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### Atlas of a manifold as a Sheaf

--Hopefully this question does not dublicate another--
In this question Tom Goodwillie pointed out, that the 'atlas part' of
the definition of a smooth manifold can be redefined in terms of
sheaves. ...

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### right adjoint functor for closed immersion of topoi

Let $i\colon (X,A)\rightarrow (Y,B)$ be a closed immersion of ringed topoi. Does functor $i_*\colon Mod(A)\rightarrow Mod(B)$ have a right adjoint?

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### Do non-subcanonical Grothendieck topologies always induce a category of fractions?

Suppose that $\mathscr{C}$ is a (possibly higher) category and $J$ is a Grothendieck topology which is not subcanonical. Denote the composite
$$\mathscr{C} \hookrightarrow ...

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### Sheaves, colimits and closure

I am considering the sheaf topos $\mathbf{Sh}(\mathfrak{X})$ on a topological space $\mathfrak{X}$.
Limits in this category are constructed pointwise, as in presheaves. Colimits, however, are not ...

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112 views

### When does prolongation preserve sheaves?

Suppose that $(C,J)$ and $(D,K)$ are two Grothendieck (possibly $\infty$-)sites and $f:C \to D$ is a functor such that $$f^*:Psh(D) \to Psh(C)$$ sends sheaves to sheaves. Under what conditions will ...

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### Non-degenerate limits of topoi

Let $\mathcal{E}$ be an elementary topos with natural numbers object $\mathbb{N}$, and let $\mathbb{C}$ be a cofiltered internal category in $\mathcal{E}$. Suppose we have a diagram of shape ...

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329 views

### What information is lost in $X \to \mathrm{Sh}(X)$?

Given a topological space or site $X$. Construct $\mathrm{Sh}(X)$ - the sheaves on $X$ with values in $\mathrm{Set}$. Is it known what information is lost in this procedure?
Thanks, Adrian.

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462 views

### Modern versions of Verdier's hypercovering theorem?

Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a ...

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340 views

### Is it possible for a theorem to be constructive only in a non-constructive metatheory?

There are several theorems in category-theoretic logic which say something like, "any proposition in X logic that is provable in topos logic assuming (the law of excluded middle and) the axiom of ...

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### Validity of equations in a topos bis

I clarify my previous question with a simple example. It is also a simpler (and more general) question whose resolution resolves the previous question on interpreting a theory in another.
The ...

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125 views

### Validity of equations in a topos

To simplify consider simple algebraic theories (universal algebra)
A and L, but the question applies to geometric theories.
1) Syntactically, we can interprete L in A if we can define the operations ...

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88 views

### Induced adjunctions

Suppose $F: C \rightarrow D$ is the left adjoint to a functor $G$. Then is it true that the functor $F^{\star}:[C : Sets]$ defined by prescomposing a functor $P: C \rightarrow Sets$ is still left ...

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469 views

### Internal logic of the topos of simplicial sets

I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, ...

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106 views

### Do coherent toposes descend along open surjection?

Let $f:\mathcal{T} \rightarrow \mathcal{S}$ be a geometric morphism between two toposes. Let $g:\mathcal{L}\rightarrow \mathcal{S}$ be an open surjection (of toposes) and assume that the map ...

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144 views

### Is the collage of two spatial toposes a spatial topos?

Consider the collage operation along a profunctor, defined between two categories ${\bf C}, \bf D$.
Suppose now that the two categories are toposes, say $Sh(X), Sh(Y)$ for two topological spaces ...

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129 views

### Cloven Kan fibrations

For each groupoid $G$ there is a monad such that the cloven fibred groupoids over $G$ are algebras for this monad. I am looking for an infinite dimensional counterpart: a monad on simplicial sets ...

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389 views

### classifying $\infty$-toposes for topological/localic groups?

Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?
More precisely, is there an $\infty$-topos $BG$ ...

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121 views

### Dedekind reals in heyting valued models

Let $V^{H}$ be a Heyting valued model of intuitionistic set theory. What conditions does $H$ have to satisfy in order for the following claim to hold? (where $\| \phi(u) \| \in H$ is the truth value ...

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### Analogy between topology and algebraic geometry

In topos theory, there are many generalizations of topological concepts. For example, open, closed, proper and etale morphisms between toposes. However, there are also such analogous concepts in ...

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685 views

### Examples of toposes for analysts

I've read that toposes are extremely important in modern mathematics, but I find the definitions and examples given on the nLab page a little too abstract to understand.
Can you provide some examples ...

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### Is there a measure / probability theory in a topos of “generalized measure spaces”?

Consider the category with the standard Lebesgue measure space $\Omega$ as its only object and measure type preserving nonincreasing (equivalence classes of) maps as morphisms.
Question: is there an ...

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103 views

### Is there a characterization of the topos of finite sets in the internal language?

The topos ${\mathcal{Set}}$, at least as axiomatized in ETCS, is a well-pointed topos that satisfies the axiom of choice and has a natural numbers object.
Is there a characterization of the topos ...

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271 views

### What properties do “large topoi” share with actual topoi?

Fix Grothendieck universes $\mathcal{U} \in \mathcal{V}$ and suppose that $C$ is a locally $\mathcal{U}$-small category which is $\mathcal{V}$-small. Denote by $Set$ the category of ...

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### Properties of the internal language of the category of sheaves

Consider a simple case of set-valued sheaves on some topological space $X$, $\operatorname{Sh}(X)$. All of these are Grothendieck toposes but clearly not all of them are equivalent.
Is there an ...

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### Is Logic/Set Theory necessary for studying Topos Theory?

I have just completed a postgraduate course, in which I studied Category Theory, without having a background in Set Theory and Logic - this probably already sounds absurd to many. This did not seem to ...

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### When is the localic reflection of a topos discrete?

Recall that the inclusion of locales into topoi has a left adjoint, called the localic reflection. It sends a topos $\mathcal{E}$ to the poset of subobjects of the terminal object, which is a locale. ...

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211 views

### Are topoi and etale geometric morphisms locally small?

This question is related to this one: Local smallness and (higher) topoi which has not yet been answered.
The $2$-category of topoi and geometric morphisms is not locally small. (As I mentioned in ...

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### Representing a small allegory in a tabular allegory?

Let $A$ be a small allegory (like in Freyd and Scedrov book, or in the Elephant of Johnstone), does it always exists a tabular allegory $B$ and a fully faithfull representation of $A$ in $B$ ?
I am ...

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### Which dense inclusions of sites are ∞-dense?

An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D.
Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ...

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357 views

### Does the internal axiom of choice imply Lagrange's theorem?

In a topos ${\mathcal{A}}$, given a group object $G$ and a subgroup $H$, the object $H\backslash G$ of right cosets is the coequalizer of two maps $G\times H\rightrightarrows G$, namely the group ...

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### Configuration topos?

Let ${\bf Fin}$ denote the category of finite sets. If $X$ is a topological space, then for any natural number $k\in{\mathbb N}$, the slice category ${\bf Fin}/X$ contains the configuration space ...

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### Local smallness and (higher) topoi

The $2$-category of topoi and geometric morphisms is not locally small. For example, if $A$ is the classifying topos for abelian groups, the category of geometric morphisms from $Set$ to $A$ is ...

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### Interaction petit topos - gros topos

I know that there are already several discussions on this topic (and they already helped me a lot, so far), but I couldn't find one completely solving my problem.
Question 1. Fix a scheme $X$. I know ...

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### Ex/reg toposes without generic monomorphisms

A generic monomorphism is a monomorphism of which every monomorphisms in the same category is a pullback; it is like a subobject classifier without uniqueness. I am looking for a locally Cartesian ...

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### Presheaves and Heyting Valued Models

I'm doing some reading on the relationship between the topos of pre-sheaves over a poset P, the topos of sheaves over the Heyting algebra H of sieves on P, and the Heyting valued model of ...

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### Generalizing prime numbers to product-indecomposable objects in toposes

Any object $A$ in any topos decomposes as $A\times1$ and $1\times A$, and in $FinSet$ objects with no other product decompositions are "prime numbers". Is there any extension of the theory of ...

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### is this a simplicial model category?

A basic result in the theory of model categories is that simplicial sets form a simplicial model category. The same is true for simplicial $k$-algebras. I have two questions related to this. One ...

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### monics in topoi

In the question Do pushouts preserve monic? it is said that monics in a topos are stable under push out. I would like a precise reference or a nice proof of this fact for elementary topoi (for ...

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### Barr's theorem and constructivity?

Barr's covering theorem assert that any Gorthendieck topos can be covered by a Grothendieck topos (even a locale) satisfying the axiom of choice (and hence also the law of excluded middle). Its ...

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### A topos-theoretic thickening of the nerve functor

Let $\Delta$ be the standard cosimplicial space sending $[n]\mapsto \Delta^n\subset \mathbb R^{n+1}$.
Then the correspondence $\delta\colon [n]\mapsto {\rm Sh}(\Delta^n)$ defines a cosimplicial ...

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### How strong is “all sets are Lebesgue Measurable” in weaker contexts than ZF?

Famously, Solovay showed that, if $\textrm{ZFC}$ plus $\textrm{IC}$ (the existence of an inaccessible cardinal) is consistent, then so is $\textrm{ZF}$ plus $\textrm{DC}$ (dependent choice) plus ...

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### change of topologies and functoriality

Let X be a scheme and $\epsilon_X:X_{FL}\to X_{et}$ be the morphism of topoi from the big flat topos to the small etale topos. Let $f:X\to Y$ be a morphism of schemes. I denote $f_{top}$ the induced ...

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### Question on existence and atomicity of a geometric morphism

I am curious to know where we can find a geometric morphism from the Zariski topos to the étale topos and more specifically when this is atomic. I would like to know, actually, in which instances is ...

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### In the topos-theoretic interpretation of Physics by Isham & Doering what role does intuitionistic logic play? [closed]

I've asked this question on Physics.SE but was advised to ask it here.
Isham & Doering have written a series of papers exploring how to ground physics in topoi. Now the internal logic of topoi is ...

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### Relation between Galois theory and Etale Cohomology

I am a graduate student working on category theory. I am familiar with categorical Galois theory, in the way developed by Janelidze - as described for example in "Galois Theories". I am now trying to ...

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### Pullback-stability of internally projective objects

An object $X$ of a category $C$ is said to be projective if the hom-functor $C(X,-)$ preserves epimorphisms (or, in general, some restricted class of epimorphisms such as the regular or effective ...

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### Bohr topos and quantization

Bohrification is a natural way to construct a quantum "phase space" (with some nice insights on foundational problems like non-contextuality through Kochen-Specker etc). I was wondering, since we get ...

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### work in a topos as in sets: disjoint coproducts

Assume a topos $\mathcal{S}$ as the base topos, and we work in this topos as in naive set theory (without choice or excluded middle). Take a Grothendieck topos
$\mathcal{E} \to \mathcal{S}$ with a ...

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### Is there a translation invariant measure on an infinite dimensional space 'without points'?

This is just a reference request. I thought I'd come across a paper demonstrating that there is a translation invariant measure on an infinite-dimensional space without 'points' whilst browsing the ...

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### For which sites are all constant presheaves separated?

I'm intererested in open surjective geometric morphisms induced by fibrations of sites $S\to T$ a la Moerdijk, but as a warm-up, let's consider the case $S \to \ast$. In the case that $S$ is a poset ...