# Questions tagged [topos-theory]

A topos is a category that behaves very much like the category of sets and possesses a good notion of localization. Related to topos are: sheaves, presheaves, descent, stacks, localization,...

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### Examples of five-adjoint systems

I'm looking into Lawvere's formulation of unities of opposites and opposites of unities, and for this I would be interested in systems of five (or more) adjoint functors $X\stackrel{\stackrel{\...

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### Does a well-pointed topos with enough projectives satisfy the internal axiom of chioice?

If yes, then I am also wondering if being well-pointed can be weakened to boolean (i.e. this is in the context of using Set as our metalogic so that well pointed Topoi are boolean). If not, then any ...

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### Unifying the $n$-truncation factorization system in a topos with the $n$-truncation factorization system of a t-structure

(This is a corrected and more detailed version of an earlier question.)
In good circumstances, e.g. the setting of an $\infty$-topos, we have for each $n\geq -1$ an orthogonal factorization system ...

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### When is the category of connective objects a topos?

There is only one stable $\infty$-topos, namely the trivial category. However, the theory of stable $\infty$-categories with $t$-structure is strikingly reminiscent of the theory of topoi, as both ...

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### Is $\mathit{Topos}^\text{op} \to \mathit{Pr}^L$ monadic?

$\newcommand\Logos{\mathit{Logos}}\newcommand\Topos{\mathit{Topos}}\newcommand\op{^\text{op}}\newcommand\Pr{\mathit{Pr}}$Let $\Logos = \Topos\op$ be the $\infty$-category of $\infty$-topoi and ...

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### Non-trivial automorphisms and descent

In this expository paper by Low it says:
Roughly
speaking, a topos in the sense of Grothendieck is the category of sheaves on a
kind of generalised space whose “points” may have non-trivial ...

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### Is there something similar to Lawvere-Tierney topologies for Abelian categories?

Lawvere-Tierney topologies generalize the notion of local operators on a Topos from Sheaf toposes over a Grothendieck site to arbitrary Toposes.
However, while the special case of Sheaves of sets or ...

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### If we replace the spectrally ringed space in the definition of a spectral scheme with an arbitrary infinity-topos, what objects do we get?

I'll phrase this in terms of spectral AG, but I'm curious about the same question in the classical context.
We define a nonconnective spectral Deligne-Mumford stack to be a spectrally-ringed topos ...

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### Groupoid of points, shape and stratified shape of $\operatorname{Sh} (X_\text{pro-ét})$

$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid ...

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### Flasque sheaves on a site

This is a cross-post from MathStackexchange.
We define a flasque sheaf on a site as one whose first Čech cohomology vanishes for every covering of every object of the site. I know this definition is ...

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### Relating properties of geometric morphisms to the inverse image of the generic model

Let $p:X\to [\mathbb{T}]$ be a morphism of Grothendieck topoi such that $[\mathbb{T}]$ is the classifying topos for some geometric theory $\mathbb{T}$; let me write $U$ for the generic model of $\...

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### Deductive system involving only geometric sequents

A geometric theory is made up of sequents of restricted form: It may only be of the form $$\phi \vdash \psi$$
possibly with free variables (which are implicitly taken universal closure). $\phi, \psi$ ...

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### Subcategory of coherent objects in an $\infty$-topos forming a local $\infty$-pretopos

My question is about the proof of Proposition A.6.1.6 in Lurie's Spectral Algebraic Geometry, which says the following:
Let $\mathcal{X}$ be any $\infty$-topos and denote by $\mathcal{X}^{coh}$ the ...

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### For which "permutation groups" is the sign homomorphism well-defined constructively?

Let $X$ be a finite set. I now have a favorite construction of the sign homomorphism $Sym(X) \to C_2$. But perhaps it shouldn't be my favorite construction.
After discussion with the experts, I've ...

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### Topological groupoids and equivariant sheaves

Some statements that are true for ordinary groupoids fail for topological groupoids (by which I mean groupoids internal to the category of topological spaces): for instance, every ordinary groupoid is ...

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### Questions about coherent topology

For each coherent category $C$, let $J_C$ be the topology on $C$ in which a sieve $\{f_i\colon U_i\to X\}_{i\in I}$ is covering if and only if there exists a finite set $I_0\subseteq I$ such that $\...

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### Hypercovers with sieves

Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\...

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### Topos with enough points but not coherent

By Deligne's theorem, each coherent topos has enough points. What would be an example of a Grothendieck topos with enough points which is not coherent?

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### When is a basis of a topological space a Grothendieck pretopology?

Bases of a topological space in point set topology will in general form a coverage on its category of inclusion on open subsets and on its category of inclusion on basic opens, but it takes a bit more ...

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### Every Grothendieck topos can be built from localic topoi

Theorem 2 in these notes[1] states that, roughly, that each Grothendieck topos can be built (using limits and colimits) from localic topoi. To what extent is that related to the theorem of Joyal and ...

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### Interview of Connes, Caramello, and L. Lafforgue about topos theory

In a recent blog post, Lieven le Bruyn, discussing an interview with Connes, Caramello, and Lafforgue on France Culture, wrote
Towards the end of the programme Connes, Caramello and Laforgue [sic] ...

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### Classifying cohomology

In his 1973 topos seminar in Buffalo (the tapes are now freely available online!), Grothendieck said:
The cohomology of a topos associated to an algebraic structure should be called the "...

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### Equivalence between geometric theories and frames internal to the free topos

What is a reference for "the equivalence between geometric theories and frames internal to the free topos"? This seems to be an extremely interesting theorem.

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### Questions about SGA 4

What is Deligne's motivation in Appendice 9 of Exposé VI to prove that every coherent topos has enough points? For instance, does that have applications in étale cohomology (or other parts of ...

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### Lefschetz for topoi

The Lefschetz fixed-point theorem is a nice theorem in topology which counts the number of fixed points (counted with multiplicities) of a continuous map $f\colon X\to X$ using the traces of the ...

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### De Rham via topoi

Étale cohomology of schemes $X$ is constructed as follows: one associates to $X$ the so-called étale topos of $X$, and then one just takes the sheaf cohomology of that topos.
Is it possible to ...

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### Lurie's applications of $\infty$-topoi in topology

Lurie's book Higher Topos Theory is extremely interesting, but pretty overwhelming. I don't have the time to read it at the moment. However, the last chapter (7) gives applications of $\infty$-topoi ...

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### Topos-theoretic Galois theory

This page is an overview of some of the types of "Galois theories" there are. One of the most basic type is the fundamental theorem of covering spaces, which says, roughly, that for each ...

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### Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?

To give an example of a peculiar feature of simplicial sets that I cannot remember encountering anywhere in the context of homotopy theory: every simplicial set $X$ possesses partial map classifier $X\...

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### nPOV: Cohomology and derived functors

In the nPOV, cohomology is realized as the connected component of the derived hom space [1]. Namely,
$$H^n(X,Y) = \pi_0 \mathbb{H}(X,B^nY),$$
where $\mathbb{H}$ is an $(\infty,1)$-topos and $B$ is the ...

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### Do pretopoi have cohomology and homotopy groups?

Grothendieck topoi have cohomology: the abelian category of abelian group objects in a topos has enough injectives, hence one can consider the right derived functors of the global sections functor ...

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### Is every topos a sheaf topos with values in a well-pointed one?

Here's a mix of heuristic and precise questions as I try to grapple with topos theory.
I try to think of topoi as two notions of "$1$" being glued at the hip. One is the "building block&...

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### What's the point of a point-free locale?

In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - Why can't we identify those as the
trivial ...

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### Compactness theorem and topos theory

The theory of classifying topoi due to Makkai, Reyes, Hakim, and Grothendieck supplies a bijection between geometric theories (up to Morita equivalence) and Grothendieck topoi, by assigning to each ...

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### Does the 2-category of topoi have exponential objects?

Does the 2-category of Grothendieck topoi have exponential objects?
There are size issues: Since Grothendieck topoi are supposed to have a small set of generators, the collection of objects of a ...

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### Major applications of the internal language of toposes

What are the major applications of the internal language of toposes?
Here are a few applications I know:
Mulvey's proof of the Serre–Swan theorem in which he interprets the intuitionistically valid ...

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### An extension of the Galois theory of Grothendieck

This question is about Joyal and Tierney's famous An extension of the Galois theory of Grothendieck. One of the main results states (see the MathSciNet review by Peter Johnstone):
Joyal and Tierney's ...

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### Direct and inverse image terminology

Let $f\colon X\to Y$ be a continuous map. Then $f$ induces a geometric morphism $f^\ast\dashv f_\ast\colon \mathrm{Sh}(X)\leftrightarrows\mathrm{Sh}(Y)$, whose left adjoint is called inverse image and ...

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### Definition of "classifying topos"

Is there a geometric theory $T$ and a Grothendieck topos $\mathcal E$ such that (2) holds but (1) doesn't:
$\mathcal E$ 2-represents the 2-functor
$$\mathbf{GrothTop}\to\mathbf{Cat}$$
which sends a ...

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### Group objects via diagrams or generalized elements — Kripke–Joyal?

The notion of a group object $G$ in a category with finite products can either be defined with a few commutative diagrams or via requiring that each hom set $\hom(X,G)$ is a group. There is a theorem ...

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### Tohoku and cohomology of toposes

In McLarty's The Rising Sea: Grothendieck on simplicity and generality I found the following quote:
The same, Grothendieck knew, would work for cases yet unimagined. He notes that Tohoku [...

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### On realizing a topos of sheaves as a topos of equivariant sheaves

This question is motivated by the following example : let $X$ be a variety over a field $k$, with algebraic closure $\bar{k}$. The Galois group $G_k:=\mathrm{Gal}(\bar{k}/k)$ acts on $X_{\bar{k}}:=X\...

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### Relative homology in Fargues-Scholze paper

if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they ...

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### Indexing categories of derivators

It is not clear to me the role of the domain and target in the definition of prederivators.
For instance, the classical references put the domain as $\mathit{Dia}$, others as $\mathit{Cat}$ itself.
...

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### Homotopical realizability

After a long story of dancing around the effective topos $ \mathcal{Eff}$, I finally resolved to get to the bottom of it. To this effect, working as it were backward, I ended up revisiting Kleene's ...

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### A non-conventional definition of topoi

In "Toward a Galoisian interpretation of homotopy theory" (2000), B. Toën wrote:
Pour expliquer notre point de vue sur la notion de champs rappelons une construction (non conventionnelle) ...

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### How to express in categorical language that in some toposes not all complex numbers have square roots

I'm trying to improve my ability to translate constructive logic into the category theoretical language of topos theory. So far, my understanding of constructive logic has been rather naive. I know ...

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### Consistency proof in topos logic

Is the consistency of classical third-order arithmetic provable in the logic of a topos with natural numbers?
(My guess would be yes, but I haven't seen this anywhere.)
Edit: in the original version I ...

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### Connections between $0$-toposes and $1$-toposes (Grothendieck and elementary)

This is a crosspost from math.stackexchange.
A Grothendieck $0$-topos is the same as a frame (see here). Another relationship between Grothendieck toposes and frames is the following: whenever $\...

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### Explicit description of the canonical $\pi_G: \mathrm{Sh}(G_0)\to B_S(\mathbf{G})$

Given a localic groupoid $\mathbf{G} = (G_1\overset{d_0}{\underset{d_1}{\rightrightarrows}}G_0)$ and letting $B\mathbf{G}$ denote its classifying topos, I'm looking for a explicit description of the ...