Questions tagged [topos-theory]
The topos-theory tag has no usage guidance.
10
votes
0answers
560 views
Toposophy vs Set theoretical multiverse philosophy
Johnstones classic topos theory book talks at some length in its introduction about how category theory/topos theory suggest that we view the 'universe' in which mathematics takes place as consisting ...
5
votes
1answer
271 views
Universal property of sheaf category
Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...
2
votes
1answer
79 views
Definition of $\in_c$ for power objects
On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any ...
5
votes
0answers
92 views
Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)
SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...
5
votes
0answers
108 views
Topos with enough projectives
It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...
8
votes
1answer
195 views
Is there a version of the “infinitary” disjunctive normal form theorem for topoi and slice categories?
According to nLab
Proposition 2.3. A complete Boolean algebra is completely distributive iff it is atomic (a CABA), i.e., is a power set as a Boolean algebra.
This is basically an infinitary ...
8
votes
2answers
312 views
What is a spectrum object in $\infty$-topoi?
For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E_n)_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra?
To ...
10
votes
1answer
294 views
Intuition for pseudo-points and the inductive step in Johnstone's proof of Deligne's completeness theorem
In Johnstone's Topos Theory appears the following lemma.
7.41 Lemma. Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two distinct element of $P(X)$. Let $(V_j\to V)...
8
votes
0answers
140 views
Are dualizable objects in the derived category of a ringed topos perfect?
Recall that an object $a$ in a symmetric monoidal category $(\mathcal{C}, \otimes, e)$
is dualizable if there exists an object $b$ and morphisms $\varepsilon\colon b \otimes a \to e$
and $\eta\colon e ...
10
votes
0answers
228 views
What are the monomorphisms of ($\infty$-)toposes?
There are standard notions of "surjections" and "embeddings" of toposes. However, not every surjection is an epimorphism, and not every regular monomorphism is an embedding. (EDIT: as Alexander ...
11
votes
1answer
265 views
Are there continua in $\infty$-topoi?
If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence....
19
votes
2answers
643 views
What is the geometric significance of fibered category theory in topos theory?
Often in topos theory, one starts with a geometric morphism $f: \mathcal Y \to \mathcal X$, but quickly passes to the Grothendieck fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$, which ...
7
votes
1answer
262 views
What are the “smallest” topoi?
Yesterday I was talking to somebody from the Haskell community.
Late in the night we found ourselves discussing possible topoi.
Lets order topoi (up to equivalence, ...) by number of objects/...
15
votes
1answer
340 views
When is the category of models of a limit theory a topos?
If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory.
Is there a characterization of ...
9
votes
1answer
276 views
Points of the big Zariski site
It's relatively simple to show that the geometric morphisms $ \mathbf{Set} \to \mathrm{Sh}(\mathbf{CRing}^\mathrm{op}_{\mathrm{fp}}, \mathrm{Zar})$ correspond to local rings.
More precisely, since ...
0
votes
0answers
73 views
A finiteness property of profinite sets
I would like to understand the canonical topology on the category of profinite sets. Unless I am making mistakes, this translates to the following question in point set topology:
Say $X$ is a ...
6
votes
0answers
76 views
Internal van Kampen colimits
Let $C$ be a category with pullbacks. Recall that a colimit in $C$ is van Kampen if it is preserved by $C/(-): C^{op} \to CAT$, $c \mapsto C/c$.
We can $C$-internalize everything in sight:
Let $\...
5
votes
1answer
160 views
What does an ideal correspond to in the internal language of sheaves?
Suppose I have a sheaf $\mathcal F$ in some topos $\mathrm{Sh}(\mathcal C)$. Then this becomes the sheaf of rings from algebraic geometry when described as a ring in the internal language of the topos....
10
votes
1answer
417 views
The étale topos of a scheme is the classifying topos of which groupoid?
[Sent here from Math.StackExchange by suggestion of an user.]
By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. ...
-4
votes
1answer
843 views
Why sheaves are important and why do we care about them? [closed]
Presheaves are contravariant functors from a category $C$ to the category $Set$, that is functors $P$:
$$P:C^{op}\to Set.$$
For every topology $J$ on $C$ we can generate a reflexive subcategory
$$Sh(...
4
votes
2answers
270 views
Categorification of spaces and models for set theory
One aspect of topos theory is that it provides an enlarged view of the classical concept of space. Indeed, one may thought that the notion of topos is a sort of categorification of the notion of space....
9
votes
2answers
463 views
2-natural operations on toposes
Any pseudonatural endomorphism $\Phi$ of the forgetful 2-functor $U:Topos^{coop}\to Cat$ is essentially determined by its component $\Phi_{Set}$. But which endofunctors of $Set$ induce such a $\Phi$? ...
6
votes
1answer
240 views
Left Kan extension along Yoneda of pullback-preserving functor preserving pullbacks
Let $F\colon C \to D$ be a functor. The Kan Extension of $y_D \circ F$ along $y_C$ yields a functor $F_!: Fun(C^{op},Set) \to Fun(D^{op},Set)$. Here, $y_C$ and $y_D$ denotes the respective Yoneda ...
10
votes
3answers
370 views
Example of non-“propositional” local operators on a topos?
Let $\mathcal{E}$ be a topos, and let $\top\colon1\to\Omega$ be its subobject classifier. We refer to global elements $P\colon 1\to\Omega$ as propositions; they form a poset, denoted $(|\Omega|,\leq)$....
8
votes
1answer
271 views
Is there something like internal language of an abelian category?
While studying topos theory I was wondering if there is something like internal logic of an abelian category. Aparently the answer is yes
(by 7º slide in https://www.mimuw.edu.pl/~gael/xxi/files/...
5
votes
1answer
160 views
How do (co)limits in posets of subobjects relate to (co)limits in ambient category?
Sorry if this is elementary.
Let $C$ be a category, $X$ an object in $C$ and let $S(X)$ denote the poset of subobjects of $X$.
According to the nlab entry, if $C$ has all limits and co-limits, so ...
7
votes
1answer
120 views
Tietze transformations for sites of toposes
I have read that people think of a site as a presentation of the corresponding sheaf topos. For instance, on page 7 of this text by Caramello: as Grothendieck observed himself, a site of definition ...
14
votes
2answers
758 views
Is there a universal way to force the Axiom of Choice to be true?
Given a model of set theory $V$ there are various ways to construct a model in which the Axiom of Choice holds, such as Gödel's constructible universe $L^V$ or by using forcing*. I'm wondering if any ...
2
votes
0answers
109 views
epimorphism of fppf sheaves is an fppf morphism
I asked this question on math.stackexchange (https://math.stackexchange.com/questions/2693471/epimorphism-of-fppf-sheaves-is-an-fppf-morphism) but didn't get an answer. Maybe someone here can help.
...
3
votes
0answers
159 views
Locality in Grothendieck Topologies
Let $\mathcal{C}$ be a category and $\mathcal{J}$ be a Grothendieck topology on it (i.e., $(\mathcal{C},\mathcal{J})$ is a site). Then what is a good notion of locality in it?
I came up with the ...
3
votes
1answer
155 views
When the global section functor is a Cartesian fibration?
Given a Cartesian fibration $p : \mathbf{E} \to \mathbf{B}$ over an $\infty$-topos the paper by Marc Hoyois mentioned in his answer to this question gives some sufficient conditions for $\mathbf{E}$ ...
13
votes
2answers
270 views
A locally presentable locally cartesian closed category that is not a quasitopos
This question asks for a locally presentable locally cartesian closed category that is not a topos. All the answers given (at least in the 1-categorical case) are quasitoposes. What is an example of ...
6
votes
1answer
210 views
Example of a locally presentable locally cartesian closed category which is not a topos?
The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a ...
4
votes
2answers
173 views
Generalizations of tangent $\infty$-topos
If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\...
7
votes
1answer
200 views
Can the subobject classifier be covered by an object in a subtopos?
Let $\mathcal{E}$ be a topos and $\Omega$ its subobject classifier.
Is it possible to have a nonidentity local operator (a.k.a Lawvere-Tierney topology) $j\colon\Omega\to\Omega$, a $j$-sheaf $X\in\...
13
votes
1answer
292 views
Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects
These are five important constructions and I would like to know how they are related.
The $n$th unordered configuration space of a space $X$ is
$$
\operatorname{UConf}_n(X):=\{\text{embeddings of $\{...
6
votes
0answers
88 views
Formal Group Laws in a lined topos
I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the ...
7
votes
1answer
181 views
How are the left and the right group of a bitorsor related?
This question arose from my answer to To what extent does a torsor determine a group: it turns out that I do not know one thing about it.
Let $G$, $G'$ be groups in some nice enough category (you may ...
15
votes
2answers
520 views
Cauchy real numbers with and without modulus
In constructive mathematics there are many possible inequivalent definitions of real numbers. The greatest variety seems to be in Dedekind-style approaches: in addition to "the" Dedekind real numbers ...
2
votes
0answers
80 views
Reference to an explixit construction of a locale from a measurable space
In A sheaf theoretic approach to measure theory shows that measures on a measurable space are equivalent to measures on some locale whose open sets are the $\sigma$-ideals of the $\sigma$-algebra. The ...
3
votes
1answer
113 views
Are there any useful conditions for a biclosed monoidal structure on presheaves to descend to a biclosed monoidal structure on sheaves?
Suppose $C$ is a small category with a monoidal structure. Then by the special case of the Day convolution theorem for presheaves, $\operatorname{Psh}(C)$ is equipped with a corresponding biclosed ...
10
votes
0answers
144 views
Can we internalize topological fixed point theorems in an effective topos?
Reflective oracles are a kind of Turing oracle that give stochastic answers about the outputs of Turing machines. This works in a self-referential way, where they can answer queries about Turing ...
7
votes
0answers
120 views
Sufficient cohesion and conservativity of “underlying stuff”
Consider a category $\mathsf C$ admitting a quadruple adjunction as below.
$$(\Pi_0 \dashv \text{disc} \dashv \Gamma \dashv \text{codisc}) :
\mathsf{C}
\stackrel{\stackrel{\longrightarrow}{\...
8
votes
1answer
188 views
Topos properties from coverage conditions
For any category $C$ and coverage $J$ on it, let $\mathcal{E}:=\mathsf{Shv}(C,J)$ denote topos of sheaves on the site $(C,J)$. What sorts of results are known about the relationship between properties ...
3
votes
0answers
65 views
Universal basis for graph analog of Boolean circuits?
The NAND gate is universal for Boolean circuits. That means that any function $2^n \to 2$ can be built out of NAND gates.
Let $\Omega$ be the subgraph classifier, i.e., the graph with two vertices $t$...
9
votes
1answer
409 views
Relationship between synthetic differential geometry and differential cohesion?
I'm a big fan of synthetic differential geometry (or smooth infinitesimal analysis), as developed by Anders Kock and Bill Lawvere. It's a beautiful and intuitive geometric theory, which gives ...
15
votes
2answers
325 views
Characterize constant objects in the internal language of a topos?
A Grothendieck topos $\mathcal{E}$ is equivalent to the category of sheaves on some site $Q$. We say a sheaf $X\colon Q^{\text{op}}\to\mathsf{Set}$ is constant if it is the sheafification of a ...
8
votes
0answers
277 views
Examples of locally-but-not-globally (pre)sheaf toposes?
Re-reading my own recently posted question What is the total space of a stack after all? I realized that I don't know something more simple and presumably more basic.
Are there (bounded if you like) ...
10
votes
1answer
387 views
What is the total space of a stack after all?
From my general experience I think for myself of what follows as some kind of taboo question for some reason: in my imagination, everybody wants an answer to this but somehow thinks it shall not be ...
5
votes
0answers
404 views
What are the most general conditions under which the inverse image of sheaves of abelian groups has a left adjoint?
If $f: E \to X$ is an étale map (a local homeomorphism), then the inverse image of sheaves of abelian groups $f^{-1}$ has a left adjoint, as shown by Roland in his answer here. This subsumes as a ...
