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Questions tagged [topos-theory]

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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
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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
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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
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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
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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
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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
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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
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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)...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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
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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
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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 ...