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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.

Topos theory has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kind of manifolds (algebraic, analytic, etc.). Sheaves also appear in logic as carriers for models of set theory as well as for the semantics of other types of logic.

Grothendieck introduced a topos as a category of sheaves for algebraic geometry. Subsequently, Lawvere and Tierney obtained elementary axioms for such (more general) categories.

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Do free modules exist in a general topos?

(Inspired by an answer given here: does monomorphism of sheaves of abelian groups imply it's injective on the level of open sets? ) I wonder if, in a general (elementary) topos $\mathcal{T}$, if you have a ring object $R$, then the underlying set…
Daniel Schepler
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Example of a small topos

I'm currently trying to understand this article by T. Noll on the topos of triads in music theory (also, this) However, I can't get past section 2.2 where Noll introduces the subobject classifier, most probably because I don't know much about topos…
AlexPof
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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 essentially a description of his overall method of…
Malice Vidrine
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Toposes and Stone-type dualities

The duality between the category Sets and the category CABool of complete atomic boolean algebras is an example of a general Stone-type duality. At the same time, Sets is a topos. Are there other known examples of Stone-type dualities involving…
Sephi
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Internal logic characterization of closed subobjects for a Lawvere-Tierney topology

I am trying to understand the relation between Lawvere-Tierney topologies and the internal logic of toposes. For a closure operator $\overline{-}$ I am trying to prove that a subobject $A$ of an object $E$ is closed (with respect to the given…
Boogie
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Is there a non-topos for which the objects having power objects form a topos?

Given any category $C$ with finite limits, let $D$ be the full subcategory of $C$ consisting of those objects $X$ for which a power object $P(X)$ exists. Then, one might ask whether the following properties hold: $1 \in D$, i.e. $C$ has a subobject…
Geoffrey Trang
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Geometric surjections are not stable under pullback

It is mentioned at the end of the introduction to Johnstone's Factorization systems for geometric morphisms, I that the pullback of a geometric surjection need not be a surjection, and hence that the surjection-inclusion factorisation system is not…
Morgan Rogers
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Does the internal logic of a topos satisfy propositional, functional, set extensionality?

Given an arbitrary topos $\mathscr{T}$ and objects $X, Y$ of $\mathscr{T}$, are the interpretations of the following terms equal to $\top : \mathrm{Hom}(1, \Omega)$? $\forall p, q : \Omega, (p \rightarrow q) \rightarrow (q \rightarrow p)…
Daniel Schepler
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Is the cartesian product of objects in an elementary topos cancellative?

My question is the internalization of this question to an elementary topos $C$. Is it true that: For objects $X,Y$ and $Z$ in an elementary topos $C$ with $X\times Y\cong X\times Z$, then also $Y\cong Z$?
user17982
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Is the category of simplicial presheaves an example of a topos?

It is known that for any small category $\mathcal{C}$ the presheaf category $[\mathcal{C}^{op}, \mathsf{Set}]$ forms a topos. What about the category of simplicial presheaves, i.e. $[\Delta^{op},[\mathcal{C}^{op}, \mathsf{Set}]]$? I would guess…
Nary
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Existential quantifier in a topos

Suppose that $E$ is an elementary topos. Thus, for every object $X\in E$ one has the corresponding existential quantifier $\exists_X:PX\rightarrow \Omega$. My question is: Whats is the subobject of the power $PX$ classified by $\exists_X$? My guess…
None
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Regular functors $F\colon\mathcal C\to\mathcal E$ factor essentially uniquely through a topos $\tilde C$.

I want to show the following statement: Given a small regular category $\mathcal C$ there exists a topos $\widetilde{\mathcal C}$ and a regular functor $F\colon\mathcal C\to\widetilde{\mathcal{C}}$ such that any regular functor…
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subtoposes induced by generating collections

(I will use the terminology in the book {\em Sheaves in Geometry and Logic} by Mac Lane and Moerdijk.) A collection ${\{ G_{\xi} \}}$ of objects in a category $\mathbf{A}$ is said to {\em generate} $\mathbf{A}$ iff any parallel pair of morphisms…
Boogie
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Topos isomorphic to the sheaf topos of the circle

When we abstract the notion of space, we get more and more isomorphisms. For example, the triangle, square and circle are not isometrical, but they are homeomorphic. If we take one step further to the sheaf toposes, do we get an isomorphism with…
V. Semeria
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Is the category of sheaves on a site a 1-category or a 2-category?

In the use of the term topos I read about today, $\mathfrak{Top}$ is a 2-category. In the first use I ever read about a topos was just a category of sheaves on a site. I didn't think if it was a 2-category instead of a 1-category though. Is the…
sheafpants4
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