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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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13 votes
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Categories in which isomorphism of stalks does not imply isomorphism of sheaves

Let $\mathcal{A}$ be a locally small category with colimits of small filtered diagrams. For the purposes of this question, an $\mathcal{A}$-presheaf on a topological space $X$ is a functor $\Omega (X)^...
Zhen Lin's user avatar
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Does this weak omniscience principle have a name?

In constructive analysis, I'm looking at principles which follow both when there exists at least one discontinuous function from $\mathbb{R}$ to $\mathbb{R}$ (equivalent to WLPO i.e. $x > 0$ or $x \...
saolof's user avatar
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Can the real numbers be constructed as/from a Hom-object in a topos?

I've been reading through Greenblatt's Topoi and while I'm still definitely over my head I'm starting to get a feel for some of the concepts at play there. I see the definitions of $\mathbb{R}_c$ and $...
Steven Stadnicki's user avatar
6 votes
0 answers
153 views

Covering categories with posets

Let $C$ be a small (1-)category. There is always a poset $D$ and a functor $p : D \to C$ such that: $p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$,...
Zhen Lin's user avatar
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17 votes
3 answers
689 views

Large "internal" categories and "finite" products

The question is basically "do we really have a good way to talk about large categories internally in an elementary topos?" An internal small category in a topos $E$ is just a category object ...
Simon Henry's user avatar
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5 votes
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Explicit description of a topos of sheaves on an internal boolean algebra

I have a question on how to calculate a topos of sheaves on an internal site. Let $F$ be the category of finite sets and functions so that the topos ${\widehat{F}}$ of presheaves on $F$ classifies ...
Mendieta's user avatar
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11 votes
3 answers
671 views

Merging single-sorted and multi-sorted theories

The general theory of single-sorted (say, algebraic) theories is very similar to the general theory of multi-sorted (algebraic) theories. Each variable gets a sort, but apart from that nothing really ...
Martin Brandenburg's user avatar
2 votes
2 answers
172 views

Can the Category of that kind of small sets in $\sf NFU$ be Cartesian closed?

Working in Quine's $\sf NFU$, with urelements being at least as many as sets. Formally the latter is: $|Ur| \geq |Set|$. Where $Ur$ is the set of all urelements and $Set$ is the set of all sets. We ...
Zuhair Al-Johar's user avatar
7 votes
1 answer
196 views

When is the category of sheaves on a site compactly assembled/a continuous category?

If $(C,J)$ is a site, what is a natural condition on the Grothendieck topology $J$ to ensure that the category $Sh(C,J)$ is compactly assembled? I am both interested in the 1-categorical as well as ...
Georg Lehner's user avatar
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Is the logic of the ($\infty$-?)topos of simplicial sets "contradictory up to homotopy"?

In a sense this is a followup to my earlier question Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?. In the topos of simplicial sets, the subobject ...
მამუკა ჯიბლაძე's user avatar
11 votes
1 answer
390 views

Giraud's axioms imply balanced

I'm stuck on the following. Let $\mathcal{E}$ be a category satisfying the Giraud axioms: namely $\mathcal{E}$ is locally presentable, has universal colimits, has disjoint coproducts, and has ...
Emilio Minichiello's user avatar
8 votes
1 answer
343 views

What topos-theoretic construction lies behind the “symmetric model” construction (used to refute AC) in Set Theory?

Suppose we want to prove that (classical!) $\mathsf{ZF}$ does not prove, say, “for every infinite set $A \subseteq \{0,1\}^{\mathbb{N}}$ there exists an injection $\mathbb{N} \to A$” (I take this ...
Gro-Tsen's user avatar
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7 votes
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Bounded geometric morphisms, origin and motivation for the terminology

Bounded geometric morphisms serve as a generalization of Grothendieck topoi; $T$ being a Grothendieck topos iff global section $T \rightarrow Set$ is bounded. I managed to only track this down to the ...
Ilk's user avatar
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4 votes
1 answer
371 views

Grothendieck topoi as a constructive property

This question is inspired by Homotopy type theory, but I believe it can be thought about also in other constructive foundations. In HoTT the question could be stated as follows: Given a definition of ...
Ilk's user avatar
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7 votes
2 answers
292 views

Quotient topoi as quotient objects

In Lawvere's Open problems in topos theory; quotient topoi are treated as connected geometric morphisms of Grothendieck topoi. Is there a good reference for where these come from? Is there any sense ...
Ilk's user avatar
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9 votes
1 answer
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Topos notions coming from topology and uniqueness of generalizations

Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call ...
Ilk's user avatar
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2 votes
1 answer
74 views

Conditions for partially applied induced product functor to preserve colimits

Let $\mathcal{C}$ have products, $A, B \in \mathcal{C}_0$, ${\times}\colon \mathcal{C}^2\to \mathcal{C}$ be the functor that sends objects to their product. Then the induced product ${\boxtimes}\colon ...
cxandru's user avatar
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3 votes
1 answer
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Do $\mathcal{U}$-small partitioned assemblies densely generate realizability toposes?

Given a partial combinatory algebra $A$, the realizability topos $\mathrm{RT}(A)$ is the ex/lex completion of the category of partitioned assemblies $\mathrm{PA}(A)$. This implies that $\mathrm{PA}(A)$...
Colin's user avatar
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6 votes
1 answer
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Adjoints to inclusion pseudofunctors from topoi to Cat

Let's consider the bicategories, LogTopos of elementary topoi, logical functors and natural transformations and GrTopos of Grothendieck topoi, geometric morphisms and natural transformations. The ...
Ilk's user avatar
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6 votes
1 answer
388 views

Decimal expansion definition of real numbers, constructively

The two most common definitions of $\mathbb{R}$ are as Dedekind cuts or Cauchy sequences of rational numbers. A real analysis student of mine is working out of the book Real Analysis and Applications ...
Alec Rhea's user avatar
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6 votes
0 answers
188 views

The shape of an ($\infty$-)topos as a monad

Let $\mathcal E$ be an $\infty$-topos, and let $t : \mathcal E \to Spaces$ be the unique geometric morphism to $Spaces$. The composite $t_\ast t^\ast : Spaces \to Spaces$ is a left-exact accessible ...
Tim Campion's user avatar
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5 votes
1 answer
415 views

Intuition for the "internal logic" of a cotopos

Let $\mathcal{E}$ be an elementary topos. By definition, $\mathcal{E}$ is a category that has finite limits, is Cartesian closed, and has a subobject classifier $\Omega$. This subobject classifier can ...
safsom's user avatar
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4 votes
0 answers
193 views

Examples of Grothendieck ($\infty$-)topoi which do / do not satisfy the law of excluded middle

I would like to create a big list of Grothendieck topoi (or Grothendieck $\infty$-topoi) which do / do not satisfy the law of excluded middle. That is, let’s list some examples of topoi whose internal ...
7 votes
1 answer
432 views

Strict toposes as a finite limit theory

For some motivation I have been wondering about generalizing the topos of coalgebras theorem to relative monads in my previous question. This brought me to wonder about topos objects. NLab on ...
Ilk's user avatar
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8 votes
0 answers
386 views

Can the p-adic be countable?

Recently arxiv submitted a new paper (Andrej Bauer, James E. Hanson, The Countable Reals) claiming an incredible theorem that Dedekind reals are not sequence-avoiding, and furthermore obtaining a ...
Ember Edison's user avatar
2 votes
1 answer
98 views

What can be said about a semigroup $M$ from its topos $\mathit{Sets}^M$?

Please recommend some review "what can be said about a semigroup $M$ from its topos $\mathit{Sets}^M$". For example, for which semigroups are de Morgan's laws true?
Pavel Shuhray's user avatar
9 votes
1 answer
2k views

What does the topos of (light) condensed sets classify?

Recall that $\mathrm{Pro}(\mathbf{FinSet}) = *_{\text{proét}}$, the category of profinite sets, forms a site with finite jointly surjective families as covers, and that the category of sheaves on this ...
xuq01's user avatar
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4 votes
0 answers
102 views

Topos as a totally cocomplete object in a 2-category CART

In the preface to Sketches of an elephant, Peter Johnstone gives a list of characterizations of topos, some applicable to elementary (ii- finite limits and power objects) other to Grothendieck (i- ...
Ilk's user avatar
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0 votes
0 answers
185 views

How might someone with a background in group theory start research into topos theory?

The Question: How might an early career mathematician with a background of research in group theory start research into topos theory? I want links between the two areas, not career advice, though it ...
Shaun's user avatar
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1 vote
1 answer
122 views

System of local isos gives system of local epis

Suppose that $W$ is a system of local isomorphisms on a presheaf topos $\mathbf{Pre}(\mathcal{C})$. We say a map in $W$ is a $W$-local isomorphism, and we say that a map of presheaves $f: X \to Y$ is ...
Emilio Minichiello's user avatar
9 votes
1 answer
261 views

Does there exist a geometric morphism between the effective and topological topoi? Does one arise from synthetic topology?

I'm presenting in final projects for my computability and computational topology courses on the connections between computability, continuity, and logic. As a mathematician/unmentored baby logician ...
Garrett Figueroa's user avatar
2 votes
1 answer
159 views

"$X$ is $n$-truncated $\iff$ $\Omega X$ is $(n-1)$-truncated" for connected pointed $X$. (HTT, 7.2.2.11)

In the proof of Lemma 7.2.2.11 of Higher Topos Theory, Lurie makes the following claim: ($\ast$) Let $n\geq1$ be an integer, let $\mathcal{X}$ be an $\infty$-topos, and let $1\to X$ be a pointed ...
Ken's user avatar
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7 votes
0 answers
306 views

The constructive Eudoxus reals

Recently arxiv submitted a new paper (Andrej Bauer, James E. Hanson, The Countable Reals) claiming an incredible theorem that Dedekind reals are not sequence-avoiding, and furthermore obtaining a ...
Ember Edison's user avatar
5 votes
1 answer
361 views

Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$ in a topos?

I am looking for ways to internalize the modal operator of necessity $\Box$, ending up with a morphism $\Box: \Omega \to \Omega$ satisfying the necessitation rule (if $\phi$, then $\Box \phi$) and the ...
Miviska's user avatar
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9 votes
1 answer
265 views

Is there a correction to the failure of geometric morphisms to preserve internal homs?

Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
Cameron's user avatar
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6 votes
0 answers
122 views

Points of the sheaf topos over Blass' category

There is a site $\textsf{Blass}$ used for (constructive) non-standard analysis, whose objects are sets equipped with a filter, and morphisms are continuous functions defined up to a small set. (It is ...
Trebor's user avatar
  • 1,263
6 votes
1 answer
590 views

Is there an English translation of Monique Hakim's thesis?

Monique Hakim's thesis, published in 1972 as Topos annelés et schémas relatifs, has been referenced on a multitude of occasions. But I struggle to find a translation into English, even an informal one....
xuq01's user avatar
  • 1,094
4 votes
1 answer
212 views

A complex version of the Cahiers topos

Has anyone tried defining a complex version of the Cahiers topos? If we take the definition of $C^\infty$-rings, replace "smooth" with "holomorphic" (of course, one has to take ...
xuq01's user avatar
  • 1,094
8 votes
0 answers
221 views

When is the Eilenberg-Moore category of a relative monad between two topoi a topos?

In the non-relative case, we have a theorem, that an Eilenberg-Moore category of algebras of a Monad $T$ on a topos is itself a topos if the monad in question has a right adjoint. Now how does this ...
Ilk's user avatar
  • 1,347
7 votes
1 answer
205 views

Variation on definition of logical functors avoiding power objects

Without power sets in meta-theory not every Grothendieck topos is an elementary topos, Set is still Grothendieck, but it lacks power objects. Now I am looking for a definition of a logical functor ...
Ilk's user avatar
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5 votes
0 answers
189 views

When are topoi of coalgebras atomic?

A geometric morphism is atomic if its inverse image is logical. Now consider a Grothendieck topos $\varepsilon$ and its terminal geometric morphism $\Gamma : \varepsilon \rightarrow Set$, topos is ...
Ilk's user avatar
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1 vote
0 answers
49 views

Defining properties of categories out of an indicial category

$\newcommand{\Hom}{\operatorname{Hom}}$Suppose we want to define the category of arrows of $S$. Below are two forms of doing it. Definition 1: If $D$ is of the following type: $\bullet \to \bullet$, ...
user234212323's user avatar
9 votes
2 answers
2k views

Truth in a different universe of sets?

I understand that provability and truth as different concepts. Provability is syntactic, it only concerns whether the given sentence can be derived by reiterating the inference rules over a collection ...
Student's user avatar
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2 votes
0 answers
94 views

What can be said about the free-forgetful adjunction of monad algebras with respect to topoi?

For a monad T on a topos E, if T has a right adjoint, then the Eilenberg-Moore category of algebras of T is equivalent to the co-Eilenberg-Moore category of co-algebras for the right adjoint comonad ...
Ilk's user avatar
  • 1,347
11 votes
1 answer
461 views

Are flat functors out of a finite category necessarily finite?

Note: I've originally asked this question on math stack exchange, but I have learnt that this is the better place to ask for research level questions, so I have deleted the original question there. ...
Lingyuan Ye's user avatar
7 votes
0 answers
156 views

Logical properties of realizability (topoi or McCarty models) defined by alpha-recursion on admissible ordinals

Setup: Let $\alpha$ be an admissible ordinal (viꝫ., one such that $L_\alpha$ is a model of Kripke-Platek set theory), identified as usual with the set of ordinals $<\alpha$. Then there is a ...
Gro-Tsen's user avatar
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3 votes
1 answer
148 views

Does every $\kappa$-compact topos embedd relatively $\kappa$-tidily into a presheaf topos?

Let $\kappa$ be a regular cardinal and say that a topos $\mathcal{E}$ is $\kappa$-compact if the global sections $\gamma_{\ast} : \mathcal{E} \to \mathsf{Set}$ preserves $\kappa$-filtered colimits. My ...
David Jaz Myers's user avatar
5 votes
0 answers
163 views

Why equaliser of product and terminal object is coproduct?

I’m reading “Sheaves in geometry and logic”, in page 80: Please refer to [1]: https://i.sstatic.net/INrU0.jpg It says “…,therefore $FU=\coprod_{x\in U} fx$. The space…”. So could anyone please explain ...
Bonan Su's user avatar
6 votes
1 answer
341 views

$\infty$-topos as an internal $\infty$-category in itself

I'm interested (both autonomously and directly related to my work) in the natural internalization of $\infty$-topos sheaves in it (as usual, assuming Grothendieck universes). Is there any literature ...
Arshak Aivazian's user avatar
1 vote
0 answers
214 views

Kan extensions in Grothendieck school

Considering both the ubiquity of Kan extensions in category theory (as MacLane stated, 'The notion of Kan extensions subsumes all the other fundamental concepts of category theory.'), its early ...
user234212323's user avatar

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