Russian); transl. in USSR Math. USSR Sbornik, vol. 70 (1991), No. 1, pp. 93–107, (MR91g:18010) (<a class="existingWikiWord" href="/nlab/files/bondalKaprEnhTRiangCat.pdf" title="Bondal-Kapranov Enhanced triangulated categories pdf">Bondal-Kapranov Enhanced triangulated categories pdf</a>)</li> </ul> <p>A bicategory refinement</p> <ul> <li>Yuki Imamura. <em>A formal categorical approach to the homotopy theory of dg categories</em> (2024). (<a href="https://arxiv.org/abs/2405.07873">arXiv:2405.07873</a>).</li> </ul> <h3 id="overviews">Overviews</h3> <p>For concise reviews of the theory, see section 1 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A.+Beilinson">A. Beilinson</a>, <a class="existingWikiWord" href="/nlab/show/V.+Vologodsky">V. Vologodsky</a>, <em>DG guide to Voevodsky’s motives</em>.</li> </ul> <p>as well as the introduction and appendices to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/V.+Drinfeld">V. Drinfeld</a>, <em>DG quotients of DG categories</em>, <a href="http://www.math.harvard.edu/~gaitsgde/grad_2009/DGquotients.pdf">pdf</a>.</li> </ul> <p>For longer surveys, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bernhard+Keller">Bernhard Keller</a>, <em>On differential graded categories</em>, International Congress of Mathematicians. <strong>II</strong> (2006) 151-190, Eur. Math. Soc., Zürich [<a href="http://arxiv.org/abs/math/0601185">arXiv:math/0601185</a>]</li> </ul> <p>and</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bertrand+To%C3%ABn">Bertrand Toën</a>, <em>Lectures on dg-categories</em> (<a href="https://perso.math.univ-toulouse.fr/btoen/files/2012/04/swisk.pdf">pdf</a>).</li> </ul> <h3 id="homotopy_theory_of_dgcategories">Homotopy theory of dg-categories</h3> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/dg-categories">dg-categories</a> is studied in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gon%C3%A7alo+Tabuada">Gonçalo Tabuada</a>, <em>Théorie homotopique des DG-caté́gories</em>, Thesis, Paris, 2007, <a href="https://arxiv.org/pdf/0710.4303.pdf">pdf</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gon%C3%A7alo+Tabuada">Gonçalo Tabuada</a>, <em>Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories</em>, C. R. Math. Acad. Sci. Paris 340 (2005), no. 1, 15–19.</p> </li> </ul> <p>The equivalence with the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-categories">stable (infinity,1)-categories</a> is discussed in</p> <ul> <li id="Cohn13"><a class="existingWikiWord" href="/nlab/show/Lee+Cohn">Lee Cohn</a>, <em>Differential Graded Categories are k-linear Stable Infinity Categories</em> (<a href="http://arxiv.org/abs/1308.2587">arXiv:1308.2587</a>)</li> </ul> <p>(Note that the proof works over any ring, even though it is stated there for <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>.)</p> <p>In the following it is shown that the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-categories">(infinity,1)-categories</a> enriched in the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> is equivalent to the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of ordinary categories strictly enriched in <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>.</p> <ul> <li id="Haugseng13"><a class="existingWikiWord" href="/nlab/show/Rune+Haugseng">Rune Haugseng</a>, <em>Rectification of enriched infinity-categories</em>, <a href="http://arxiv.org/abs/1312.3881v2">arXiv:1312.3881</a>.</li> </ul> <h3 id="derived_noncommutative_geometry">Derived noncommutative geometry</h3> <p>The following references discuss the use of dg-categories in <a class="existingWikiWord" href="/nlab/show/derived+noncommutative+algebraic+geometry">derived noncommutative algebraic geometry</a> and <a class="existingWikiWord" href="/nlab/show/noncommutative+motives">noncommutative motives</a>.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gon%C3%A7alo+Tabuada">Gonçalo Tabuada</a>, <em>Invariants additifs de DG-catégories</em>, Int. Math. Res. Not. 2005, no. 53, 3309–3339; Addendum in Int. Math. Res. Not. 2006, Art. ID 75853, 3 pp. ; Erratum in Int. Math. Res. Not. IMRN 2007, no. 24, Art. ID rnm149, 17 pp.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marco+Robalo">Marco Robalo</a>, <em>Théorie homotopique motivique des espaces noncommutatifs</em>, <a href="http://webusers.imj-prg.fr/~marco.robalo/these.pdf">pdf</a>.</p> </li> <li> <p>S. Mahanta, <em>Noncommutative geometry in the framework of differential graded categories</em>, (<a href="http://arxiv.org/abs/0805.1628">arXiv:0805.1628</a>)</p> </li> <li> <p>D. Orlov, <em>Smooth and proper noncommutative schemes and gluing of DG categories</em>, Adv. Math. 302 (2014) <a href="https://dx.doi.org/10.1016/j.aim.2016.07.014">doi</a></p> </li> </ul> <h3 id="other_aspects">Other aspects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bernhard+Keller">Bernhard Keller</a>, <em>Deriving DG categories</em>, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102 (<a href="http://www.numdam.org/item?id=ASENS_1994_4_27_1_63_0">numdam</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dmitry+Tamarkin">Dmitry Tamarkin</a>, <em>What do dg-categories form?</em>,</p> <p>Compos. Math. 143 (2007), no. 5, 1335–1358.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Batanin">M. Batanin</a>, <em>What do dg-categories form</em> (after Tamarkin), talks at Paris 7 and Australian category seminar (<a href="http://www.maths.usyd.edu.au/u/AusCat/abstracts/060726mb.html">abstract</a>), <a href="http://arxiv.org/abs/math.CT/0606553">math.CT/0606553</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Oren+Ben-Bassat">Oren Ben-Bassat</a>, <a class="existingWikiWord" href="/nlab/show/Jonathan+Block">Jonathan Block</a>, <em>Cohesive DG categories I: Milnor descent</em>, <a href="http://arxiv.org/abs/1201.6118">arxiv/1201.6118</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 14, 2024 at 11:34:38. 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