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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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Intuition for pseudo-points and the inductive step in Johnstone's proof of Deligne's completeness theorem

In Johnstone's Topos Theory appears the following lemma. 7.41 Lemma. Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two distinct element of $P(X)$. Let $(V_j\to V)...
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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 presheaves

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 ...
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Show that a certain functor preserves colimits and finite limits by verifying it on the stalks of sheaves

In Moerdijk, Classifying spaces and classifying topoi, on page 22, we have a functor from the topos $\mathcal BG$ of right $G$-sets ($G$ is a group) to the topos $Sh(X)$, the sheaves (étale spaces) ...
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Continuous functor for a Grothendieck topology

In MacLane Moerdijk, pag. 393, lemma VII.7.3, the following is stated: Let $f:\mathcal{E}\rightarrow\widehat{\mathbb{C}}$ be a geometric morphism and $J$ a Grothendieck topology over $\mathbb{C}$, ...
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Site for slice sheaf topos

The fundamental theorem of topos theory states that the slice categories $\mathcal{E}/A$ of a (Grothendieck) topos $\mathcal{E}$ are again (Grothendieck) toposes. If $\mathcal{E}$ is of the form $\...
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Reference for "topos obtained by adjoining an indeterminate set' theorem

From Lawvere's Continuously variable sets; algebraic geometry = geometric logic: The following illuminating fact about topoi (long known for the case $\mathsf S$=constant sets) was (conjectured ...
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When sheafification functor is open

Let $\mathcal{G}$ be a Grothendiek topos and let $(\mathcal{C},J)$ be a site for $\mathcal{G}$. Is it true that if the sheafification functor $$ \mathcal{G} \leftrightarrows \text{Set}^{\mathcal{C}^...
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Is $P(M)$ internally complete in the category of $M$-sets?

Let $\textsf{MSet}$ be the category of $M$-sets. The powerset of $M$, $P(M)$, is an $M$-set with action given by division: $iX = \{j \mid j\circ i \in X\}$ Suppose that $B$ is a Boolean algebra with ...
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Classifying topos for higher order logic

Chapter X of Mac Lane and Moerdijk's Sheaves in Geometry and Logic focuses on Classifying topoi. The basic concept in the early pages is the one of geometric formula, which is by definition a first-...
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How is a Scott domain Cartesian-closed?

I read the following excerpt from nLab but I need further explanation: The problem Scott solved is to faithfully model untyped lambda calculus; in categorical terms, the basic problem is to ...