# 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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### Using Yoneda lemma to get isomorphisms

Let’s assume we have two functors $F:C\longrightarrow D$ and $G:C\longrightarrow D$ such that the functor $Hom_D(F(-),-)$ is naturally isomorphic to $Hom_D(G(-),-)$. Can we get from this that F is ...
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### Transpose of a diagram under an adjunction

By a diagram of type I (I is also a category)in a category C, we mean a functor $F:I\longrightarrow C$. Now, let’s assume we have an adjunction $F:C\longrightarrow D$ and $G:D\longrightarrow C$ such ...
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### Compatible family in a presheaf

Let C be a category. Let c be an object in C. One definition I know for a compatible family on c in a presheaf F is that it is a natural transformation from R to F where R is a sieve on c. Another ...
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### What are the prerequisites for topos theory?

This might be too vague or too broad if we're not careful. Therefore, let's focus on the basics. According to this MSE search, this is new to MSE. Some Background: I have read Goldblatt's, "Topoi: A ...
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### Direct image in a topos

I am trying to solve exercise 10 of chapter 1 “topos theory” by Johnstone. I have a very long solution for that and I am trying to solve it in an easier way. To do this, I came up with this question ...
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### Defining an arrow as its composition

I'm working through the book Algebraic Set Theory by Joyal & Moerdijk. However, I stumbled upon some contruction which I did not manage to understand yet. In the proof of Proposition 4.2, we ...
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### Two equivalent notions of compactness?

Let $X$ be a topological space $Sh(X)$ sheaves of $X$. Then it is stated that compactness of $X$ can be expressed equivalently as Every cover of $1$ by subobjects has a finite subcover. ...
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### . . . and what about $\Gamma$ in $\S II.5$ of Mac Lane and Moerdijk?

This is about $\S II.5$ of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]" and is a follow-up to this: Just what is Mac Lane & Moerdijk's $\Lambda$ from $\S II.5$? The ...
### In a topos, retracts of objects of the form $\Omega^Y$ are injective
I am trying to prove the statement from the title. Let $X$ be a retract of $\Omega^Y$, say $r: \Omega^Y \to X$ with right inverse $s$. Suppose we have a diagram $B \xleftarrow{f} A \xrightarrow{m} X$...