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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="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> </div> </div> <h1 id="big_and_little_toposes">Big and little toposes</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#relationships'>Relationships</a></li> <li><a href='#the_big_and_little_topos_of_an_object'>The big and little topos of an object</a></li> <li><a href='#axiomatizations'>Axiomatizations</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>There are two different (related) relationships between <a class="existingWikiWord" href="/nlab/show/Grothendieck+topoi">Grothendieck topoi</a> and a notion of <em>generalized <a class="existingWikiWord" href="/nlab/show/space">space</a></em>. (Recall that a Grothendieck topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a <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>T</mi><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T = Sh(S)</annotation></semantics></math> on some <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.)</p> <p>On the one hand, we can regard the topos <em>itself</em> as a generalized space. This tends to be a useful point of view when the site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Op(X)</annotation></semantics></math> of 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> (or some <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> or the like), or some other site which we regard as containing data from only “one space.” In this case, we refer to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> as a <strong>little topos</strong>, or (if we fail to translate the original French) a <strong>petit topos</strong>.</p> <p>On the other hand, we can view a topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> as a well-behaved category whose <em>objects</em> are generalized spaces. This tends to be a useful point of view when the site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a category of <em>all test <a class="existingWikiWord" href="/nlab/show/space">space</a>s</em> in some sense, such as <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>, or <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>. In this case, we refer to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> as a <strong>big topos</strong>, or (in French) a <strong>gros topos</strong>.</p> <p>These distinctions carry over in a straightforward way to higher topoi such as <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topoi">(∞,1)-topoi</a>.</p> <h2 id="relationships">Relationships</h2> <p>Objects in a big 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>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(S)</annotation></semantics></math> may be thought of as <a class="existingWikiWord" href="/nlab/show/space">space</a>s <em>modeled on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></em>, in the sense described at <a class="existingWikiWord" href="/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks">motivation for sheaves, cohomology and higher stacks</a> and at <a class="existingWikiWord" href="/nlab/show/space">space</a>.</p> <p>On the other hand, the objects of a petit topos, such as <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>, can also be regarded as a kind of generalized spaces, but generalized spaces <em>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></em> on which the rigid structure of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Op(X)</annotation></semantics></math> (only inclusions of subsets, no more general maps) induces a correspondingly rigid structure so that they are not all that general. In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(Op(X))</annotation></semantics></math> is equivalent to the category of <a class="existingWikiWord" href="/nlab/show/etale+space">etale 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>—i.e. spaces “modeled on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>” in a certain sense. More generally, for 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>, the objects of <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> can be identified with <a class="existingWikiWord" href="/nlab/show/local+homeomorphisms+of+toposes">local homeomorphisms of toposes</a> into <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>.</p> <p>From the “little topos” perspective, it can be helpful to think of a “big topos” as a “fat point,” which is not “spread out” very much spatially itself, but contains within that point lots of different types of “local data,” so that even spaces which are “rigidly” modeled on that point can have a lot of interesting cohesion and local structure. (One should not be misled by this into thinking that a big topos has <em>only</em> one <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">point</a>, although it is usually a <a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a> and hence has an <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> point.)</p> <h2 id="the_big_and_little_topos_of_an_object">The big and little topos of an object</h2> <p>If <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> is a topological space, then the canonical little topos associated to <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> is the sheaf 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>. On the other hand, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a site of probes enabling us to regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as an object of a big topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H = Sh(S)</annotation></semantics></math>, then we can also consider the topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">H/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>. These two toposes are often called the <strong>little topos 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></strong> (or <strong>petit topos 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></strong>) and the <strong>big topos 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></strong> (or <strong>gros topos 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></strong>) respectively.</p> <p>There might be some debate about whether <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">H/X</annotation></semantics></math> is, itself, “a little topos” or “a big topos.” While it certainly contains information about the 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> specifically, its objects are not “spaces locally modeled on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>” but rather spaces locally modeled on the big site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> which happen to have a map to <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>. The standard phrase “the big topos 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>” is the most descriptive.</p> <p>Note that if <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> is actually an <em>object</em> of the site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">H/X</annotation></semantics></math> can be identified with the topos of sheaves on the <a class="existingWikiWord" href="/nlab/show/slice+category">slice</a> site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S/X</annotation></semantics></math> (and otherwise, it can be identified with the topos of sheaves on the <a class="existingWikiWord" href="/nlab/show/category+of+elements">category of elements</a> of <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>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\in Sh(S)</annotation></semantics></math>). This site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S/X</annotation></semantics></math> is often referred to as the <a class="existingWikiWord" href="/nlab/show/big+site">big site</a> 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>, as compared to the <a class="existingWikiWord" href="/nlab/show/little+site">little site</a>, which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Op(X)</annotation></semantics></math> (or appropriate replacement). 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>S</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(S/X)</annotation></semantics></math> can thus be viewed as spaces modelled on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, but parameterised by the representable sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Note that when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">S=Top</annotation></semantics></math> with its local-homeomorphism topology, there is a canonical functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>S</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Op(X) \to S/X</annotation></semantics></math> which preserves finite limits and both <a class="existingWikiWord" href="/nlab/show/cover-preserving+functor">preserves</a> and <span class="newWikiWord">reflects<a href="/nlab/new/cover-reflecting+functor">?</a></span> covering families. Therefore, it induces both a geometric morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">/</mo><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">H/X \to Sh(X)</annotation></semantics></math> and one <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>H</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Sh(X) \to H/X</annotation></semantics></math>, of which the latter is the left adjoint of the former in <a class="existingWikiWord" href="/nlab/show/Topos">Topos</a>. In other words, the geometric morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">/</mo><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">H/X \to Sh(X)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local</a>, and in particular a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence+of+toposes">homotopy equivalence of toposes</a>. This fact relating the big and little toposes 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> also holds in other cases.</p> <h2 id="axiomatizations">Axiomatizations</h2> <ul> <li> <p>If a site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is given by a <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">Grothendieck pretopology</a>, then one can define an associated notion of a <a class="existingWikiWord" href="/nlab/show/little+site">little site</a> associated to any object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, and hence both a little topos and a big topos, which are related as above.</p> </li> <li> <p>One proposed axiomatization of the notion of big topos is that of a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a>.</p> </li> <li> <p>In his early papers in the 80s, <a class="existingWikiWord" href="/nlab/show/Lawvere">Lawvere</a> emphasized the existence of a contractible <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>, a concept which together with the <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> goes under the name <a class="existingWikiWord" href="/nlab/show/sufficiently+cohesive+topos">sufficiently cohesive topos</a> in the later axiomatization (modulo some fineprint).</p> </li> </ul> <h2 id="examples">Examples</h2> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the <em>little topos</em> that it defines is 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>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X) := Sh(Op(X))</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> 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>. A general object in this topos can be regarded as an <a class="existingWikiWord" href="/nlab/show/etale+space">etale space</a> 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>. The 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> itself is incarnated as the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo>*</mo><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = * \in Sh(X)</annotation></semantics></math>.</p> <p>On the other hand, a <em>big topos</em> in which <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> is incarnated is a <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> on a <a class="existingWikiWord" href="/nlab/show/site">site</a> of test spaces with which <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> may be probed. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, or <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> or <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> with their standard <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>s, <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 such a big topos. See for instance, <a class="existingWikiWord" href="/nlab/show/topological+topos">topological topos</a> and the <a class="existingWikiWord" href="/nlab/show/quasi-topos">quasi-topos</a> of <a class="existingWikiWord" href="/nlab/show/quasitopological+space">quasitopological space</a>s.</p> <p>In good cases, the intrinsic properties 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> do not depend on whether one regards it as a little topos or as an object of a gros topos. For instance at <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in the section <a href="http://ncatlab.org/nlab/show/cohomology#NonabelianSheafCohomology">Nonabelian sheaf cohomology with constant coefficients</a> it is discussed how the <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> of a <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</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> with constant coefficients gives the same answer in each case.</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a>, <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+site">topological site</a>, <a class="existingWikiWord" href="/nlab/show/continuous+truth">continuous truth</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sufficiently+cohesive+topos">sufficiently cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesive+%28infinity%2C1%29-topos">infinitesimal cohesive (infinity,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quality+type">quality type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tendue">étendue</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+decidable+topos">locally decidable topos</a></p> </li> </ul> <h2 id="references">References</h2> <p>The notion of a <em>gros topos</em> of a <em>topological space</em> is due to <a class="existingWikiWord" href="/nlab/show/Jean+Giraud">Jean Giraud</a>. Some early results from the Grothendieck school appear in</p> <ul> <li id="SGA4"><a class="existingWikiWord" href="/nlab/show/M.+Artin">M. Artin</a>, <a class="existingWikiWord" href="/nlab/show/A.+Grothendieck">A. Grothendieck</a>, <a class="existingWikiWord" href="/nlab/show/J.+L.+Verdier">J. L. Verdier</a>, <em>Théorie des Topos et Cohomologie Etale des Schémas (<a class="existingWikiWord" href="/nlab/show/SGA4">SGA4</a>)</em>, Springer LNM <strong>269</strong> (1972). (exposé IV, 2.5 pp.316-318, 4.10 pp.358-365)</li> </ul> <p>In this context see also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Saunders+Mac+Lane">S. Mac Lane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">I. Moerdijk</a>, pp. 113, 325, 416 in: <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em>, Springer (1994) [<a href="https://link.springer.com/book/10.1007/978-1-4612-0927-0">doi:10.1007/978-1-4612-0927-0</a> ]</li> </ul> <p>In the context of a discussion of the <a class="existingWikiWord" href="/nlab/show/big+Zariski+topos">big Zariski topos</a> <a href="#Lawvere76">Lawvere (1976, p. 110)</a> calls the gros-petit distinction ‘<em>a surprising twist of logic that is not yet fully clarified</em>’:</p> <ul> <li id="Lawvere76"><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, <em>Variable quantities and variable structures in topoi</em>, in <a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a> (eds.), <em>Algebra, Topology and Category Theory – A Collection of Papers in Honor of <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a></em>, Academic Press New York (1976) 101-131 [<a href="https://doi.org/10.1016/C2013-0-10841-0">doi:10.1016/C2013-0-10841-0</a>]</li> </ul> <p>The suggestion that a <em>general notion</em> of gros topos is needed goes back to some remarks in <em><a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">Pursuing Stacks</a></em>. A precise axiom system capturing the notion is first proposed in</p> <ul> <li id="Lawvere86"><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, <em>Categories of spaces may not be generalized spaces, as exemplified by directed graphs</em>, Revista Colombiana de Matematicas <strong>XX</strong> (1986) 179-186, reprinted as: Reprints in Theory and Applications of Categories, <strong>9</strong> (2005) 1-7 [<a href="http://www.tac.mta.ca/tac/reprints/articles/9/tr9abs.html">tac:tr9</a>]</li> </ul> <p>“Axiom 0” (<a class="existingWikiWord" href="/nlab/show/local+topos">locality</a>) used in <a href="#Lawvere86">Lawvere 1986</a> for gros toposes is argued in <a href="#Lawvere94">Lawvere 1994</a> to be essentially an insight due to <a class="existingWikiWord" href="/nlab/show/Georg+Cantor">Georg Cantor</a> and is called the <em>Cantorian Contrast</em> (namely between <a class="existingWikiWord" href="/nlab/show/discrete+spaces">discrete spaces</a> and <a class="existingWikiWord" href="/nlab/show/codiscrete+spaces">codiscrete spaces</a>) in <a href="#LawvereRosebrugh03">Lawvere & Rosebrugh (2003)</a>, <a href="http://patryshev.com/books/Sets%20for%20Mathematics.pdf#page=260">p. 245</a>.</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>, <a href="http://patryshev.com/books/Sets%20for%20Mathematics.pdf#page=260">p. 245</a> in: <em><a class="existingWikiWord" href="/nlab/show/Sets+for+Mathematics">Sets for Mathematics</a></em>, Cambridge University Press (2003) [<a href="https://doi.org/10.1017/CBO9780511755460">doi:10.1017/CBO9780511755460</a>, <a href="http://www.mta.ca/~rrosebru/setsformath/">book homepage</a>, <a href="http://patryshev.com/books/Sets%20for%20Mathematics.pdf">pdf</a>]</li> </ul> <p>The axioms 0 and 1 for <em>toposes of generalized spaces</em> given in <a href="#Lawvere86">Lawvere 1986</a> later became called the axioms for a <em><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></em></p> <ul> <li id="Lawvere94"><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, <em><a class="existingWikiWord" href="/nlab/show/Cohesive+Toposes+and+Cantor%27s+%22lauter+Einsen%22">Cohesive Toposes and Cantor's "lauter Einsen"</a></em>, Philosophia Mathematica <strong>2</strong> 1 (1994) 5-15 [<a href="https://doi.org/10.1093/philmat/2.1.5">doi:10.1093/philmat/2.1.5</a>, <a class="existingWikiWord" href="/nlab/files/LawvereCohesiveToposes.pdf" title="pdf">pdf</a>]</li> </ul> <p>together with axiom 2 they make out a <a class="existingWikiWord" href="/nlab/show/sufficiently+cohesive+topos">sufficiently cohesive topos</a>.</p> <p>Further discussion of this axiomatics for gros toposes is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a>, <em>Categories of space and quantity</em> in: J. Echeverria et al (eds.), <em>The Space of mathematics</em>, de Gruyter, Berlin, New York (1992) [<a href="https://raw.githubusercontent.com/mattearnshaw/lawvere/master/pdfs/1992-categories-of-space-and-quantity.pdf">pdf</a>]</li> </ul> <p>where a proposal for a general axiomatization of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>/<a class="existingWikiWord" href="/nlab/show/homology">homology</a>-like “extensive quantities” and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>-like “intensive quantities”) as covariant and contravariant functors out of a distributive category are considered.</p> <p>The following two papers contain Lawvere’s early view of a trichotomy between big toposes vs. étendue and locally decidable toposes as paradigmatic “generalized spaces” with “infinitesimally cohesive” in between, with the latter subsumed into the fine structure of cohesion in more recent versions</p> <ul> <li id="Law89a"> <p><a class="existingWikiWord" href="/nlab/show/F.+W.+Lawvere">F. W. Lawvere</a>, <em>Qualitative Distinctions between some Toposes of Generalized Graphs</em>, in <em>Categories in Computer Science and Logic</em>, Cont. Math. <strong>92</strong> (1989) 261-299 [<a href="http://dx.doi.org/10.1090/conm/092">doi:10.1090/conm/092</a>, <a href="https://github.com/mattearnshaw/lawvere/raw/master/pdfs/1989-qualitative-distinctions-between-some-toposes-of-generalized-graphs.pdf">pdf</a>]</p> </li> <li id="Law91a"> <p><a class="existingWikiWord" href="/nlab/show/F.+W.+Lawvere">F. W. Lawvere</a>, <em><a class="existingWikiWord" href="/nlab/show/Some+Thoughts+on+the+Future+of+Category+Theory">Some Thoughts on the Future of Category Theory</a></em>, 1-13 in Springer LNM <strong>1488</strong> (1991).</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> in a cohesive topos is also mentioned in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/8/tr8.pdf#page=14">page 14</a> of: <em>Taking categories seriously</em>, Revista Colombiana de Matematicas <strong>XX</strong> (1986) 147-178, Reprints in Theory and Applications of Categories, <strong>8</strong> (2005) 1-24. [<a href="http://www.tac.mta.ca/tac/reprints/articles/8/tr8abs.html">tac:tr8</a>, <a href="http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf">pdf</a>]</li> </ul> <p>Under the term <em>categories of cohesion</em> these axioms are discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a>, <em>Axiomatic cohesion</em>, Theory and Applications of Categories <strong>19</strong> 3 (2007) 41-49 [<a href="http://www.tac.mta.ca/tac/volumes/19/3/19-03abs.html">tac:19-03</a>, <a href="http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf">pdf</a>]</li> </ul> <p>Another definition of gros vs petit toposes and remarks on applications in <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a> is in</p> <ul> <li>Nick Duncan, <em>Gros and petit toposes</em> (<a href="http://www.cheng.staff.shef.ac.uk/pssl88/pssl88-duncan.pdf">pdf</a>)</li> </ul> <p>and yet another one is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Spectral+Algebraic+Geometry">Spectral Algebraic Geometry</a></em>, chapter 20 “Fractured <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>-Topoi” (<a href="http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf">pdf</a>)</li> </ul> <p>There is also something relevant in this article:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mathieu+Anel">Mathieu Anel</a>, <em>Grothendieck topologies from unique factorization systems</em> (<a href="http://arxiv.org/abs/0902.1130">arXiv:0902.1130</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mamuka+Jibladze">Mamuka Jibladze</a>, <em>Homotopy types for “gros” toposes</em>, thesis, <a href="http://www.rmi.ge/~jib/pubs/thesis.pdf">pdf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Calibrated Toposes</em>, Bull. Belgian Math. Soc. - Simon Stevin <strong>19</strong> 5 (2012) 889-907. [<a href="http://projecteuclid.org/euclid.bbms/1354031555">euclid:1354031555</a>]</p> </li> </ul> <p>A discussion and comparison of big vs little approaches to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos theory began at these blog entries:</p> <ul> <li><a href="http://golem.ph.utexas.edu/category/2010/10/cohesive_toposes.html">Cohesive (∞,1)-toposes</a> and <a href="http://golem.ph.utexas.edu/category/2010/10/petit_1toposes.html">Petit (∞,1)-toposes</a>.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 14, 2022 at 09:18:01. 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