# 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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### Introduction Of Mitchell-Bénabou-Language

I'm confused about the introduction of the Mitchell-Bénabou-language in Sheaves in Geometry and Logic: Here are the inductive clasues which simultaneously define the terms of the language and their ...
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### Constructing injective resolution on big sites.

Let's work over a category of varieties on some field. Let $\mathcal{F}$ be a presheaf on varieties that satisfies Zariski/etale sheaf conditions whenever restricted to the opens of a specific variety....
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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 $F\colon\mathcal C\to\widetilde{\mathcal{C}}$ ...
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### Indexed Yoneda in the Elephant

I've been trying to gather definitions and results to understand the internal Diaconescu theorem, for I wish to read Joyal-Tierney's An Extension of the Galois Theory of Grothendieck. My question ...
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### What is the connection between algebraic groups and topoi?

I have a longstanding interest in topos theory. (See this MSE search of my questions about topos theory.) I am studying for a postgraduate research degree in linear algebraic group theory. Naturally, ...
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### Presheaf problem

I need to show that the global element of a presheaf $P\in \widehat{\mathcal{O}(X)}$ is equivalent to global section of bundle of germs $\Lambda_P\xrightarrow{p}X$. In particular, I need help to show ...
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### Is the functor category of a small category into a topos itself a topos?

I am reading the book "Sheaves in Geometry and Logic" by MacLane and Moerdijk. Early in the book, the authors show that $Sets^{\mathscr{C}^{op}}$ where $\mathscr{C}$ is a small category is ...
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### On Atomistic Categories

Being impressed by Barr and Diaconescu paper entitled "Atomic Toposes" (https://www.math.mcgill.ca/barr/papers/atom.top.pdf), I would like to ask whether it makes sense to investigate "...
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### Reconstructing isomorphisms via the bijection between the corresponding posets of subobjects - Part 2

In this post Reconstructing isomorphisms via the bijection between the corresponding posets of subobjects I asked for the possibility of constructing an isomorphism via the order-preserving bijection ...
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### Order relations on natural number objects in topoi, and symmetry

Goldblatt in "Topoi" defines the relation ≤ on an NNO $N$ as the subobject $\langle \pi_1, \oplus \rangle : N \times N \rightarrowtail N \times N$, where $\oplus : N \times N \rightarrow N$ ...
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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 ...
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### How to construct a group-valued sheaf from set-valued sheaf

Suppose that I have a set-valued sheaf $S$ on a site $(\mathcal C, J)$. Question 1.) Is there a canonical way to turn $S$ into a sheaf with Abelian group values? I considered the following: for each ...
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### Uniqueness of arrows between Heyting-valued sets

I’m going through Goldblatt’s “Topoi”, in particular through the subchapter 11.9 on Heyting-valued sets, and it feels overly… concise. In short, I’m not sure how to prove uniqueness of arrows. For ...
The paragraph reads: Let $T$ be a topos, and let $\Lambda$ be a ring in $T$. Denote by $\operatorname{Mod}_{\Lambda}$ the category of $\Lambda$-modules in $T$. My question is how is the category of \$\...