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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="topos_theory">Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="toc_internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="category_theory">Category Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <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> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <ul> <li><a href='#what_a_topos_is_like'>‘What a topos is like:’</a></li> </ul> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#ElementaryTopos'>Elementary toposes</a></li> <li><a href='#SheafToposes'>Grothendieck/sheaf toposes</a></li> <li><a href='#WToposes'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-toposes</a></li> <li><a href='#toposes_over_a_base'>Toposes over a base</a></li> <li><a href='#morphisms_of_toposes'>Morphisms of toposes</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#extensivity'>Extensivity</a></li> <li><a href='#adhesiveness'>Adhesiveness</a></li> <li><a href='#epimorphisms'>Epimorphisms</a></li> <li><a href='#relation_to_abelian_categories'>Relation to abelian categories</a></li> <li><a href='#reasoning_in_a_topos'>Reasoning in a topos</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#specific_examples_and_key_results'>Specific examples and key results</a></li> <li><a href='#classes_of_examples'>Classes of examples</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#Introductions'>Introductions</a></li> <li><a href='#ReferencesMonographs'>Monographs</a></li> <li><a href='#classifying_toposes'>Classifying toposes</a></li> <li><a href='#CategoricalLogicAndElementaryToposes'>Categorical logic and elementary toposes</a></li> <li><a href='#algebra_ringed_toposes_algebraic_geometry'>Algebra, ringed toposes, algebraic geometry</a></li> <li><a href='#quantum_theory'>Quantum theory</a></li> <li><a href='#original_articles'>Original articles</a></li> <li><a href='#history'>History</a></li> <li><a href='#general_theory'>General theory</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>There are various different perspectives on the notion of <em>topos</em>. One is that a topos is a <a class="existingWikiWord" href="/nlab/show/category">category</a> that looks like a category of <a class="existingWikiWord" href="/nlab/show/space">space</a>s that sit by <a class="existingWikiWord" href="/nlab/show/local+homeomorphisms">local homeomorphisms</a> over a given base <a class="existingWikiWord" href="/nlab/show/space">space</a>: all spaces that are <em>locally modeled on</em> a given base space.</p> <p>The archetypical class of examples is that of <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>es <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Et</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X) = Et(X)</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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>: these are the categories of <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+space">étale space</a>s over <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>, topological spaces <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> that are equipped with a <a class="existingWikiWord" href="/nlab/show/local+homeomorphisms">local homeomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math>.</p> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X = *</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/point">point</a>, this is just the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of all <a class="existingWikiWord" href="/nlab/show/sets">sets</a>: spaces that are <em>modeled on the point</em>. This is the archetypical topos itself.</p> <p>What makes the notion of topos powerful is the following fact: even though the general topos contains objects that are considerably different from and possibly considerably richer than plain sets and even richer than étale spaces over a topological space, the general abstract <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic</a> properties of every topos are essentially the same as those of <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. For instance, in every topos, all finite <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> exist, and it is <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a> (even <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+category">locally</a>). This means that a large number of constructions in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> have immediate analogs <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> every topos, and the analogs of the statements about these constructions that are true in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> are true in <em>every</em> topos.</p> <p>On the one hand, this may be thought of as saying that toposes are <em>very nice categories of spaces</em>, in that whatever construction on spaces one thinks of – for instance, formation of <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a>s or of <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>s or of <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a>s – the resulting space with the expected general abstract properties will exist in the topos. In this sense, toposes are <em>convenient categories</em> for geometry – as in: <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>, but even more convenient than that.</p> <p>On the other hand, by de-emphasizing the geometric interpretation of their objects and just using their good abstract properties, this means that toposes are contexts with a powerful <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>. The internal logic of toposes is <a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic</a> <a class="existingWikiWord" href="/nlab/show/higher+order+logic">higher order logic</a>. This means that, while the <a class="existingWikiWord" href="/nlab/show/law+of+excluded+middle">law of excluded middle</a> and the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> may fail, apart from that, every logical statement not depending on these does hold <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <em>every</em> topos.</p> <p>For this reason, toposes are often studied as abstract contexts “in which one can do mathematics”, independently of their interpretation as categories of spaces. These two points of view on toposes, as being about geometry and about logic at the same time, is part of the richness of topos theory.</p> <p>On a third hand, however, we can de-emphasize the role of the objects of the topos and instead treat the topos itself as a “generalized space” (and in particular, a <a class="existingWikiWord" href="/nlab/show/vertical+categorification">categorified</a> space). We then consider the topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X)</annotation></semantics></math> as a representative 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> itself, while toposes not of this form are “honestly generalized” spaces. This point of view is supported by the fact that the assignment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\mapsto Sh(X)</annotation></semantics></math> is a full embedding of (sufficiently nice) topological spaces into toposes, and that many topological properties of a space <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> can be detected at the level of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X)</annotation></semantics></math>. (This is even more true once we pass to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a>.)</p> <p>From this point of view, the objects of a topos (regarded as a category) should be thought of instead as <em>sheaves on</em> that topos (regarded as a generalized space). And, just as sheaves on a topological space can be identified with local homeomorphisms over it, such “sheaves on a topos” (i.e., objects of the topos <em>qua</em> category) can be identified with other <em>toposes</em> that sit over the given topos via a <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">local homeomorphism of toposes</a>.</p> <p>Finally, mixing this point of view with the second one, we can regard toposes over a given topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> instead as “toposes in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-world of mathematics.” For this reason, the theory of toposes over a given base is formally quite similar to that of arbitrary toposes. And, coming full circle, this fact allows the use of “base change arguments” as a very useful technical tool, even if our interest is only in one or two particular toposes <em>qua</em> categories.</p> <h3 id="what_a_topos_is_like">‘What a topos is like:’</h3> <p>(i) ‘A topos is a category of sheaves on a site’</p> <p>(ii) ‘A topos is a category with finite limits and power-objects’</p> <p>(iii) ‘A topos is (the embodiment of) an intuitionistic higher-order theory’</p> <p>(iv) ‘A topos is (the extensional essence of) a first-order (infinitary) geometric theory’</p> <p>(v) ‘A topos is a <a class="existingWikiWord" href="/nlab/show/total+category">totally cocomplete</a> object in the meta-2-category CART of cartesian (i.e. , finitely complete) categories’</p> <p>(vi) ‘A topos is a generalized space’</p> <p>(vii) ‘A topos is a semantics for intuitionistic formal systems’</p> <p>(viii) ‘A topos is a Morita equivalence class of continuous groupoids’</p> <p>(ix) ‘A topos is the category of maps of a power allegory’</p> <p>(x) ‘A topos is a category whose canonical indexing over itself is complete and well-powered’</p> <p>(xi) ‘A topos is the spatial manifestation of a giraud frame’</p> <p>(xii) ‘A topos is a setting for synthetic differential geometry’</p> <p>(xiii) ‘A topos is a setting for synthetic domain theory’,</p> <p>And so on. But the important thing about the elephant is that ‘however you approach it, it is still the same animal’. <a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a></p> <h2 id="definitions">Definitions</h2> <p>The general notion of <em>topos</em> is that of</p> <ul> <li><a href="#ElementaryTopos">Elementary toposes</a>.</li> </ul> <p>A specialization of this which is important enough that much of the literature implicitly takes it to be the general definition is the notion of</p> <ul> <li><a href="#SheafToposes">Grothendieck/sheaf toposes</a></li> </ul> <p>This is the notion relevant for applications in <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> and <a class="existingWikiWord" href="/nlab/show/geometric+logic">geometric logic</a>, whereas the notion of elementary toposes is relevant for more general applications in <a class="existingWikiWord" href="/nlab/show/logic">logic</a>.</p> <p>For standard notions of mathematics to be available <a class="existingWikiWord" href="/nlab/show/internal+logic">inside</a> a given topos, one typically at least needs a <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a>. Its existence is guaranteed by the axioms of a sheaf topos, but not by the more general axioms of an elementary topos. Adding the existence of a natural numbers object to the axioms of an elementary topos yields the notion of a</p> <ul> <li><a href="#WToposes">W-topos</a>.</li> </ul> <h3 id="ElementaryTopos">Elementary toposes</h3> <p>A quick formal definition is that an <strong>elementary topos</strong> is a <a class="existingWikiWord" href="/nlab/show/category">category</a> which</p> <ol> <li> <p>has <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finite limits</a>,</p> </li> <li> <p>is <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a>, and</p> </li> <li> <p>has a <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>.</p> </li> </ol> <p>There are alternative ways to state the definition; for instance,</p> <ol> <li>has <a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a>s and</li> <li>has <a class="existingWikiWord" href="/nlab/show/power+objects">power objects</a>.</li> </ol> <p>In a way, however, these concise definitions can be misleading, because a topos has a great deal of other structure which plays a very important role but just happens to follow automatically from these basic axioms. Most importantly, an elementary topos is all of the following:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+category">locally cartesian closed</a>,</li> <li><a class="existingWikiWord" href="/nlab/show/finitely+cocomplete+category">finitely cocomplete</a>,</li> <li>a <a class="existingWikiWord" href="/nlab/show/Heyting+category">Heyting category</a>, and</li> <li>a <a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a>.</li> </ul> <p>The last two imply that it has an <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> that resembles ordinary mathematical reasoning, and the presence of exponentials and power objects means that this logic is <a class="existingWikiWord" href="/nlab/show/higher+order+logic">higher order</a>.</p> <h3 id="SheafToposes">Grothendieck/sheaf toposes</h3> <p>The above is the definition of an <em>elementary</em> topos. We also have the (historically earlier) notion of <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a>: a <strong>Grothendieck topos</strong> is a topos that is neither too small nor too large, in that it is:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete</a> (not too small), and</li> <li>has a <a class="existingWikiWord" href="/nlab/show/small+set">small</a> <a class="existingWikiWord" href="/nlab/show/generating+set">generating set</a> (not too large).</li> </ul> <p>Equivalently, a Grothendieck topos is any category <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> on some <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/site">site</a>.</p> <h3 id="WToposes"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-toposes</h3> <p>There is a further elementary property of <a class="existingWikiWord" href="/nlab/show/Set">Set</a> that might have gone into the definition of elementary topos, but historically did not: the existence of a <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a>. Any topos with this property is called a <strong>topos with NNO</strong> or a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-topos</strong>. The latter term comes from the result that any such topos must have (not only an NNO but also) all <a class="existingWikiWord" href="/nlab/show/W-types">W-types</a>.</p> <h3 id="toposes_over_a_base">Toposes over a base</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a></li> </ul> <h3 id="morphisms_of_toposes">Morphisms of toposes</h3> <p>There are two kinds of <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> between toposes that one considers:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>– this is the kind of morphism that regards a topos as a generalized <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a>– this is the kind of morphism that regards a topos in terms of its <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>.</p> </li> </ul> <p>Accordingly there is a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Topos">Topos</a> of toposes, whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/object">object</a>s are toposes,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s are <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>s, and</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a>s are <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s between the <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s underlying the geometric morphisms.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="extensivity">Extensivity</h3> <p>Every topos is an <a class="existingWikiWord" href="/nlab/show/extensive+category">extensive category</a>. For <a class="existingWikiWord" href="/nlab/show/Grothendieck+toposes">Grothendieck toposes</a>, infinitary extensivity is part of the characterizing <a class="existingWikiWord" href="/nlab/show/Giraud%27s+theorem">Giraud's theorem</a>. For elementary toposes, see <a class="existingWikiWord" href="/nlab/show/toposes+are+extensive">toposes are extensive</a>.</p> <h3 id="adhesiveness">Adhesiveness</h3> <p>Every topos is an <a class="existingWikiWord" href="/nlab/show/adhesive+category">adhesive category</a>. For <a class="existingWikiWord" href="/nlab/show/Grothendieck+toposes">Grothendieck toposes</a> this follows immediately from the adhesion of <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, while for elementary toposes it is due to (<a href="#LackSobocinski">Lack-Sobocinski</a>).</p> <h3 id="epimorphisms">Epimorphisms</h3> <p>In a topos, <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>s are stable under <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, and hence the <a class="existingWikiWord" href="/nlab/show/%28epi%2C+mono%29+factorization+system">(epi, mono) factorization system</a> in a topos is a <a class="existingWikiWord" href="/nlab/show/stable+factorization+system">stable factorization system</a>.</p> <h3 id="relation_to_abelian_categories">Relation to abelian categories</h3> <p>While crucially different from <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a>, there is some intimate relation between toposes and abelian categories. For more on that, see <a class="existingWikiWord" href="/nlab/show/AT+category">AT category</a>.</p> <h3 id="reasoning_in_a_topos">Reasoning in a topos</h3> <p>Any result in <a class="existingWikiWord" href="/nlab/show/ordinary+mathematics">ordinary mathematics</a> whose proof is <a class="existingWikiWord" href="/nlab/show/finite+mathematics">finitist</a> and <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> automatically holds in any topos. If one removes the restriction that the proof be finitist, then the result holds in any topos with a <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a>. If one removes the restriction that the proof be constructive, then the result holds in any <a class="existingWikiWord" href="/nlab/show/boolean+topos">boolean topos</a>. On the other hand, if one adds the restriction that the proof be predicative in the weaker sense used by constructivists, then the result may fail in some toposes, but holds in any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Pi-pretopos">pretopos</a>. If one adds the restriction that the proof be predicative in a stronger sense, then the result holds in any <a class="existingWikiWord" href="/nlab/show/Heyting+pretopos">Heyting pretopos</a>.</p> <p>Therefore, one can prove results in toposes and similar categories by reasoning, not about the <a class="existingWikiWord" href="/nlab/show/objects">objects</a> and <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in the topos themselves, but instead about <a class="existingWikiWord" href="/nlab/show/sets">sets</a> and <a class="existingWikiWord" href="/nlab/show/functions">functions</a> in the normal language of structural <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>, which is more familiar to most mathematicians. One merely has to be careful about what axioms one uses to get results of sufficient generality.</p> <p>The <a class="existingWikiWord" href="/nlab/show/internal+logic">internal language</a> of a topos is called <a class="existingWikiWord" href="/nlab/show/Mitchell-B%C3%A9nabou+language">Mitchell-Bénabou language</a>.</p> <h2 id="Examples">Examples</h2> <h3 id="specific_examples_and_key_results">Specific examples and key results</h3> <ul> <li> <p>The archetypical topos is <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. Notice that this happens to be a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a>: it is the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> on the <a class="existingWikiWord" href="/nlab/show/point">point</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</a> is also an elementary topos, and the inclusion functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FinSet</mi><mo>↪</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">FinSet \hookrightarrow Set</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a>. This is not a Grothendieck topos.</p> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cardinal">strong limit cardinal</a>, the full subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Set</mi> <mi>κ</mi></msub></mrow><annotation encoding="application/x-tex">Set_\kappa</annotation></semantics></math> of sets of <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> less than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> is a topos.</p> </li> <li> <p>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> any (small) <a class="existingWikiWord" href="/nlab/show/site">site</a>, the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a>. Either by definition or by <a class="existingWikiWord" href="/nlab/show/Giraud%27s+theorem">Giraud's theorem</a>, every Grothendieck topos arises in this way. Important examples include:</p> <ul> <li> <p>The case where the <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a> is the trivial one, so that also all categories of <a class="existingWikiWord" href="/nlab/show/presheaf">presheaves</a> (on small categories) are (Grothendieck) toposes.</p> </li> <li> <p>The case of sheaves on (the <a class="existingWikiWord" href="/nlab/show/site">site</a> given by the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> of) a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, or more generally a <a class="existingWikiWord" href="/nlab/show/locale">locale</a>.</p> </li> <li> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Set</mi></mrow><annotation encoding="application/x-tex">G Set</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/set">set</a>s equipped with the <a class="existingWikiWord" href="/nlab/show/action">action</a> of a <a class="existingWikiWord" href="/nlab/show/group">group</a> <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>. This is the topos of presheaves on the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>, which is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> of <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>.</p> </li> </ul> </li> <li> <p>Continuing the previous example, if <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 topological group, then the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Set</mi></mrow><annotation encoding="application/x-tex">G Set</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/category+of+G-sets">sets with a continuous action of G</a> (that is, the action map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">G\times X\to X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous</a>, where <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> has the <a class="existingWikiWord" href="/nlab/show/discrete+topology">discrete topology</a>) is a topos, and in fact a Grothendieck topos (though this may not be obvious).</p> </li> <li> <p>More generally, <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> may be a <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological groupoid</a>, or even a <a class="existingWikiWord" href="/nlab/show/localic+groupoid">localic groupoid</a>. In fact, it is a theorem that every Grothendieck topos can be presented as the topos of “continuous sheaves” on a localic groupoid.</p> </li> <li> <p>If <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 again a topological group, then the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Unif</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Unif(G)</annotation></semantics></math> of <em><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+map">uniformly continuous</a></em> <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>-sets is also a topos, but not (in general) one of Grothendieck’s. For example, if <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 the <a class="existingWikiWord" href="/nlab/show/profinite+completion">profinite completion</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>, then a continuous <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>-set is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-set whose orbits are all finite, while a uniformly continuous one is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-set with a finite upper bound on the size of all its orbits.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> a topos and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">X \in E</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/object">object</a>, the <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</a> or <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E/X</annotation></semantics></math> is also a topos. (<a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a>, A.2.3.2). See also <a class="existingWikiWord" href="/nlab/show/over-topos">over-topos</a>.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a topos and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">j \colon E \to E</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/exact+functor">lex</a> idempotent monad, then the category of <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>-algebras is a topos. Each such <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> corresponds to a <a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">Lawvere-Tierney topology</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, and the category of <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>-algebras is equivalent to the category of sheaves for this topology. An example is the <a class="existingWikiWord" href="/nlab/show/double+negation">double-negation topology</a>.</p> </li> <li> <p>An obvious example of an elementary topos that is not a Grothendieck topos is the category <a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</a> of finite sets. More generally, for any <a class="existingWikiWord" href="/nlab/show/strong+limit+cardinal">strong limit cardinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>, the category of sets of cardinality less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> is an elementary topos, and as long as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo>></mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">\kappa \gt \omega</annotation></semantics></math>, it has a <a class="existingWikiWord" href="/nlab/show/NNO">NNO</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Set</mi> <mrow><mo><</mo><mi>κ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Set_{\lt \kappa}</annotation></semantics></math> does not even admit a geometric morphism to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>.</p> </li> <li> <p>Since the definition of elementary topos is “algebraic,” there exist <a class="existingWikiWord" href="/nlab/show/free+topos">free topos</a>es generated by various kinds of data. The category of toposes (and <a class="existingWikiWord" href="/nlab/show/logical+functor">logical functor</a>s) is <a class="existingWikiWord" href="/nlab/show/2-monadic">2-monadic</a> over the 2-category of categories, functors, and natural isomorphisms. It has an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, which is sometimes called <em>the free topos</em>. More generally, any <a class="existingWikiWord" href="/nlab/show/higher-order+logic">higher-order</a> <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> (of the sort which can be interpreted in the <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> of a topos) generates a free topos containing a model of that theory. Such toposes (for a consistent theory) are never Grothendieck’s.</p> </li> <li> <p>If <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 <a class="existingWikiWord" href="/nlab/show/large+category">large</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> with a small set of objects, then the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[G,Set]</annotation></semantics></math> (which makes sense if we define “large” and “small” relative to a <a class="existingWikiWord" href="/nlab/show/Grothendieck+universe">Grothendieck universe</a>) is a topos, but not in general a Grothendieck topos. Note that this topos is in fact <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a> and cocomplete, but it does not have a small generating set, and so is an <a class="existingWikiWord" href="/nlab/show/unbounded+topos">unbounded topos</a>.</p> </li> <li> <p>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">\mathcal{F}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/filter">filter</a> of <a class="existingWikiWord" href="/nlab/show/subterminal+objects">subterminal objects</a> in any topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, then there is a <span class="newWikiWord">filterquotient<a href="/nlab/new/filterquotient">?</a></span> topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">E//\mathcal{F}</annotation></semantics></math>. Grothendieck-ness (and completeness and cocompleteness) are not generally preserved by this construction.</p> </li> <li> <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> and <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> are toposes and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C\to D</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/lex+functor">lex functor</a>, then there is a topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gl</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gl(F)</annotation></semantics></math> called the <a class="existingWikiWord" href="/nlab/show/Artin+gluing">Artin gluing</a> 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> and <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> along <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>, and defined to be 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>D</mi><mo stretchy="false">/</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D/F)</annotation></semantics></math>. 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> and <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> are Grothendieck toposes, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gl</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gl(F)</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>-topos. 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 <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gl</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gl(F)</annotation></semantics></math> is again Grothendieck (hence <a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded</a>), but in general it may not be. (Note, though, that it is not clear whether it can be proven in ZFC that there exist any inaccessible lex functors between Grothendieck toposes, although from a proper class of <a class="existingWikiWord" href="/nlab/show/measurable+cardinal">measurable cardinal</a>s one can construct an inaccessible lex endofunctor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>.)</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">category of coalgebras</a> for any <a class="existingWikiWord" href="/nlab/show/exact+functor">lex</a> <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> on a topos is again a topos: a <a class="existingWikiWord" href="/nlab/show/topos+of+algebras+over+a+monad">topos of coalgebras</a>, and if the comonad is <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible</a>, then this construction preserves Grothendieck-ness. If the comonad is not accessible, then this topos is unbounded.</p> <p>For instance, the <a class="existingWikiWord" href="/nlab/show/Artin+gluing">Artin gluing</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gl</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gl(F)</annotation></semantics></math> is equivalent to the category of coalgebras for the comonad on the topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>×</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C\times D</annotation></semantics></math> defined by <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>d</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(c,d) \mapsto (c, d\times F(c))</annotation></semantics></math>.</p> </li> <li> <p>More generally, the category of coalgebras for any <a class="existingWikiWord" href="/nlab/show/preserved+limit">pullback-preserving</a> comonad on a topos is again a topos. This covers both the preceding result and also the overcategory (slice category) result above, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E/X</annotation></semantics></math> is the category of coalgebras for the pullback-preserving comonad given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">X \times - \colon E \to E</annotation></semantics></math>. It also covers Artin gluing along a pullback-preserving functor.</p> </li> <li> <p>More generally still, <a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd Trimble</a> has a notion called a “modal operator” on a topos, from which one can construct a new topos of “<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>-structures”: see <a class="existingWikiWord" href="/toddtrimble/published/Three+topos+theorems+in+one">Three topos theorems in one</a>. Special cases include the pullback-preserving comonad result just mentioned, and the result that the category of algebras for a lex <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent monad</a> is a topos. A related idea is Toby Kenney’s notion of lex distributive <span class="newWikiWord">diad<a href="/nlab/new/diad">?</a></span>, from which one can also construct a topos.</p> </li> <li> <p>From any <a class="existingWikiWord" href="/nlab/show/partial+combinatory+algebra">partial combinatory algebra</a> one can construct a <a class="existingWikiWord" href="/nlab/show/realizability+topos">realizability topos</a>, whose <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> is “computable” or “effective” mathematics relative to that PCA. In particular, for <a class="existingWikiWord" href="/nlab/show/Kleene%27s+first+algebra">Kleene's first algebra</a>, one obtains the <a class="existingWikiWord" href="/nlab/show/effective+topos">effective topos</a>, the most-studied of realizability toposes. Realizability toposes are generally not Grothendieck toposes.</p> </li> <li> <p>A topos can also be constructed from any <a class="existingWikiWord" href="/nlab/show/tripos">tripos</a>. This includes realizability toposes as well as toposes of sheaves on locales.</p> </li> </ul> <h3 id="classes_of_examples">Classes of examples</h3> <p>For various applications, one uses toposes that have <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">extra structure or properties</a>.</p> <ul> <li> <p>In the <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a> of mathematics, one often studies <a class="existingWikiWord" href="/nlab/show/well-pointed+toposes">well-pointed toposes</a>, especially models of <a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a> as potential replacements for the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>, one studies <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>es as a context for axiomatic <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a>, <a class="existingWikiWord" href="/nlab/show/Giraud%27s+theorem">Giraud's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+topos+theory">fundamental theorem of topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/predicative+topos">predicative topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/boolean+topos">boolean topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/predicative+topos">predicative topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a>, <a class="existingWikiWord" href="/nlab/show/%CE%A0-pretopos">Π-pretopos</a>, <a class="existingWikiWord" href="/nlab/show/%CE%A0W-pretopos">ΠW-pretopos</a>, <a class="existingWikiWord" href="/nlab/show/list-arithmetic+pretopos">list-arithmetic pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure+on+a+topos">smooth structure on a topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+topos">monoidal topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropy+group+of+a+topos">isotropy group of a topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/test+topos">test topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> </ul> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher toposes</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/1-topos">1-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-topos">(1,1)-topos</a> = <a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(n,1)</annotation> </semantics> </math>-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> (<a class="existingWikiWord" href="/nlab/show/n-localic+%28infinity%2C1%29-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-localic</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\;\;\;</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/model+topos">model topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C2%29-topos">(2,2)-topos</a> (<a class="existingWikiWord" href="/nlab/show/n-localic+2-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-localic</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,n)</annotation> </semantics> </math>-topos</a></li> </ul> </li> </ul> </div><div> <p><strong>Locally presentable categories:</strong> <a class="existingWikiWord" href="/nlab/show/cocomplete+category">Cocomplete</a> possibly-<a class="existingWikiWord" href="/nlab/show/large+categories">large categories</a> generated under <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a> by <a class="existingWikiWord" href="/nlab/show/small+object">small</a> <a class="existingWikiWord" href="/nlab/show/generators">generators</a> under <a class="existingWikiWord" href="/nlab/show/small+colimit">small</a> <a class="existingWikiWord" href="/nlab/show/relations">relations</a>. Equivalently, <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible</a> <a class="existingWikiWord" href="/nlab/show/reflective+localizations">reflective localizations</a> of <a class="existingWikiWord" href="/nlab/show/free+cocompletions">free cocompletions</a>. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a> localization.</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/toposes">toposes</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th>locally presentable</th><th>loc finitely pres</th><th>localization theorem</th><th><a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a></th><th>accessible</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locales">locales</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/suplattice">suplattice</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+lattices">algebraic lattices</a></td><td style="text-align: left;"><a href="algebraic+lattice#RelationToLocallyFinitelyPresentableCategories">Porst’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/powerset">powerset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/poset">poset</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+finitely+presentable+categories">locally finitely presentable categories</a></td><td style="text-align: left;"><a href="locally+presentable+category#AsLocalizationsOfPresheafCategories">Gabriel–Ulmer’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+toposes">model toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dugger%27s+theorem">Dugger's theorem</a></td><td style="text-align: left;">global <a class="existingWikiWord" href="/nlab/show/model+structures+on+simplicial+presheaves">model structures on simplicial presheaves</a></td><td style="text-align: left;">n/a</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-categories">locally presentable (∞,1)-categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a href="locally+presentable+infinity-category#Definition">Simpson’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf+%28%E2%88%9E%2C1%29-categories">(∞,1)-presheaf (∞,1)-categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-categories">accessible (∞,1)-categories</a></td></tr> </tbody></table> </div> <h2 id="References">References</h2> <h3 id="Introductions">Introductions</h3> <p>Brief expositions:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Background in topos theory</em>, chapter I in: <em>Classifying Spaces and Classifying Topoi</em>, Lecture Notes in Mathematics <strong>1616</strong>, Springer (1995) [<a href="https://doi.org/10.1007/BFb0094441">doi:10.1007/BFb0094441</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Luc+Illusie">Luc Illusie</a>, <em>What is… a Topos?</em>, Notices of the AMS <strong>51</strong> 9 (2004) [<a href="https://www.ams.org/notices/200409/what-is-illusie.pdf">pdf</a>, full volume:<a href="https://www.ams.org/notices/200409/200409FullIssue.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em><a class="existingWikiWord" href="/nlab/show/Leinster2010">An informal introduction to topos theory</a></em> (2010) [<a href="https://arxiv.org/abs/1012.5647">arXiv:1012.5647</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em>Topos Theory in a Nutshell</em> (2021) [<a href="http://math.ucr.edu/home/baez/topos.html">web</a>]</p> </li> <li> <p>MathProofsable, Category Theory - Toposes <a href="https://www.youtube.com/watch?v=gKYpvyQPhZo&list=PL4FD0wu2mjWM3ZSxXBj4LRNsNKWZYaT7k">video playlist</a></p> </li> </ul> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean+B%C3%A9nabou">Jean Bénabou</a>, <em>Problèmes dans les topos</em>, Séminaire de mathématique pure <strong>34</strong>, Université de Louvain (1973) [<a class="existingWikiWord" href="/nlab/files/Benabou-ProblemesDansLesTopos.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>A survey of topos theory</em>, lecture notes (1978) [<a href="http://www.math.mq.edu.au/~street/ToposSurvey.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Street-SurveyToposTheory.pdf" title="pdf">pdf</a>]</p> </li> <li id="Wyler"> <p><a class="existingWikiWord" href="/nlab/show/Oswald+Wyler">Oswald Wyler</a>, <em>Lecture Notes on Topoi and Quasitopoi</em>, World Scientific (1991) [<a href="https://doi.org/10.1142/1047">doi:10.1142/1047</a>]</p> </li> <li id="Joyal15"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em><a class="existingWikiWord" href="/nlab/show/A+crash+course+in+topos+theory+--+The+big+picture">A crash course in topos theory – The big picture</a></em>, lecture series at <a href="https://indico.math.cnrs.fr/event/747/">Topos à l’IHES</a>, Paris (November 2015)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>Geometric aspects of topos theory in relation with logical doctrines</em>, talk at <em><a class="existingWikiWord" href="/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a></em>, IHP Paris 2015 (<a href="https://www.youtube.com/watch?v=kaZpOEOAUzE">video recording</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Jaap+van+Oosten">Jaap van Oosten</a>, <em>Topos Theory</em>, Master class notes (2007) [<a href="http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jaap+van+Oosten">Jaap van Oosten</a>, <em>Topos Theory</em> (2018) [<a href="https://webspace.science.uu.nl/~ooste110/syllabi/topostheory.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/vanOosten-ToposTheory.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <em>Some glances at topos theory</em>, lecture notes, Como (2018) [<a href="https://tcsc.lakecomoschool.org/files/2018/01/Borceux.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Borceux-ToposTheory.pdf" title="pdf">pdf</a>, <a href="https://www.youtube.com/watch?v=s_fN9euuVAY&list=PLh_3Q6ZRqWs0LBptMGClJ8OArR0fBT6Pp&index=11">video playlist</a>]</p> </li> </ul> <p>A monograph that aims to be more expository, focusing on <a class="existingWikiWord" href="/nlab/show/presheaf+toposes">presheaf toposes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marie+La+Palme+Reyes">Marie La Palme Reyes</a>, <a class="existingWikiWord" href="/nlab/show/Gonzalo+E.+Reyes">Gonzalo E. Reyes</a>, <a class="existingWikiWord" href="/nlab/show/Houman+Zolfaghari">Houman Zolfaghari</a>, <em>Generic figures and their glueings. A constructive approach to functor categories</em>, Polimetrica (2008) [<a href="https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf">pdf</a>, ISBN:8876990046]</li> </ul> <h3 id="ReferencesMonographs">Monographs</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Topos theory</em>, London Math. Soc. Monographs <strong>10</strong>, Acad. Press 1977, xxiii+367 pp. (Dover reprint 2014)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Wells">Charles Wells</a>, <em><a class="existingWikiWord" href="/nlab/show/Toposes%2C+Triples%2C+and+Theories">Toposes, Triples, and Theories</a></em>, Springer Heidelberg 1985. (<a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf">TAC reprint</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a>, <em><a class="existingWikiWord" href="/nlab/show/Elementary+Categories%2C+Elementary+Toposes">Elementary Categories, Elementary Toposes</a></em>, Oxford University Press 1992 (<a href="https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732?cc=ae&lang=en&">ISBN:9780198514732</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em>, 1992</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a> 3 - Categories of Sheaves</em>, Cambridge UP 1994 (<a href="https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/handbook-categorical-algebra-volume-3?format=PB">ISBN:9780521061247</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Elephant">Sketches of an elephant: a topos theory compendium</a></em> Oxford University Press 2002, Volume 1 (<a href="https://global.oup.com/academic/product/sketches-of-an-elephant-9780198534259">ISBN:9780198534259</a>), Volume 2 (<a href="https://global.oup.com/academic/product/sketches-of-an-elephant-9780198515982">ISBN:9780198515982</a>)</p> </li> <li id="Warner12"> <p><a class="existingWikiWord" href="/nlab/show/Garth+Warner">Garth Warner</a>, <em>Homotopical Topos Theory</em>, EPrint Collection, University of Washington (2012) [<a href="http://hdl.handle.net/1773/19722">hdl:1773/19722</a>, <a href="https://sites.math.washington.edu//~warner/HTT_Warner.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Waner-HomotopicalTopos.pdf" title="pdf">pdf</a>]</p> </li> </ul> <h3 id="classifying_toposes">Classifying toposes</h3> <p>Specifically on <a class="existingWikiWord" href="/nlab/show/classifying+toposes">classifying toposes</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Classifying Spaces and Classifying Topoi</em>, Lecture Notes in Mathematics 1616, Springer 1995 (ISBN: 978-3-540-60319-1, <a href="https://doi.org/10.1007/BFb0094441">doi:10.1007/BFb0094441</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Olivia+Caramello">Olivia Caramello</a>, <em>Theories, Sites, Toposes</em>, Oxford University Press, 2017.</p> <p><a href="https://doi.org/10.1007/BFb0094441">doi</a></p> </li> </ul> <h3 id="CategoricalLogicAndElementaryToposes">Categorical logic and elementary toposes</h3> <p>On <a class="existingWikiWord" href="/nlab/show/categorical+logic">categorical logic</a> via toposes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a>, <em>Elementary Categories, Elementary Toposes</em>, Oxford University Press 1992 (reprinted in 2005) (<a href="https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732">ISBN:9780198514732</a>)</p> </li> <li> <p>Jonathan Chapman, Frederick Rowbottom, <em>Relative Category Theory and Geometric Morphisms. A Logical Approach</em>, Oxford University Press 1992 (<a href="https://global.oup.com/academic/product/relative-category-theory-and-geometric-morphisms-9780198534341">ISBN:9780198534341</a>)</p> </li> <li> <p>J. L. Bell, <em>Toposes and Local Set Theories. An Introduction</em>, Oxford University Press 1988 (<a href="https://doi.org/10.2307/2274680">doi:10.2307/2274680</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+Goldblatt">Robert Goldblatt</a>, <em>Topoi. The categorial analysis of logic</em>, Studies in Logic and the Foundations of Math. <strong>98</strong>, North-Holland Publ. Co., Amsterdam, 1979, 1984; (Rus. transl. Mir Publ., Moscow 1983).</p> </li> <li id="LambekScott86"> <p><a class="existingWikiWord" href="/nlab/show/Joachim+Lambek">Joachim Lambek</a>, <a class="existingWikiWord" href="/nlab/show/Philip+J.+Scott">Philip J. Scott</a>, <em>Introduction to higher order categorical logic</em>, Cambridge Studies in Advanced Mathematics 7 (1986) (<a href="https://www.cambridge.org/ae/academic/subjects/mathematics/logic-categories-and-sets/introduction-higher-order-categorical-logic?format=PB&isbn=9780521356534">ISBN: 0-521-24665-2</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bart+Jacobs">Bart Jacobs</a>, <em>Categorical logic and type theory</em>, Studies in Logic and the Foundations of Mathematics 141 (1999). North-Holland Publishing Co.</p> <p>ISBN: 0-444-50170-3</p> </li> </ul> <h3 id="algebra_ringed_toposes_algebraic_geometry">Algebra, ringed toposes, algebraic geometry</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Monique+Hakim">Monique Hakim</a>, <em>Topos annelés et schémas relatifs</em>, Ergebnisse der Mathematik und ihrer Grenzgebiete 64 (1972) (<a href="https://www.springer.com/gp/book/9783540055730">ISBN:9783540055730</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Gilberte van den Bossche, <em>Algebra in a localic topos with applications to ring theory</em>, Lecture Notes in Mathematics 1038 (1983), ISBN: 3-540-12711-9 (<a href="https://www.springer.com/gp/book/9783540127116">ISBN:9783540127116</a>)</p> </li> </ul> <h3 id="quantum_theory">Quantum theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Cecilia+Flori">Cecilia Flori</a>, <em>A first course in topos quantum theory</em>. Lecture Notes in Physics 868 (2013).</p> <p>ISBN: 978-3-642-35712-1; 978-3-642-35713-8</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cecilia+Flori">Cecilia Flori</a>, <em>A second course in topos quantum theory</em>. Lecture Notes in Physics 944 (2018).</p> <p>ISBN: 978-3-319-71107-2; 978-3-319-71108-9</p> </li> </ul> <h3 id="original_articles">Original articles</h3> <p>Original source of (Grothendieck) topoi:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/SGA4">SGA4</a> Exposé IV by Grothendieck and Verdier (and Exposé V for cohomology in a (Grothendieck) topos)</li> </ul> <p>That every topos is an <a class="existingWikiWord" href="/nlab/show/adhesive+category">adhesive category</a> is discussed in</p> <ul id="LackSobocinski"> <li><a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a>, Pawel Sobociński, <em>Toposes are adhesive</em> (<a href="http://users.ecs.soton.ac.uk/ps/papers/toposesAdhesive.pdf">pdf</a>)</li> </ul> <h3 id="history">History</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a>, <em>The Uses and Abuses of the History of Topos Theory</em> , Brit. J. Phil. Sci., 41 (1990) (<a href="http://www.jstor.org/stable/687825">JSTOR</a>) <a href="http://bjps.oxfordjournals.org/content/41/3/351.full.pdf">PDF</a> (<a href="http://www.cwru.edu/artsci/phil/UsesandAbuses%20HistoryToposTheory.pdf">direct link</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+W.+Gray">John W. Gray</a>, <em>Fragments of the history of sheaf theory</em>,</p> <p>in: Applications of Sheaves, pp. 1–79, Lecture Notes in Mathematics 753, ISBN: 978-3-540-09564-4, <a href="https://doi.org/10.1007/BFb0061812">doi</a>.</p> </li> </ul> <p>According to appendix C.1 in</p> <ul> <li id="LawvereRosebrugh03"><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Rosebrugh">Robert Rosebrugh</a>, <em><a class="existingWikiWord" href="/nlab/show/Sets+for+Mathematics">Sets for Mathematics</a></em> , Cambridge UP 2003. (<a href="http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics">web</a>)</li> </ul> <blockquote> <p>“Topos” is a Greek term intended to describe the objects studied by “<a href="http://en.wikipedia.org/wiki/Analysis_Situs_%28paper%29">analysis situs</a>,” the Latin term previously established by Poincaré to signify the science of place [or situation]; in keeping with those ideas, a <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">\mathcal{U}</annotation></semantics></math>-topos was shown to have presentations in various “sites”.</p> </blockquote> <p>A historical analysis of Grothendieck’s 1973 Buffalo lecture series on toposes and their precedents is in</p> <ul> <li id="McLarty18"><a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a>, <em>Grothendieck’s 1973 topos lectures</em>, Séminaire Lectures grothendieckiennes, 3 May (2018) (<a href="https://www.youtube.com/watch?v=hhWT5V0oaSI">YouTube video</a>)</li> </ul> <h3 id="general_theory">General theory</h3> <p>On the application of <span class="newWikiWord">diads<a href="/nlab/new/diads">?</a></span> to constructing toposes:</p> <ul> <li>Toby Kenney, <em>Diads and their Application to Topoi</em>, Applied Categorical Structures 17 (2009): 567-590.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 17, 2025 at 22:29:06. 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