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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Intuition on direct image and inverse image

A geometric morphism $f:\mathcal{C}\to \mathcal{D}$ between toposes is defined as an adjoint pair of functors $f_*:\mathcal{C}\to \mathcal{D}$ and $f_*:\mathcal{D}\to \mathcal{C}$. The adjunction is $...
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Is there a name for the fragment of (typed) first-order logic which works “pointwise” in any topos?

Recently while thinking about the interpretation of the internal language of a topos, it occurred to me to wonder what would happen if we restricted to the fragment of typed first-order logic which ...
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Let $\epsilon$ be a topos and let $f:Y\to X$ be a morphism. Let $M,N$ be two subobjects of $X$. Is it true that $f^*(M\cap N)=f^*(M)\cap f^*(N)$?

Let $\epsilon$ be a topos and let $f:Y\longrightarrow X$ be a morphism in $\epsilon$. Now, let $M, N$ be two subobjects of $X$. Is it true that $f^*(M\cap N)=f^*(M)\cap f^*(N)$? For $\epsilon=Set^{C^...
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Comonoids in coslices of a topos

Is it true that there are no nontrivial comonoids (with respect to the cocartesian monoidal structure, of course) in any coslice category of a topos? Proof that the answer is "Yes" for the case of $...
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Right adjoints preserve j-dense monos

Let’s assume $\epsilon$ is a topos and $L:\epsilon\longrightarrow \epsilon$ and $R:\epsilon \longrightarrow\epsilon$ are two adjoint functors such that $L\dashv R$. Now, let j be a Lawvere-Tierney ...
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j-closed monomorphisms in a topos with a Lawvere-Tierney topology j

Let’s assume $\epsilon$ is a topos and j is a Lawvere-Tierney topology in $\epsilon$. Now, let $Sh_j(\epsilon)$ be the full subcategory of $\epsilon$ on the sheaves for j. Now, let $i:Sh_j(\epsilon)\...
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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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Partial maps are representable in the category of presheaves on C

I know $Set^{C^{op}}$ is a topos and in toposes partial maps are representable but indeed I want to construct the partial map classifier $\tilde{F}$ for a presheaf F and a mono $\eta:F\longrightarrow \...
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Syntactic category of a geometric theory has finite limits.

Let $T$ be a geometric theory. Consider the syntactic category $C_T$. I want to show that $C_T$ has all finite limits. To show this, it is enough to show that it has finite products and equalizers. ...
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Do “non-full subtoposes” exist?

For the purpose of the following question, a "subtopos" of a topos $\mathcal{E}$ means a (not necessarily full) replete subcategory of $\mathcal{E}$ that is itself a topos and for which the inclusion ...
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Classifying map of a graph of a morphism

Let $\epsilon$ be a topos. Let $f:Y\longrightarrow X$ be a morphism in $\epsilon$. By the graph of f we mean the mono $<id_Y,f>:Y\longrightarrow Y\times X$. Let $\Delta$ be the classifying map ...
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Motivating results about the category of (simple) graphs?

My background is in group theory and algebraic geometry. I am due to give an introduction to the topos of graphs to some graph theorists. I understand the material, but what I am missing is some sort ...
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Slice categories and dependent types

How are slice categories related to dependent types ? Here is a quote from Mclarty's book Elementary Categories, Elementary Toposes that made me wonder about this: high-level programming languages ...
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Generalizations of topos theory. [closed]

Is there a generalisation for topos theory? If so; then can you define that particular generalisation in brevity?
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Showing some object is initial

This is a problem from logic. I'll first phrase it in general categorical terms since it is rather obscure, and then with context. Let $\mathcal{E}$ be a regular category where all epi's are regular. ...
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Computing the truth values in $\mathbf{Set}^2$.

This is part of Exercise 4.5.2 of Goldblatt's, "Topoi: A Categorial Analysis of Logic". Context: Here is an old question of mine on the preceding exercise: Verifying a Construction Satisfies the $\...
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Equalizer of character arrows in a topos

I have been working through Goldblatt's Topoi and in Section 7.5 he proves the following result: If $f,g$ and $h$ are monic with codomain $d$ in a topos, then $f\cap h\simeq g\cap h$ if and only if $\...
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How to show that $a^1 \simeq a$ for all $C$-objects $a$ in a cartesian closed category? (only one direction of isomorphism proof is needed)

I have already proven that $a \times 1 \simeq 1 \times a \simeq a$ given a terminal object $1$ of a cartesian closed category $C$ (CCC). By CCC I mean that $C$ is finitely complete and has ...
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Universal closure operation examples

A universal closure operation on a topos $\mathcal{E}$ is a family of functions $c_{(-)}: Sub(-) \to Sub(-)$ such that for $A$ a subobject of $X$ we have $A \leq c_X(A)$ $c_X(c_X(A)) = c_X(A)$ For ...
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In a topos, an internally functional relation induces a partial morphism

As a simple exercise, I wanted to write up a proof that in any topos $\mathcal{E}$, given a subobject $r:R\rightarrowtail X\times Y$ in $\mathcal{E}$, if $$(\forall x:X)(\forall y,z:Y)([R(x,y)\wedge R(...
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Category of sheaves generated by a subcategory?

Let $C$ be a category and $D$ a full subcategory of $[C^{op}, \mathrm{Set}]$. Can we form something like Grothendieck topos generated by $D$? By that I mean some full subcategory $\mathcal{E}$ of $[C^{...
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Examples of co-implication (a.k.a co-exponential)

In Dual Intuitionistic Logic and a Variety of Negations: The Logic of Scientific Research Yaroslav Shramko, inspired by Popper, makes an interesting case that co-constructive logic as the logic of ...
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Interesting Lawvere-Tierney topologies on Set

When doing topos theory I like working out hard theorems for Set first, and then translating back to general topoi. For stuff like subobjects and regular epi-mono factorization this works great, but I ...
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Geometric morphisms and presheaves

I would like concrete descriptions of the adjoint functors that arise in a geometric morphism induced by a functor between the base categories of categories of presheaves. Suppose we have functor $F$...
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Pulling back along a closed inclusion induces a bijection on dense subobjects.

Let $\mathcal{E}$ be a topos, $X$ and $Y$ objects and $i$ a closed inclusion $Y \to X$. Let $c_{(-)}$ be a universal closure operation. The function $i^*: Sub(X) \to Sub(Y)$ is a well defined ...
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Equivalence between Lawvere-Tierney topologies and universal closure operations in general topoi

Please don't be discouraged by the long post, most of it is definitions and stuff I have been able to prove, and not relevant to my questions (I typed it out for sake of completeness). I am stuck at ...
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Mac Lane & Moerdijk's Exercise II.7.

This is Exercise II.7 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to the first few pages of this Approach0 question, it is new to MSE. The Details: From p. 17 ibid....
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The legitimacy of topos theory and intuitionism.

This is an exercise in critical thinking. I am not looking, therefore, for opinions on the matter; rather: I would like to know the evidence (whatever that might mean). Background: I have a ...
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What is the best reference for real number objects?

Are there textbooks that, assuming a decent grasp of category theory and topos theory in particular, start with a natural numbers object and build the integers, the rationals and the real numbers (...
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Does $\pi_0$ preserve fibered products and effective epimorphisms?

There is an adjunction $$ (\pi_0 \dashv disc): Spaces \rightarrow Set $$ where $\pi_0$ sends a space to its path components, and $disc$ sends a set to the space with discrete topology. (i) Does $...
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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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Exponents in a slice category of presheaves

Let $\mathcal{C}$ be a category and denote $\hat{\mathcal{C}}$ for the category of presheaves on $\mathcal{C}$. For $k: K \to F$, denote $k^*$ for the pullback functor $\hat{\mathcal{C}}/F \to \hat{\...
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Equivalence of toposes $\mathcal{E}/(X + Y) \simeq \mathcal{E}/X \times \mathcal{E}/Y$

Notation: I write $A + B$ for the coproduct. Nota bene: I have no idea if the pseudo-inverses I cooked up are the right ones, but I think those are the only ones you can define. For $\mathcal{E}$ a ...
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Definition of a compact topos

This is in continuation of the previous post. I know what a compact topos: $F$ in $Set$ is: $F$ is compact if the geometric morphism $\gamma: F \rightarrow Set$ preserves directed suprema of ...
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Show that $\Gamma$, $\Lambda$, and the associated sheaf functor are all left exact.

This is Exercise II.6 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to the first few pages of this Approach0 search, it is new to MSE. The Details: The functors $\...
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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 ...
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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 $B \xleftarrow{f} A \xrightarrow{m} X$...
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Just what is Mac Lane & Moerdijk's $\Lambda$ from $\S II.5$?

This is a question concerning Exercise II.6 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, it is new to MSE. The Details: I'm not going to relay all of $...
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Prove that $F$ is locally constant iff the associated étale space over $X$ is a covering. [duplicate]

This is Exercise II.5 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, it is new to MSE. The Details: On p. 66, ibid. . . . Definition 1: A sheaf of ...
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For basis $\mathcal{B}$ of the topology on $X$, the restriction functor $\mathbf{r}:{\rm Sh}(X)\to{\rm Sh}(\mathcal{B})$ is an equivalence.

This is Exercise II.4 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, it is new to MSE. The Details: Adapted from p. 13, ibid. . . . Definition 1: A ...
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Prove that if $G$ acts properly on $X$, the quotient map $\pi: X\to X/G$ to the space $X/G$ of orbits is a covering.

This is Exercise II.3 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, it is new to MSE. The Details: From Mac Lane and Moerdijk, p. 37 Definition 1: ...
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A presheaf $P$ on $X$ is a sheaf iff for every covering sieve $S$ on an open set $U$ of $X$ one has $PU=\varprojlim_{V\in S}PV.$

This is Exercise II.2 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". The Details: Adapted from Adámek et al.'s, "Abstract and Concrete Categories: The Joy of Cats", p. 48 . . ....
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A sieve $S$ on $U$ in the category $\mathcal{O}(X)$ is principal iff the corresponding subfunctor $S\subset 1_U\cong{\rm Hom}(-,U)$ is a sheaf.

This is Exercise II.1 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, this is new to MSE. The Details: On p. 36, ibid. . . . Definition 0: For an ...
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The gist of Section II.3: “Sheaves and Manifolds” in Mac Lane and Moerdijk.

I'm reading Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]" for fun. It says readers may skip Section II.3 on Sheaves and Manifolds. Since I have very little experience with ...
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Proving Proposition I.5.1 of Mac Lane and Moerdijk.

This is Exercise I.11 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". The Details: Adapted from p. 25, ibid. . . . Definition: Let $\mathbf{C}$ be a category. Then $\hat{\...

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