Questions tagged [infinity-topos-theory]
The infinity-topos-theory tag has no usage guidance.
136
questions
91
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10
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Reflection principle vs universes
In category-theoretic discussions, there is often the temptation to look at the category of all abelian groups, or of all categories, etc., which quickly leads to the usual set-theoretic problems. ...
51
votes
2
answers
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What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?
My question is as in the title:
Does anyone have an example (supposing one exists) of an
$\infty$-topos which is known not to be equivalent to sheaves on a
Grothendieck site?
An $\infty$-topos is as ...
29
votes
5
answers
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Homotopy groups of spheres in a $(\infty, 1)$-topos
Let $H$ be an $(\infty,1)$-topos (seen as a generalization of the homotopy category of spaces).
You can define the suspension of an object $X$ as the (homotopy) pushout of $*\leftarrow X \to *$, ...
26
votes
1
answer
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Is the $\infty$-topos $Sh(X)$ hypercomplete whenever $X$ is a CW complex?
It can be shown (see Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?) that if $X$ is a locally contractible paracompact Hausdorff space such that the $\...
25
votes
0
answers
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$\infty$-topos and localic $\infty$-groupoids?
It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales).
For the record, this is proved by, starting ...
24
votes
1
answer
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Examples of $(\infty,1)$-topoi that are not given as sheaves on a Grothendieck topology
An $(\infty,1)$-topos according to Lurie is defined as (accessible) left exact localization of a presheaf $(\infty,1)$-category $\text{P}(\mathcal C)$.
Those $(\infty,1)$-topoi $\text{Sh}(\mathcal C)$ ...
21
votes
2
answers
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$\infty$-categorical interpretation of type theory
One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity ...
20
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2
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Commutative rings : Topoi = Fields :?
The following is probably a bad question, but hopefully, it might have a very good answer.
In category theory there is a quite famous analogy between topoi and commutative rings, I was never ...
20
votes
0
answers
344
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Homotopic version of Freyd's AT category observations
Freyd was the first to formalize a striking comparison between abelian categories and topoi, showing that their exactness properties can be jointly captured by the axioms of AT categories, and the ...
19
votes
2
answers
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A "universally non Hypercomplete" $\infty$-topos via Goodwillie calculus?
My question is :
Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ?
What I mean by $\infty$-connected ...
18
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1
answer
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A sheaf is a presheaf that preserves small limits
There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough.
However when reading ...
18
votes
1
answer
612
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Equivalences of categories of sheaves vs categories of $\infty$-Stack
Let say I have two different sites $(\mathcal{C},I)$ and $(\mathcal{D},J)$ for an ordinary topos $\mathcal{T}$. I.e.
$$Sh(\mathcal{C},I) \simeq \mathcal{T} \simeq Sh(\mathcal{D},J)$$
And we want to ...
18
votes
1
answer
859
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 ...
17
votes
1
answer
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Forcing in Homotopy Type Theory
I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? ...
17
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1
answer
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What would be an infinity-groupoid analogue of the duality between sets and complete atomic boolean algebras?
Consider the object classifier of the $\infty$-topos of $\infty$-groupoids. For the role it plays in homotopy type theory as the type of types, let’s denote it as $Type = \coprod_{[F]} B Aut(F)$, the ...
16
votes
0
answers
197
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Characterization of geometric morphisms without referring explicitly to the left adjoint?
Recall that a functor $f_\ast : \mathcal E \to \mathcal F$ between toposes is called a geometric morphism if it has a left exact left adjoint $f^\ast$. Is there an intrinsic characterization of such ...
15
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2
answers
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Modern versions of Verdier's hypercovering theorem?
Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a ...
15
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2
answers
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Is the site of (smooth) manifolds hypercomplete?
By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...
15
votes
2
answers
633
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How to formulate the univalence axiom without universes?
The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =_U Y) \to (X \simeq Y)$ is an equivalence.
As we (usually) cannot ...
15
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1
answer
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Examples of differential cohomology in cohesive $\infty$ topos
I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public.
The formulation of differential cohomology in ...
15
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1
answer
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Is there a condensed / pyknotic refinement of the shape of an $\infty$-topos?
Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the shape of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to_\leftarrow ...
15
votes
1
answer
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$\infty$-topoi versus condensed anima
Let $ExDisc_\kappa$ denote the category of $\kappa$-small extremally disconnected topological spaces (for now fix a strong limit cardinal $\kappa$). There's a functor $ExDisc_\kappa \to \mathsf{RTop}$ ...
14
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2
answers
713
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Examples of topoi that are not ordinary spaces
In [SGA6] we find:
Mais nous lui conseillons néanmoins, de préférence, de s'assimiler le langage des topos, qui fournit un principe d'unification extrêmement commode. (DeepL translate: However, we ...
14
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2
answers
656
views
When is a stable $\infty$-category the stabilization of an $\infty$-topos?
Let $\mathcal X$ be a presentable $\infty$-category. Then the stabilization $Stab(\mathcal X)$ of $\mathcal X$ is the universal presentable stable category on $\mathcal X$.
Conversely, if $\mathcal A$ ...
13
votes
1
answer
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Learning higher differential geometry
I have read parts of the motivation on nlab and all the posts on MO I could find on the subject, and by now there are a few questions on my mind. If they trivial for someone who understands the theory,...
13
votes
1
answer
571
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....
13
votes
1
answer
453
views
Comonadicity of spaces over spectra?
As connective spectra are equivalent to group-like $E_{\infty}$ algebras in spaces, the $\infty$-category of connective spectra is monadic over the $\infty$-category of spaces though the usual $\Sigma^...
13
votes
0
answers
268
views
Which topoi are local with respect to Stone-Cech compactification?
Compact Hausdorff spaces $X$ are characterized among all topological spaces by the fact that for any topological space $S$, the embedding $S \to \beta S$ into its Stone-Cech compactification induces a ...
13
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0
answers
243
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Is every colimit-generator dense in an $\infty$-topos?
Recall that there are various senses in which a full subcategory $G \subseteq C$ may "generate" a category $C$. For example, in order of increasing strength (under reasonable conditions):
$G$ is a ...
12
votes
1
answer
522
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Are $\infty$-topoi determined by their localic points ?
Hello !
If $T$ is an infinity topos, then you can consider the infinity category of geometric morphism from $Sh_{\infty}(\mathcal{L})$ to $T$ for any locale $\mathcal{L}$. This associate to $T$ an ...
12
votes
0
answers
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What's wrong with the obvious argument that the unstable motivic category is an $\infty$-topos?
Fix a base scheme $S$, and let $Sm_S$ be the (ordinary) category of smooth schemes over $S$.
Denote by $Psh(Sm_S)$ the $\infty$-category of $\infty$-presheaves on $Sm_S$.
Let $Sh_{Nis}(Sm_S)\...
11
votes
2
answers
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Relation between topos and $\infty$-topos
I'm currently reading The book of Jacob Lurie, 'Higher Topos Theory', and I'm a little confused by the relation between classical topos and $\infty$-topos :
to an $\infty$-topos I can attach the ...
11
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1
answer
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Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves
Consider the global projective model category
of simplicial presheaves on some category
(the category of smooth manifolds is particularly interesting to me).
In Section 9.1 of Dugger's paper “...
11
votes
1
answer
624
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Geometric morphism of $\infty$ topos
I have a very simple question regarding geometric morphisms of $\infty$ topoi, but have been unable to find the answer in Lurie's HTT (although it seems likely that its there somewhere and I just can'...
11
votes
1
answer
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What is an Elementary "Homotopy, Model" Topos?
Context:
Def (Rezk): A (Grothendieck) homotopy topos is a homotopy left exact Bousfield localization of the model category of simplicial presheaves sPsh(C) on a small simplicial category C.
Thm (...
11
votes
1
answer
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What is the connection between Lurie's definition of shape and Čech homotopy?
It seems there are many subtly different notions of the shape of a topological space (and, more generally, toposes).
For instance, Lurie [Higher topos theory] defines this one:
Definition 1.
The ...
10
votes
2
answers
565
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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 ...
10
votes
3
answers
907
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classifying $\infty$-toposes for topological/localic groups?
Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?
More precisely, is there an $\infty$-topos $BG$ ...
10
votes
2
answers
817
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
1
answer
570
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Boolean non-hypercomplete $(\infty,1)$-toposes
Let's say that an $(\infty,1)$-topos is Boolean if for every object $X$, the lattice $Sub(X)$ of subobjects (i.e. $(-1)$-truncated morphisms into $X$) is a Boolean algebra. I think this is equivalent ...
10
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2
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Canonical topology for infinity topoi revisited.
A while ago I asked this quetion: Canonical topology for big infinity topoi
and this question: How to resolve size issues with the regular epimorphism topology
Let me first summarize some of what I ...
10
votes
1
answer
552
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Lex $\infty$-colimits
In the paper lex colimits, Garner and Lack gave a general characterization of "exactness properties" for categories (and enriched categories). A "lex-weight" is a weight for colimits whose domain ...
10
votes
0
answers
579
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A model category for descent?
Recall that an $(\infty,1)$-category $C$ is said to have descent if for any small diagram $X:I\to M$ with (homotopy) colimit $\overline{X}$, the adjunction between $C/\overline{X}$ and "equifibered" $...
9
votes
2
answers
582
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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 $\Phi$? ...
9
votes
1
answer
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Cohesion relative to a pyknotic/condensed base
Something that usefully emerged for me from this discussion and follow-up MO question is that rather than see cohesiveness and condensedness/pyknoticity in rivalry with one another, as my initial ...
9
votes
2
answers
732
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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 ...
9
votes
1
answer
543
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Homotopy left-exactness of a left derived functor
Let
$$
F: \mathcal{C} \leftrightarrows \mathcal{D} :G
$$
be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors
$$
\mathbb{L}F: \mathrm{Ho}(\...
9
votes
1
answer
314
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A "small" definition of sub-(∞,1)-topoi
Suppose I have an $(\infty,1)$-topos $\mathcal{X}$ and a (small) set of maps $S$ in $\mathcal{X}$, which therefore generates an accessible localization $S^{-1}\mathcal{X}$. Is there any "small" ...
9
votes
1
answer
550
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Higher-dimensional version of the "Magic Cube Lemma" for homotopy pushouts/pullbacks
The "Magic Cube Lemma" is a surprising (to me) relationship between (homotopy) pushouts and (homotopy) pullbacks of spaces:
Consider a cubical diagram $I^3\to \mathcal{S}$ in the $\infty$-category of ...
9
votes
1
answer
442
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Base change in homotopy type theory
Recall that with the internal language of 1-toposes, we have the nice, basic, and useful result that geometric sequents are stable under base change along geometric morphisms: If $\varphi$ and $\psi$ ...