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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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 ...
6 votes
1 answer
164 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 ...
16 votes
3 answers
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Recommendations to learn about the use of toposes in logic?

I'd like to learn about the use of toposes in logic. The "logic" side I know quite well, but of the "topos" side I am totally ignorant. Which books/articles (formal and/or casual) ...
6 votes
1 answer
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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....
4 votes
1 answer
188 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 ...
8 votes
0 answers
149 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 ...
6 votes
2 answers
297 views

Is there a notion of a complex/analytic diffeological space?

I have a bit of a general question. This seems like something you can do, but I can't seem to find much reference for this.. Perhaps something like this already exists in a different guise. But, is ...
5 votes
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164 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 ...
1 vote
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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: $\bullet \to \bullet$, ...
9 votes
2 answers
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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 ...
26 votes
6 answers
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What is a topos?

According to Higher Topos Theory math/0608040 a topos is a category C which behaves like the category of sets, or (more generally) the category of sheaves of sets on a topological space. Could one ...
2 votes
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73 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 ...
11 votes
1 answer
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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. ...
4 votes
2 answers
455 views

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 ...
8 votes
1 answer
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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 ...
7 votes
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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 ...
28 votes
3 answers
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Is there a good general definition of "sheaves with values in a category"?

Let $\mathcal{A}$ be a category. There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf ...
3 votes
1 answer
134 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 ...
5 votes
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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.stack.imgur.com/INrU0.jpg It says “…,therefore $FU=\coprod_{x\in U} fx$. The space…”. So could anyone please ...
7 votes
1 answer
291 views

Does a tight apartness relation on a subobject classifier imply the elementary topos is Boolean?

Given a set $S$, a tight apartness relation on $S$ is a relation $\#$ which is tight, irreflexive, symmetric, and weakly linear, or more specifically, a relation $\#$ such that for all elements $a \...
6 votes
1 answer
217 views

Dissolution of a topos

The dissolution of the locale associated with a frame $F$ is the locale associated with the frame $N(F)$ of nuclei of $F$ (see, e.g., Johnstone, “Stone Spaces” (1982), §2.5). Note that there is a ...
6 votes
1 answer
306 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 ...
1 vote
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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 ...
4 votes
1 answer
248 views

Interesting Grothendieck topologies or coverages on the category Prob

I am currently trying to understand Grothendieck Topologies and coverages and want to endow the category Prob, consisting of finite probability spaces and measure preserving maps, with a Grothendieck ...
3 votes
1 answer
182 views

Transitivity axiom for a Grothendieck Topology

I am currently trying to define a Grothendieck Topology on the category Prob which consists of finite probability spaces with measure preserving maps between them. I declared the covering sieves of an ...
8 votes
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219 views

What is the exact definition of the $\infty$-topos of sheaves on a localic $\infty$-groupoid?

The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ . The 2-category $\mathrm{LocalicGroupoid}$ (with suitable localization) is equivalent to the 2-category $1\text{...
0 votes
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Can 2 coverages generate the same Grothendieck Topology if the category is large?

I am currently analyzing a category which is not small, but locally small. I have seen that any coverage on any small category $\mathcal{C}$ generates a unique Grothendieck Topology on $\mathcal{C}$ ...
23 votes
1 answer
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Condensed vs pyknotic vs consequential

As is probably clear from my previous questions, I am coming to "condensed mathematics" from the naive perspective of a category theorist, without much knowledge of the intended applications ...
0 votes
0 answers
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Alternative definition of $\chi_{k}(x)$

Assume $q:B\rightarrow I$ is a local homeomorphism, and $A\subseteq B$ is open. Consider arbitrary $x\in B$, and $S$ is an open nbhd of $x$ such that $q\upharpoonright S$ is homeomorphism (locally). ...
6 votes
1 answer
283 views

Relationship between canonical topology on a topos and its site of definition

The canonical (Grothendieck) topology for a category $C$ is the largest (finest) topology such that every representable presheaf over $C$ is a sheaf. According to First Order Categorical Logic Lemma 1....
2 votes
1 answer
129 views

Is the slice of a subcanonical site also subcanonical?

A subcanonical site is one for which every representable functor is a sheaf. For a subcanonical site $C$, the fundamental theorem of topos theory says that there is an equivalence $Sh(C/c)\cong Sh(C)/...
5 votes
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Is Vopěnka's principle inherited by Grothendieck topoi?

I call the Vopěnka's principle: Every subfunctor of an accessible functor is accessible but other formulations (which may lose equivalence in weak contexts?) are also interesting to me. If this is ...
8 votes
1 answer
315 views

G-topological spaces and locales

Consider the following generalization of topological spaces: Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...
12 votes
2 answers
1k views

Intuitionistic consistency of surjection from naturals to reals

Is it consistent intuitionistically (in the sense of topos theory) for there to be a surjection from the natural numbers to the (Dedekind, let us say) real numbers? [I've managed to convince myself ...
8 votes
0 answers
387 views

Sheaf of compact Hausdorff spaces but not a condensed anima

Consider the site $\mathbf{CHaus}$ of compact Hausdorff spaces together with the finitely jointly surjective families of maps as coverings. Restriction induces an equivalence of categories $$ \mathbf{...
20 votes
4 answers
2k views

Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?

Last year, Andrej Bauer gave a talk showing that there is a topos in which the set of Dedekind reals is (sub)countable, and thus, you cannot prove that $\mathbb{R}$ is uncountable without LEM. He ...
2 votes
0 answers
147 views

Topos of sheaves on a scheme considered as a functor

The spectrum of a ring $R$ can be defined as $\operatorname{Spec} R := \operatorname{Hom}(R, -)\colon \mathrm{fpRing} \to \mathrm{Set}$ ($\mathrm{fpRing}$ are commutative finitely presentable rings). ...
15 votes
3 answers
1k views

Ordinals in constructive mathematics ? (references)

I'm looking for references presenting a constructive treatment of the theory of ordinals. By constructive I mean valid in the internal logic of a topos (so no axiom of choice and no law of excluded ...
5 votes
2 answers
400 views

Topos semantics of constructive higher order logic

I would like to find a reference that describes the semantics of constructive higher order logic with function types in toposes. In particular, it seems that if we are to take function types as ...
10 votes
2 answers
565 views

Effective epimorphisms and 0-truncations (HTT, 7.2.1.14)

In Proposition 7.2.1.14 of Higher Topos Theory, Lurie asserts the following: Let $\mathcal{X}$ be an $\infty$-topos and let $\tau_{\leq0}:\mathcal{X}\to\tau_{\leq0}\mathcal{X}$ denote a left adjoint ...
6 votes
1 answer
363 views

Anafunctors vs the plus construction

Given a Lie groupoid $G$, we can view it as representing a prestack on $\text{Mfld}$ by sending and manfold $M$ to the groupoid of smooth functors and smooth natural transformations $$G(M) := \text{...
4 votes
0 answers
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Relationship between coarse objects, separated objects, and sheaves

I would like to better understand the relationship between quasitopoi and topoi. Here are two relationships that I am aware of: Given a local topos $E \to S$, i.e. such that $S$ is equivalent to the ...
9 votes
1 answer
553 views

What is known about the homotopy type of the classifier of subobjects of simplicial sets?

For the presheaf topos $\mathrm{PSh}(C)$, the subobject classifier is the presheaf $\Omega$ such that For $c \in C$, $\Omega(c)$ is the set of all subobjects of the functor $\mathrm{Hom}(-, c)$ For $...
21 votes
1 answer
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Logical endofunctors of Set?

What set-theoretic assumptions are necessary and sufficient to ensure the existence of a nontrivial (i.e. not isomorphic to the identity) endofunctor of the category Set which is logical (i.e. ...
6 votes
0 answers
343 views

Can Langlands correpondence be restated using topos?

Langlands correspondence describes an equivalence between Galois representations and automorphic representations under some conditions. Laurent Lafforgue applying Olivia Caramello thesis described in ...
2 votes
1 answer
158 views

Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts?

Let $\mathcal{R}$ be a $1$-category. Assume that one has a pushout of representable $1$-presheaves $\mathrm{y} A \cup_{\mathrm{y} B} \mathrm{y} C$ in $\mathsf{PSh}(\mathcal{R})$. Under which ...
8 votes
2 answers
416 views

Condition for an equivalence of functor categories to imply an equivalence of categories

Given small categories $\mathcal{C}$ and $\mathcal{D}$, we have that $[\mathcal{C}^\text{op},\textbf{Set}]\simeq[\mathcal{D}^\text{op},\textbf{Set}]$ if and only if the Cauchy-completions of $\...
4 votes
1 answer
187 views

Property of pushouts in the category of unital $C^{\ast}$-algebras

Let $A$ be a unital $C^{\ast}$-algebra and $\{ f_i: A \rightarrow A_i \}_i$ a finite collection of morphisms of unital $C^{\ast}$-algebras, such that the associated map $A \rightarrow \prod_i A_i$ is ...
4 votes
0 answers
135 views

Do presheaf toposes satisfy the full fan theorem?

Presheaf toposes satisfy LPO and (edit: if over categories with binary products) PAx and countable choice internally, so they automatically satisfy the stable fan theorem (every bar which is the ...
3 votes
0 answers
188 views

How to read the definition of Grothendieck Pretopology in SGA4?

In SGA4, the first axiom of a Grothendieck pretopology is given as: PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(𝑋)$ sont quarrables. (Rappelons qu’un morphisme ...

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