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 ...
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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....
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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 ...
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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 ...
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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 ...
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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 ...
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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: $\bullet \to \bullet$, ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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$\infty$-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 ...
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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 ...
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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{...
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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 ...
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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 $\mathcal{C}$ ...
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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).
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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 ...
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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....
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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)/...
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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 ...
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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 ...
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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{...
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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). ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 $\...
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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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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 ...
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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{...
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Local isomorphism of condensed sets and étale condensed groupoids
Is there a notion of local isomorphism for condensed sets?
$\textbf{Motivation:}$ I am trying to define what an étale condensed groupoid would be.
A topological groupoid $\mathcal{G}$ is said to be ...
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Constructive theory of Lie algebras
I'm looking for references on constructive Lie algebra theory, e.g. the sort of theory you could develop in Martin-Löf type theory or internal to some topos with a NNO. Obviously excluded middle is ...
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A kind of “weak” filtered colimit in the effective topos
I was recently reminded that even filtered colimits in the effective topos generally do not exist. However, there is an important (albeit restrictive) situation that looks a lot like them and that I ...
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(When) do filtered colimits exist in the effective topos?
(My apologies if this is well-known: I feel that I'm missing something very obvious here.)
Basic question: Do filtered colimits exist in the effective topos?
The reason I feel I'm missing something ...
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Tensor product of sites
Let $C, D$ two Grothendieck sites. Since the corresponding toposes $E, F$ are locally presentable categories, then (by Gabriel-Ulmer duality) they correspond to limit theories $X, Y$ (that is, small ...
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One-point compactification of a condensed set
Is there a notion of a 'one-point compactification of a condensed set'?
$\textbf{Motivation:}$ For a locally compact space $X$, there is a notion of maps that vanish at infinity. A continuous function ...
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Does the Zariski spectrum of a ring arise formally from the inclusion of the big Zariski topos into the classifying topos for rings?
Let $\iota_\ast : \mathcal A \to \mathcal B$ be a geometric morphism. I'm looking for some functor
$$F_{\mathcal A \to \mathcal B} : \mathrm{Topos}_{//\mathcal B} \to \mathrm{Topos}_{//\mathcal A}$$
...
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Can we encode a torsor as a binary function on the isomorphism classes of objects?
Let $G$ be a group object in a topos $\mathcal{T}$. Then we have the notion of a $G$-torsor in $\mathcal{T}$, and the set of isomorphism classes of such objects is denoted $H^1(\mathcal{T};G)$. For ...
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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) ...
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Do Grothendieck topoi with enough points satisfy the fan theorem internally?
Fourman and Hylland proved in the 80s that all spatial topoi satisfy the full fan theorem internally, while there are examples of localic topoi that do not satisfy it.
This leads one to conjecture a ...
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Cohesive structure of Cahiers and Dubuc topoi
The inclusion of commutative rings into supercommutative rings has two adjoints, one projecting out the even part and the other quotienting out the ideal generated by odd elements. After passing to ...
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Do all toposes satisfy the internal Zorn's lemma?
I came up with this question when trying to give a more detailed answer to a question by Tim Campion in a comment to Ingo Blechschmidt's answer to Examples of statements that are valid in every ...