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Questions tagged [topos-theory]

A topos (plural topoi, toposes) is a category that behaves like the category of sheaves of sets on a topological space. Topos theory consists of the study of Grothendieck topoi, used in algebraic geometry, and the study of elementary topoi, used in logic.

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Clarify definition of 'enough points'

I have checked the stacks project, Sketches of an Elephant, and Sheaves in Geometry and Logic but cannot find a satisfactory answer. If a site or topos has enough points, does that mean, that there is ...
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Maps from $\mathcal{Cat}$ into arbitrary categories : Can you Map a Functor to a Morphism? [closed]

I am wondering about mapping functors into morphisms. What do I mean? I think I mean: I want to take a functor between arbitrary categories and map it into some specific category. I am starting to ...
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Johnstone, Topos theory: families of arrows inducing the same sheaf condition

Johnstone, Topos theory, 0.3, page 13, asserts that, given a Grothendieck pretopology $P$, if the equalizer condition on a presheaf $F$ is satisfied for a family of arrows $R=\{U_i\to U\}$, then it is ...
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Motivation for topological categories

Define a topological category as a small category with topologies on the set of objects $C_0$ and on the set of arrows $C_1$, such that the domain map and the codomain map are continuous. One can ...
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Boolean algebra homomorphisms with adjoints

I'm interested in a particular class of maps between boolean algebras: the homomorphisms $f:A\to B$ such that there exist functions (not necessarily homomorphisms) $u,d:B\to A$ such that $$f(a)\leq b\...
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Does sheafification of bundles have a right adjoint?

Given a topological space $X$, let $\mathbf{Bundle}(X)=\mathbf{Top}/X$ be the category of bundles over $X$, and let $\mathbf{Sh}(X)$ be the category of sheaves over $X$. Then there's a sheafification ...
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Well Definedness of the Representation of the Subobject Functor

Let $\mathcal{C}$ be a category with a subobject classifier $\Omega$. I want to show that the suboject functor $Sub_{\mathcal{C}}(\cdot): \mathcal{C}^{op} \to \textbf{Set}$ is representable via $\...
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Does a sheaf of abelian groups on a scheme $X$ induce a sheaf of abelian groups on the étale site $X_{ét}$?

Fixed a scheme $X$, étale cohomology $H^\bullet(X,F)$ is defined for all sheaves of abelian groups $F$ over the étale site $X_{ét}$. Now, just to understand, I tried to see if this covers sheaves of ...
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Different descriptions of internal languages of a topos

I wanted to learn more about the internal language of of toposes and to do so I have been reading both Sheaves in Geometry and Logic (Sheaves) by Mac Lane and Moerdijk and Introduction to Higher Order ...
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How to categorically characterize the structure of all grounded first-order logic formulas?

If the notion of topos corresponds to set theory or first-order logic, then what if one is only interested in grounded logic formulas, ie, formulas that don't contain variables? In a topos one has ...
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Is ZFC a geometric theory? Does it have a classifying topos?

Given a language $L$, the set of geometric formulae in $L$ is the smallest set of formulae containing atomic formulae and closed by finite conjunction, arbitrary disjunction, and existential ...
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Topos with sets as subobject classifier

In the computer science literature I have seen cases in which they move from booleans to $\mathsf{Set}$ for obtaining a "proof-relevant" version of a concept (especially in formalisations). As the ...
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Is there a classifying topos for locales?

Is there a Grothendieck topos $F$ such that, for any Grothendieck topos $E$, the category of geometric morphisms $$E\rightarrow F$$ is equivalent to the category of locales internal to $F$? I suspect ...
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Presheaves are sheaves for the trivial topology

I've just started learning about toposes and I have a stupid question to ask. Suppose we are given a small category $\mathcal{C}$ with trivial topology $T$ on it, (where the trivial topology $T$ on ...
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Is the interpretation of the Morleyization of a first order theory in a Heyting category automatically classical?

In Johnstone's Sketches of an Elephant (D1.5.13), he describes how given a first order theory $T$ in a signature $\Sigma$, one can devise a coherent theory $T'$ over a new signature $\Sigma'$ (the ...
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“Category” of (pre)stacks over a fixed category/site

I am wondering if prestacks over a fixed category, or stacks over a fixed site, form an (interesting) category, equipped with prestacks morphisms. First of all, do they form a proper class or a set? ...
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Logic for physics

I'm looking for references on applications of logic (e.g. intuitionistic logic, toposes) to physics (especially quantum physics, as this is the area that I'm aware can be helped by logical ...
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Looking for the Deligne's example of a non trivial topos with no points

In the topos entry of wikipedia, says: there is an example due to Pierre Deligne of a nontrivial topos that has no points (see below for the definition of points of a topos). I'm looking for the ...
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Nontrivial example of topos in the context of logic

I am reading Awodey's paper Structure in Mathematics and Logic: A Categorical Perspective and he is currently (p. 228-9) talking about how propositions, that is, elements $1\rightarrow P$, correspond ...
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A topos with enough points is the topos of equivariant sheaves on a topological groupoid (Butz-Moerdijk): where do continuous actions come from?

In Butz, Moerdijk, Representing topoi by topological groupoids, it is proven that given a topos $\mathcal E$ with enough points there exists a topological groupoid $G\rightrightarrows X$ such that $\...
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The étale topos is coherent: does the scheme need to be quasicompact?

It is known that the little étale topos of a scheme $X$ is coherent, i.e. the Grothendieck topology of the étale site is generated by a basis of finite coverings. For example, Butz and Moerdijk say ...
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Adjoint for pullback of sheaves on a topological groupoid

Consider a topological groupoid $G\xrightarrow{s} X$, with the arrow representing the domain map that associates to each morphism its domain. This induces a pullback functor $s^*:Sh(X)\to Sh(G)$. I ...
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All sheaves in a Grothendieck topos are generated by subobjects of powers of a suitable sheaf

Consider a topos $\mathcal E$. Butz and Moerdijk, in Representing topoi by topological groupoids, par. 2, say that one can find an object $S\in \mathcal E$ such that the subobjects of its powers (i.e.,...
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Homotopy classes of maps inside a Grothendieck topos

Moerdijk, Classifying spaces and classifying topoi, defines the Verdier cohomology of a Grothendieck topos $\mathcal E$ by taking a limit on $HC$, the homotopy of hypercoverings in $\mathcal E$. I ...
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Meaning of Grothendieck quote: recover a topological space by its category of sheaves

In "Récoltes et Semailles"(- Grothendieck), there is a moment when the author talks about the idea of sheaves of sets over a topological space, then taking the category of sheaves (of sets over a ...
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Three definitions of hypercoverings

The notion of hypercovering (or hypercover) was introduced by J.L. Verdier in SGA 4. In his definition, a hypercovering $K_\bullet$ is a semirepresentable semisimplicial presheaf on a site $C$ such ...
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Is a topological category a Grothendieck site?

I take the definition of a topological category from Segal and (later) Moerdijk. So this is a small category with topologies on the set of objects and the set of morphisms, such that the domain map, ...
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Internal frame homomorphisms and sheafification

Let $\mathcal{E}$ be an elementary topos with subobject classifier $\Omega$, and let $j : \Omega \to \Omega$ be a Lawvere-Tierney topology. $\Omega$ is naturally seen as a frame object internal to $\...
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How is it possible to recover properties of compactness, connectedness, hausdorff, of topological space by using the category of sheaves

I will rephrase the title : How is it possible to effectively recover/express all classical properties stated above of a topological space by using the category of presheaves of open subsets ?
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Morita-equivalence of groupoids and classifying topoi: correct definition

The first comment to this post points out that, given two (topological or localic) groupoids, they can be non-Morita-equivalent evenif their classifying topoi (topoi of equivariant sheaves) are ...
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Points of affine scheme and points of topos

Let $X =$Spec$(R)$ be an affine scheme and $\mathscr{E}_{X}$ denote the topos of sheaves of sets on $X$. Is it true that the (geometric) points of $\mathscr{E}_{X}$ are in bijection with the prime ...
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Constructive proof of a classical result

In a document about constructive maths, I saw the following exercise : prove that $f^{-1}: \mathcal{P}(Y) \to \mathcal{P}(X)$ is injective if and only if $f$ is surjective. There is an easy ...
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Strong homotopy equivalence on classifying spaces via weak homotopy equivalence on classifying topoi

In his book Classifying spaces and classifying topoi Moerdijk proves that there exists a wieak homotopy equivalence between the classifying topos of the Haefliger space $\Gamma^q$ and that of the ...
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Relationship between Haefliger structures and principal $\Gamma^q$-bundles

A Haefliger structure on a smooth manifold is a cocycle with coefficient in the Haefliger groupoid $\Gamma^q$. This generalises the notion of foliation of codimension $q$. We know that the classifying ...
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The étale topos of a scheme is the classifying topos of…?

By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. Butz. and I. Moerdijk. Representing topoi by topological ...
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Subfunctor of a hom functor in a Boolean topos

There is a wrong argument below. The conclusion is not correct but I can't figure out why. Could anyone tell me why it is not correct? Let X be any space and suppose the category of all sheaves $Sh(X)...
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epimorphism of fppf sheaves is an fppf morphism

Suppose $0\to F\to G \to H\to 0$ is an exact sequence of group schemes (over some base scheme $S$) by which I mean that the corresponding sequence of fppf-sheaves is exact. I read somewhere that the ...
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What has more potential: Differential algebraic geometry or Diffiety theory?

Algebraic geometry is said to be useful to study not only specific solutions of polynomial equations but to understand the intrinsic properties of the totality of solutions of a system of equations. ...
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Locality in Grothendieck Topologies

Let $\mathcal{C}$ be a category and $\mathcal{J}$ be a Grothendieck topology on it (i.e., $(\mathcal{C},\mathcal{J})$ is a site). Then what is a good notion of locality in it? I came up with the ...
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Construction of Presheaf category such that subobjects of the terminal functor give a desired Heyting Algebra

Given any Heyting algebra $H$, can you construct a category $\mathcal{C}$ such that the the subfunctors of the terminal presheaf on $\mathcal{C}$ (assign singleton to every object) form H? That is $$\...
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Coincidence of classical notion of covering in classical topology and covering sieve in grothendieck topology

Let $X$ a topological space, $\mathcal{O}(X)$ the usual category of the open sets of $X$ and $(\mathcal{O}(X),J)$ any site. Do we necessarly have for each open set U and for each sieve in $J(U)$, $(...
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Lattice subobjects of the power object in a topos

When proving that the subobject classifier of a topos is an internal Heyting algebra, we exploit the natural isomorphism $$Sub(X)\simeq Hom(X,\Omega)$$ Therefore the intersection of subobjects induces ...
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Classifying topos of a topological group

Define the classifying topos of a topological category $C$ as the category of $C$-sheaves on $Obj(C)$. The classifying topos of a discrete group (represented as discrete one-object category) is the ...
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$\infty$-categories definition disambiguation

I am currently reading Lurie's - Higher Topos Theory, but am having difficulties from the very beginning. I know that the topic is too broad and different mathematicians may define differently some of ...
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Is there a classifying topos for schemes?

Is there a topos $\mathcal E$ such that, for any sober topological space $X$, the geometric morphisms $$\mathrm{Sh}\left(\mathcal O\left(X\right)\right)\rightarrow \mathcal E$$ are in correspondance ...
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Can 2 different coverages *on the same category* yield the same sheaf topos?

It is possible to have distinct sites $(\mathbf{C},J)$ and $(\mathbf{D},K)$ such that $Sh(\mathbf{C},J)\simeq Sh(\mathbf{D},K)$. Is this still the case when $\mathbf{C} \simeq \mathbf{D}$? That is, ...
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Geometric interpretation of Lawvere-Tierney topology

A Lawvere-Tierney topology $j$ on a sheaf topos $\mathcal{G}$ yelds a closure operator $\text{Sub}(1) \to \text{Sub}(1)$. If the topos is localic $\mathcal{G} \cong \text{Sh}(L)$ this yields a closure ...
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Cardinal collapse and (higher) toposes

I've been reading some of the more philosophical papers by John Lane Bell and Steve Awodey and I'd like some conceptual clarification. I know very little model theory so apologies if this is all too ...
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(From Awodey) Find the subobject classifier for $\sf Sets^{P}$ for a poset $\sf P$

The definition of a subobject classifier is given by: Definition 8.16. Let $\cal E$ be a category with all finite limits. A subobject classifier in $\cal E$ consists of an object $Ω$ together with an ...
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Prove that a “tensor product” principal $G$-bundle coincides with a “pullback” via topos morphism

From Moerdijk, Classifying spaces and classifying topoi, page 22. Consider a right $G$-set $S$ with the discrete topology. Let $E$ be a principal $G$-bundle over the topological space $X$. One can ...