5
votes
1answer
104 views

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 ...
3
votes
0answers
106 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 ...
5
votes
0answers
82 views

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 ...
6
votes
1answer
321 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.
8
votes
1answer
329 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 ...
2
votes
1answer
86 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 ...
16
votes
2answers
435 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, ...
6
votes
0answers
103 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 ...
3
votes
0answers
142 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 ...
8
votes
1answer
575 views

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 ...
8
votes
2answers
269 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 ...
5
votes
1answer
399 views

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 ...
3
votes
0answers
66 views

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. ...
7
votes
1answer
208 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 ...
2
votes
0answers
59 views

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 ...
5
votes
0answers
97 views

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 ...
3
votes
1answer
342 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 ...
5
votes
4answers
433 views

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 ...
9
votes
1answer
332 views

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 ...
2
votes
0answers
59 views

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 ...
1
vote
1answer
129 views

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 ...
6
votes
0answers
227 views

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 ...
0
votes
0answers
191 views

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 ...
3
votes
2answers
108 views

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 ...
2
votes
0answers
202 views

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 ...
1
vote
0answers
111 views

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 ...
3
votes
0answers
245 views

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 ...
5
votes
1answer
372 views

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 ...
10
votes
1answer
208 views

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 ...
1
vote
2answers
171 views

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 ...
3
votes
1answer
120 views

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 ...
1
vote
0answers
94 views

Non spatial atomic topos

Hello ! If I'm not mistaken, an atomic topos decompose as a disjoint sum of connected atomic topos, and Connected Atomic topos with a point corresponds to classifying topos of localic groups. But ...
0
votes
1answer
159 views

Cocontinuous functor out of the terminal category

Let $\mathcal{C}$ be a small finitely complete category equipped with a Grothendieck (pre)topology $\tau$. For $\ast$ the terminal category (one object, one morphism), denote by $i: \ast \to ...
16
votes
1answer
2k views

Is Lemma A.1.5.7 in Higher Topos Theory correct?

Hello to everyone, I am studying the properties of combinatorial model categories, following the exposition given by Jacob Lurie in Higher Topos Theory ([HTT] from now on), in section A.2.6. At some ...
8
votes
3answers
922 views

Is there a categorical proof of Gödel's incompleteness theorem?

A significant result in set theory was shown by Cohen when he showed that the continuum hypothesis was independent of ZFC using a new technique called forcing. In Topos theory, this result has a new ...
4
votes
1answer
332 views

Is there a nice characterisation of topoi with nice meta-logical properties?

First-order order classical logic with standard semantics has a proof theory: it is complete, sound and effective. In higher order logic with standard semantics one cannot obtain a proof theory - ...
1
vote
2answers
181 views

Definition of subobject classifier in presheaves

I am reading Awodey (Category Theory, 1st edition), p 175, and I have difficulties to understand the paragraph about the subobject classifier of $\mathbf{Sets}^{\mathbf{C^{op}}}$. First let me quote ...
5
votes
2answers
579 views

Can Inequivalent Topologies Have Same Sheaves/Cohomology?

Let $C$ be a fixed category, and let $T_1$ and $T_2$ be two Grothendieck (pre)topologies on $C$. We say $T_1$ is subordinate to $T_2$ if every covering in $T_1$ has a refinement in $T_2$. We say $T_1$ ...
4
votes
1answer
288 views

What is a higher derived constructible sheaf

Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define the category of "$\infty$-local systems on $X$" to be DG representations of the ring $C_*(\Omega X,k)$ of ...
3
votes
0answers
80 views

How does the machinery of left-exact comonads generalize from sheaves to stacks?

Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This ...
3
votes
5answers
466 views

Does this kind of endofunctor ever have an initial algebra?

Let $C$ be a topos with subobject classifier $\Omega$. Let $F$ be the endofunctor $x \mapsto \Omega^{\Omega^x}$ on $C$. Does there exist $C$ such that $F$ has an initial algebra? What if $\Omega$ is ...
9
votes
1answer
719 views

Are there non-categorical notions in topos theory?

Suppose that $\mathcal{T}$ is an abstract $2$-category we know is equivalent to the $2$-category of Grothendieck topoi via some equivalence $$\phi:\mathcal{T} \to \mathfrak{Top},$$ and let $E$ be an ...
2
votes
1answer
110 views

questions of localization of topos

Let $T$ be a topos, and $F \in T$, $T/F$ a localization of $T$. So we have a natural morphism $i: T/F \longrightarrow T$. My questions are: 1.What are the definitions of $i_{\ast}$ and $i^{\ast}$ ...
4
votes
0answers
133 views

Extracting internal sites of definition

Given sites $(C,J)$ and $(D,K)$, and a functor $f\colon C\to D$ satisfying the covering lifting property: For every object $c$ of $C$ and $K$-covering sieve $S$ of $f(c)$, there is a $J$-covering ...
18
votes
2answers
786 views

What can be expressed in and proved with the internal logic of a topos?

The title of this post expresses what I really want, which is to learn how to wield the internal logic of a topos more effectively. However, to bring it down to earth, I'll ask a few basic questions ...
5
votes
2answers
259 views

When does the direct image functor nicely push past the power/exists functor?

Let $D$ and $E$ be toposes and let $f_{\ast}\colon D\to E$ be the direct image part of a geometric morphism $(f^{\ast},f_{\ast})$ between them. Considered as categories, we have (covariant) ...
9
votes
4answers
711 views

Connections between topos theory and topology

What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the ...
6
votes
0answers
176 views

High browed proof that a topos is sheaves over itself via the adjoint functor theorem?

Suppose that $\mathcal{E}$ is a (Grothendieck) topos. It carries a (large) Grothendieck topology (in fact the canonical topology), a basis of which is determined by declaring a collection of morphisms ...
20
votes
4answers
1k views

What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?

In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...
9
votes
1answer
284 views

Boolean non-hypercomplete $(\infty,1)$-toposes

Let's say that an $(\infty,1)$-topos is Boolean if for every object $X$, the lattice $Sub(X)$ of subobjects (i.e. $(-1)$-truncated morphisms into $X$) is a Boolean algebra. I think this is equivalent ...