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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Characteristic map in presheaf category

Let $\mathbf P:=[{\mathbf{C}}^{\operatorname{op}},\mathbf{Set}]$ be the category of presheaves on $\mathbf C$. Let $*,\Omega:\mathbf C^{\operatorname{op}}\to \mathbf{Set}$ be, respectively, the ...
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Constructive Proof that Intuitionistic Higher-Order Logic is Conservative over Constructive First-Order Logic

In the classical setting, we know that higher-order logic is conservative over first-order logic. More precisely, consider a classical first-order many-sorted theory $T$, and consider some sentence $\...
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Is it possible to define this object using Kripke-Joyal Semantics

I've been looking into Kripke-Joyal semantics and, for each formula $\phi(x)$ defining an object $\{ x \, | \, \phi(x) \}$. Is it possible to, somehow, "fix a variable"? Let's say I've got a ...
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Topos-theoretic proof of the consistency of CH

I hope this is not a duplicate, if so please let me know. So in MacLane Moerdijk‘s book Sheaves in Geometry and Logic it is shown that there is a boolean, two-valued topos with nno and choice, in ...
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Motivating Grothendieck Toposes generated by Souslin Objects

I am currently reading about the independence of the continuum hypothesis from ZFC following the topos theoretic proof given in Chapter VI.3 of MacLane Moerdijk's Sheaves in Geometry and Logic. The ...
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Schröder–Bernstein theorem in category.

In set theory, Schröder–Bernstein theorem assert for every set $A$ and $B$ if there exists injections from $A$ into $B$, from $B$ into $A$. Then there exists a bijection from $A$ onto $B$. I want to ...
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How to phrase the ability to cut up a space into pieces and glue them together again in a site?

For a topological space $X$, you can cut up $X$ into opens via an open cover $U_i : i\in I$ and take the "gluing data" of inclusions $f_{ij}:U_i\cap U_j\rightarrow U_i$, and this data is ...
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A Reference From Andrej Bauer's Recent Talk on Countable Reals

Andrej Bauer gave a talk today in the topos institute colloquium (video here) announcing a proof that the dedekind reals can be countable in the absence of LEM and CC. At roughly the 27 minute mark, ...
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How to generate Grothendieck Topologies from families of sieves

If $\mathcal{C}$ is a small category with pullbacks, a basis for a Grothendieck topology on $\mathcal{C}$ is a function which assigns to each object $C$ a collection $K(C)$ of families of morphisms ...
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Join in Lattice of Subobjects

In an elementary topos the join $A \vee B$ of two subjects $A \to X$ and $B \to X$ is defined to be the image of the induced morphism $A \sqcup B \to X$. For sets it holds, that this is the same as ...
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Externalizing Statements about the Effective Topos (or: what is it good for?)

Say we prove some theorem constructively (by which I mean, without choice and without LEM). Then it's true inside an arbitrary topos, and in the case of sheaf topoi we can externalize our theorem into ...
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Does the Yoneda extension of a "representably flat" functor preserve finite limits?

Let me first recap the definitions of flatness, using the nLab's terminology: A functor into $F : \mathcal{C} \to \mathbf{Sets}$ is Set-valued flat when the category of elements $\int E$ is filtered. ...
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Universal relation in an elementary topos

Let $\bf E$ be an elementary topos, and let $t:1\to\Omega$ be its subobject classifier. Then, for every $x\in \mathrm{ob} (\mathbf E)$, there is a monic $\tau_x:u\to x\times \Omega^x$, universal in ...
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Essentially small Grothendieck toposes

The terminal category is a Grothendieck topos which is essentially small. Are there other examples of Grothendieck toposes which are essentially small? Are there other examples of Grothendieck toposes ...
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Functor from topological spaces / schemes to topoi faithful?

Is the functor from the category of topological spaces to the 2-category of Grothendieck topoi (and geometric morphisms, of course), which sends a space to its topos of sheaves, faithful? (I already ...
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How can I make sense of global subset relation in a topos

Let $\mathcal{I}$ be the class of all functions $\iota: S \rightarrow X$ inserting a subset $S \subseteq X$ into a larger set. The presence of the family $\mathcal{I}$ underlies the subset relation $\...
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Finite étale topos = étale topos of a field

This implies/suggests that the finite étale topos $\mathrm{Sh}(\mathrm{Spec}(k)_{\mathrm{f, ét}})$ is equivalent to $G\mathbf{Set}$ for $G$ the absolute Galois group of $k$. The quote here, on the ...
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What is $\mathrm{Sh}(\mathrm{CRing}^{op})$ a classifying topos of?

$\mathrm{Sh}(\mathrm{CRing}_{fp}^{op})$(here $\mathrm{CRing}_{fp}^{op}$ is the opposite category of the category of finitely presented rings) will be the classifying topos of the theory of local rings....
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Name of functor from "Propositions"/"Properties" to "Power set" implied (in part?) by axiom schema of specification?

$\newcommand{\Inj}{\operatorname{Inj}}$$\newcommand{\Prop}{\operatorname{Prop}}$Consider the "category of propositions" or "of properties" $\Prop$, where objects are logical ...
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How $\Omega _X$ is related to $\Omega$ as subobject classifier - trying to understand toposes of deep neural networks

My aim is to understand how the notion of su-bobject classifier is used in the article Topos and Stacks of Deep Neural Networks about the categorical formalization of deep neural networks. This ...
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Externalizing A Concrete Application of Double Negation Toposes

I'm trying to come up with concrete problems which can be solved via topos theory, and I've found a good case study which has been really instructive. I've spent the past few weeks trying to ...
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Why "cocomplete topos"?

This is a quote from Mac Lane and Moerdijk's Sheaves in Geometry and Logic: A classifying topos for $T$-models is then a topos $\mathcal B(T)$ over $\mathbf{Sets}$ with the property that for every ...
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(Co)Homology of topoi

In Moerdijk's Classifying Spaces and Classifying Topoi, he defines the cohomology groups $H^n(\mathcal E, A)$ for each Grothendieck topos $\mathcal E$ and each abelian group object $A$ in $\mathcal E$ ...
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cocomplete implies geometric morphism

How cocompleteness of a topos $\cal E$ implies that $\cal E$ is equipped with a geometric morphism $\gamma:{\cal E}\to {\text {Set}}$ ?
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Proving that $\text{Set}^{\mathcal{C}}$ is cartesian closed

Let $\mathcal{C}$ be a small category. I am trying to prove that $\text{Set}^\mathcal{C}$ is cartesian closed. Given functors $F, G: \mathcal{C} \rightarrow \text{Set}$, define their product to be the ...
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the direction of a point of a topos

I would like to understand what is the direction of a point of a topos. A point of a topos $\cal T$ in one source (2nd snippet below) is a functor preserving finite limits and all colimits $$p^*:{\cal ...
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Does the (pseudo)functor that assigns a commutative monoid $M$ to the topos of $M\text{-Sets}$ preserve limits? [closed]

Let $\mathrm{CMon}$ be the category of commutative monoids, and $\mathrm{Topos}$ be the bicategory of (Grothendieck) toposes with geometric morphisms. Consider the (pseudo)functor $\mathrm{CMon}\to\...
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Reference request: Galois categories

In SGA1 Grothendieck introduced the notion of a Galois category to study/define/characterise the category of finite etale morphisms over a scheme and the associated etale fundamental group of a scheme....
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Models of pretoposes vs. models of coherent categories

I'm confused about these lecture notes. Theorem 4 states, roughly, that for each coherent category $C$ there is a pretopos $C^\mathrm{eq}$ such that for any pretopos $D$, morphisms of coherent ...
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How do you derive the adjoint's "naturality" condition as seen in MacLane & Moerdijk's book "Sheaves in Geometry and Logic"?

It's tag (7) as pictured below. I also included the definition of adjoint that they use. I know that by definition of adjunction (using the natural homset isomorphism), we have two naturality ...
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2 answers
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What is the $\textbf{Set}$-theoretical intuition behind understanding how power objects work in Topos Theory?

Definition. Let $\mathcal{E}$ be an elementary topos, $B \in \text{Ob}(\mathcal{E})$. Then a power object for $B$ is an object $PB \in \text{Ob}(\mathcal{E})$ together with a morphism $B \times PB \...
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4 votes
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2-morphisms between geometric morphisms

Let $\mathcal E$ and $\mathcal F$ be two toposes. A geometric morphism $f\colon \mathcal E\to\mathcal F$ consists of an adjunction $f^\ast\dashv f_\ast$. Let $f,g\colon \mathcal E\to\mathcal F$ be two ...
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7 votes
1 answer
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Characterization of equality of terms in the internal logic of toposes

This is a nice note explaining the internal logic of a topos. I'll use the notation and terminology defined in that article. Let $\Sigma$ be a higher-order signature and $M$ an interpretation of $\...
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6 votes
1 answer
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Exponential of monos over terminal object

Consider a cartesian closed category $\mathcal E$. Let $1$ be the terminal object and $U, V\hookrightarrow 1$ to monomorphisms into $1$. I wanted to prove that $U^V\rightarrow 1$ is also a ...
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Why are toposes usually denoted by $\mathcal E$?

Why are toposes usually denoted by $\mathcal E$? My guess would be that $\mathcal E$ stands for elementary topos. But I often see people denoting even Grothendieck toposes by $\mathcal E$ (in contexts ...
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7 votes
1 answer
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Which presheaf toposes satisfy the axiom of choice?

Which toposes of presheaves $\mathbf{Set}^{C^\mathrm{op}}$ satisfy the axiom of choice (every epimorphism splits)? Can one formulate a condition on $C$ that yields a necessary or sufficient condition ...
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What does finding a "free local ring" have to do with finding the spectrum of a ring?

In Tierney's 1976 paper On the Spectrum of a Ringed Topos (which you can find here) at the top of section 2 we read Let $A$ be a commutative ring in [a topos] $\mathbf{E}$. We look at the problem of ...
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2 votes
2 answers
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Internal logic characterization of closed subobjects for a Lawvere-Tierney topology [closed]

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 ...
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Is there a topos-theoretic notion of "dimension"?

It seems like almost any "topological" phenomenon has a generalization to toposes. For instance, in Site Characterizations for Geometeric Properties of Toposes, Olivia Caramello shows how we ...
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2 votes
1 answer
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Does replacing "strong" with "regular" in the definition of "quasitopos" give any new examples?

A quasitopos is a finitely cocomplete locally cartesian closed category with a terminal object (hence finite limits) and a classifier for strong subobjects. A monomorphism $m:A \to B$ is called strong ...
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1 answer
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All elementary toposes have finite colimits

An elementary topos is a category with finite limits, exponential objects, and a subobject classifier. Here a quote from Leinster's An informal introduction to topos theory: More spectacularly, the ...
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3 votes
1 answer
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Defining the notion of embedding of Grothendieck toposes as a quotient in the opposite category

The category of locales is defined to be the opposite of the category of frames. That's why the notion of a sublocale is defined to be a quotient in the category of frames (subobjects are dual to ...
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5 votes
1 answer
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Satisfaction of Peano postulates in topos with natural numbers object

Let $\mathcal{T}$ a topos with a natural numbers object, noted $N$. Also assume that $\mathcal{T}$ is not degenerate, meaning that its initial and terminal objects $\emptyset$ and $*$ are not ...
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*Sheaves in Geometry and Logic* Chapter V Section 7 Theorem 2: $U$ has right adjoint, or $T$ has right adjoint?

In Mac Lane and Moerdijk's Sheaves in Geometry and Logic, the following theorem is stated in Chapter V Section 7: Theorem 2: For a category object $C$ in a category $\mathcal{E}$ with pullbacks, the ...
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4 votes
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Proof that a category is cartesian closed

Suppose that a small category $\mathbf C$ has all the finite products. Then the category $\mathbf{Set}^{\mathbf{C}^{\mathrm {op}}}$ is cartesian closed. Fix a $P\in \mathbf{C}^{\mathrm {op}}$. Then we ...
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$F_2\circ F_1\dashv G_1\circ G_2$ and $F_2\dashv G_2$ but not $F_1\dashv G_1$

What are examples of pairs of adjoint functors the form $F_2\circ F_1\dashv G_1\circ G_2$ (where $F_1$ and $G_1$ are functors between, say, $C$ and $D$, and $F_2$ and $G_2$ are functors between $D$ ...
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3 votes
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Error in the proof of $\textit{Sheaves in Geometry and Logic}$ Lemma VI.3.8 - Cohen Poset satisfies Countable Chain Condition

Context Chapter VI sections 1-3 in Sheaves deals with proving that there is a certain Grothendieck topos $Sh_{\neg \neg}(\mathbb{P})$ in which the continuum hypothesis is false in the internal logic. ...
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Internal logic of topoi

There is a distinction often made between the internal and external logic of topoi, for example in Goldblatt's book Topoi, or in the nLab. I have the impression that I understand the external logic of ...
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1 answer
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Characteristic Function for Subobject Classifier in the Topos of Presheaves

Let $\mathbb{C}$ be a locally small category with pullbacks and $\hat{\mathbb{C}} = \mathrm{Set}^{\mathbb{C}^{\mathrm{op}}} = [\mathbb{C}^{\mathrm{op}}, \mathrm{Set}]$ the category of presheaves over $...
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Topoi as models of constructive mathematics

In this question, it is argued that constructive mathematics cannot prove the existence of a discontinuous real function, because there is a topos $\mathcal{E}$ where all real functions are continuous....
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