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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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Alternative sources for a topos theory description of zeroth order logic

I have recently been reading Robert Goldblatt's fantastic book Topoi: The Categorial Analysis of Logic. Through chapters 6-8, Goldblatt produces a topos theoretic approach to zeroth order logic, where ...
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Forcing in sheaf models of set theory - where do the “generics” disappear to?

I am studying "Sheaves in Geometry and Logic" by Mac Lane & Moerdijk. In their construction of a topos satisfying $\lnot CH$ they work entirely in the Grothendieck topos of double negation ...
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Is a name a sheaf?

The technique of forcing, in set theory, can be expressed in topos theory as a form of reasoning about sheaves on the notion of forcing, $\mathbb{P}$, equipped with a "double negation" Grothendieck ...
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What does “topological heuristics” refer to here?

This quote is from the wikipedia page on topoi: Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea ...
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Topological space represented as a topos?

Wikipedia states that Toposes "are in a sense a generalization of point-set topology". This suggests to me that any topological space can be represented as a topos. Is this true? How do we do this?
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Well-Pointed categories vs well-pointed topoi

When reading the definition of a well-pointed category on for example Wikipedia, only the following condition is given: The terminal object 1 is a generator. However the general definition of a well-...
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What does it mean for two morphisms with different sources and targets to be isomorphic?

The standard definition of a subobject relies on the following definition: we call two morphisms $f : X \rightarrow A, g : Y \rightarrow A$ with the same target isomorphic if there exists an ...
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A question about Ščedrov's “Forcing and Classifying Topoi”

I'm trying to read Ščedrov's Forcing and Classifying Topoi, and there's a bit in 1.1 that is frequently references, but I don't quite understand its import. If I'm not missing the point, 1.1 is ...
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The big étale and Zariski topoi are generated by small sites

Grothendieck (SGA 4, Exposé VII, Corollaire 3.2) proves that if $X$ is a quasiseparated scheme then, although the étale site is not small, there exists a subsite of it which is small and has the same ...
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A certain colimit of representables sheaves, namely group actions, is a sheaf: why?

In the first answer to this post on MO, one finds that When you look at the category of sheaves on the category of finite action with the natural topology (covering are surjection of finite ...
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The small étale topos of a scheme is equivalent to the category of finite $\pi_1(X,x)$-sets for every scheme $X$ and every geometric point $x$

Recall Milne, Etale cohomology, Theorem I.5.3: Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/X\to Sets$ ($FEt/X$ being the category of $X$-schemes finite and ...
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Are topoi like topological spaces or like set theory?

Background: I have very little intuition about category theory but I’m trying to understand the motivation behind it. Wikipedia states: In mathematics, a topos [...] is a category that behaves ...
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In algebraic geometry, what kind of theory can only be described by topos but not a site?

A (Grothendieck) topos is defined to be a category equivalent to the category of sheaves on some site. The main difference for a topos and a site is about their morphisms. I noticed that some books ...
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If a Grothendieck topos has a geometric morphism to $Sets$, then it is unique

In Caramello, Theories, Sites, Toposes, I read that Every Grothendieck topos $\mathcal E$ admits a unique (up to isomorphism) geoemtric morphism $\gamma_{\mathcal E}:\mathcal E\to \bf Set$. The ...
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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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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 ...