Questions tagged [infinity-topos-theory]

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4
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0answers
140 views

A couple of points in a proof about of $\infty$-toposes

I wanted to have a better understanding of the geometric interpretation of $\infty$-toposes, and in particular learn something about étale morphisms, but I got stuck trying to unravel two points in ...
2
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1answer
185 views

Abelian versions of straightening and unstraightening functors

Let $X$ be a quasi-category (an inner Kan complex), let $\mathfrak{C}(X)$ be its rigidification (its associated simplicial category). J. Lurie in "Higher Topos Theory" proved the following theorem 2.1....
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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 ...
7
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1answer
234 views

When is an $\infty$-categorical localization left exact?

Let $L: \mathcal C^\to_\leftarrow L\mathcal C : i$ be an adjunction with $i$ fully faithful. In ordinary category theory, $L$ is left exact iff the class of $L$-local morphisms is stable under base ...
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765 views

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 ...
7
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1answer
249 views

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 ...
6
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140 views

Is there an $\infty$-topos of monochromatic spaces?

Fix (a prime $p$ and) a chromatic height $h$. Recall that the Bousfield-Kuhn functor $\Phi_h: \mathcal M_h^f \to Sp_{T(h)}$ is monadic, where $\mathcal M_h^f \subseteq Top_\ast$ is a certain ...
5
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1answer
141 views

Are constant $\infty$-sheaves constant on connected components?

Let $C$ be an $\infty$-category endowed with a Grothendieck topology $J$ and consider the $\infty$-topos $\infty\text{Sh}(C, J)$. There is a natural geometric morphism to $\infty\text{Grpd}$ whose ...
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159 views

A categorial PCF theory?

I'm not an expert in PCF theory, so please forgive me if this question makes no sense. I'm looking for a categorial version of PCF theory. Specifically, if we replace $Set$ with another category, ...
12
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2answers
425 views

Examples of topos 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 ...
5
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1answer
243 views

Are semisimplicial hypercoverings in a hypercomplete $\infty$-topos effective?

A hypercovering in an $\infty$-topos $E$ is a simplicial object $U \in {\rm Fun}(\Delta^{\rm op},E)$ such that each map $U_n \to ({\rm cosk}_{n-1} U)_n$ is an effective epimorphism. Theorem 6.5.3.12 ...
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278 views

Roadmap to understand gerbe in the sense of Lurie’s Higher Topos Theory

Definition $7.2.2.20$ : Let $\mathfrak{X} $ be an $\infty$-topos. An $n$-gerbe on $\mathfrak{X}$ is an object in $\mathfrak{X}$ which is $n$-connective and $n$-truncated. Above is the definition of ...
18
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2answers
1k views

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 ...
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131 views

Stability of accessible $\infty$-categories under some operations

I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories. In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see ...
17
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1answer
373 views

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 ...
4
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1answer
218 views

A finite Whitehead Theorem for $\infty$-topos

Let's consider in an $\infty$-topos, we have an object $X$ of homotopy dimension $\leq n$ (in the sense of Lurie HTT), let $f: A\to B$ be an $n$-equivalence morphism. Can we conclude that $f$ induces ...
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2answers
410 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 ...
16
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1answer
513 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 ...
12
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1answer
362 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....
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452 views

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)\...
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89 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 $\...
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2answers
505 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 $\Phi$? ...
9
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2answers
471 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 ...
12
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1answer
302 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^...
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183 views

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 ...
4
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1answer
446 views

Is an ∞-topos of local homotopy dimension $\leq n$ of homotopy dimension $\leq n$?

[All references are wrt to Lurie's "Higher Topos Theory" in its latest online available version (March 10, 2012)] Definition 7.2.1.8: An ∞-topos $X$ is locally of homotopy dimension $\leq n$ if there ...
17
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1answer
740 views

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 ...
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1answer
194 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 $\mathbf{E}$ ...
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2answers
212 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 $\...
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1answer
710 views

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 $\text{Sh}(\mathcal C)$ ...
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107 views

Some operations on categories - nomenclature question

Suppose that we have a "diagram of categories", i.e. a map from some small category $J$, viewed as a 2-category (with trivial two-morphisms) to the category of categories, $j\mapsto \mathcal{C}_j.$ ...
5
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2answers
428 views

Monomorphisms, epimorphisms, (co-)images and factorizations in $\infty$-categories

Several of the many notions that don't work the same way when passing to $\infty$-categories are the ones mentioned in the title. I'm trying to understand the conceptual picture around these notions ...
6
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1answer
294 views

How does Cech cohomology get around computing a delooping?

I'm trying to understand how the nlab's definition of cohomology works concretely. There, it is (very convincingly) claimed that every incarnation of "cohomology" in mathematics is a special case of ...
19
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279 views

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 ...
6
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1answer
338 views

Understanding model independently the equivalence of two ways of obtaining homotopy types from categories

It is well known that any homotopy type can be obtained as the classifying space of a ($1$-)category. The classifying space of a category $\mathcal{C}$ can be interpreted in at least two ways: We ...
2
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1answer
173 views

What is a non-example of and $(\infty,1)$-topos where disjointness fails?

One of the axioms for $(\infty,1)$-topoi is that the topos is disjoint, meaning that we have the following pullback diagram $$ \begin{matrix} 0 & \rightarrow & A \\ \downarrow & & \...
2
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1answer
165 views

Symmetric spectra for simplicial sheaves

Let $C$ be a site. Localizing either the projective or injective model structure on simplicial presheaves at the Cech nerves of covers in $C$ gives a new model structure on simplicial presheaves which ...
11
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1answer
668 views

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 (...
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1answer
1k views

Model independent proof of colimit formula for left Kan extensions

I am interested in finding a proof of the colimit formula for left Kan extensions $(\infty,1)$-categories which does not rely on a chosen model. (By a model inpendent proof I mean a proof which uses ...
13
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1answer
831 views

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 ...
9
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1answer
448 views

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'...
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135 views

Finite covers in complex analytic geometry

Given a complex manifold or complex analytic space, one has the standard notion of open set. There are two different Grothendieck topologies that one can define using this notion, one where covers ...
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2answers
683 views

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 ...
2
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1answer
167 views

Can hypercompletion be an essential localization?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory, i.e. its inclusion functor has a left-exact left ...
6
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1answer
458 views

Can hypercomplete objects be coreflective?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory. Can it ever happen that this subcategory is ...
4
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1answer
292 views

higher Eilenberg-Moore-toposes of left exact derived comonads

It is a classical fact (e.g. here) that for an (accessible) right adjoint comonad on a (sheaf) topos, its Eilenberg-Moore category of coalgebras is itself a (sheaf) topos. I suppose this remains true ...
4
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1answer
350 views

Is the infinity-groupoid of a finite CW complex finitely-presented?

An infinity-groupoid is finitely-presented when it is equivalent to the free infinity-groupoid on a finite family of generators, possibly of different dimensions. Is the infinity-groupoid of a finite ...
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1answer
898 views

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 ...
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552 views

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" $...
3
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1answer
634 views

unbounded derived category of a $\infty$-topos

In HTT(Higher Topos Theory) Remark7.3.1.19, it it sketched that the proper base change theorem for $\infty$-topos implies the usual proper base change theorem in (unbounded) derived category. However, ...