Questions tagged [infinity-categories]

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Is there an accepted definition of $(\infty,\infty)$ category?

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...
Theo Johnson-Freyd's user avatar
59 votes
1 answer
32k views

If I want to study Jacob Lurie's books "Higher Topoi Theory", "Derived AG", what prerequisites should I have?

I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's ...
Chuang 's user avatar
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30 votes
4 answers
3k views

How should I think about presentable $\infty$-categories?

Let me start out with a confession. I have never cared much for set-theoretic size issues, for they seem not to cause much trouble in my day-to-day mathematical life. Despite that, I have always been ...
Patriot's user avatar
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30 votes
3 answers
4k views

DG categories in algebraic geometry - guide to the literature?

Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric ...
Saal Hardali's user avatar
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28 votes
5 answers
5k views

What is the motivation for infinity category theory?

To my understanding, most mathematical theories can be simply understood in the view point of Category theory and its derivative theories. But what exactly is the motivation to study infinity category ...
tryst with freedom's user avatar
27 votes
4 answers
3k views

(∞, 1)-categorical description of equivariant homotopy theory

I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak ...
Reid Barton's user avatar
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26 votes
1 answer
3k views

Why stable $\infty$-categories?

I begin by saying that while I understand what a triangulated / derived category is pretty well, I know nothing about Higher Algebra stuff and not even $\infty$-categories. I've heard some people say ...
Gabriel's user avatar
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25 votes
3 answers
3k views

Grothendieck's Homotopy Hypothesis - Applications and Generalizations

Grothendieck's homotopy hypothesis, is, as the $n$lab states: Theorem: There is an equivalence of $(∞,1)$-categories $(\Pi⊣|−|): \mathbf{Top} \simeq \mathbf{\infty Grpd}$. What are the ...
user avatar
25 votes
2 answers
3k views

The Dold-Thom theorem for infinity categories?

Let $\mathcal{M}$ denote the category of finite sets and monomorphisms, and let $\mathcal T$ denote the category of based spaces. For a based space $X \in \mathcal T$, one has a canonical funtor $S_X ...
Eric Finster's user avatar
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25 votes
2 answers
3k views

generalisations of the Seifert-van Kampen Theorem?

I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available. My attention was ...
Ronnie Brown's user avatar
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25 votes
1 answer
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Infinity-categorical analogue of compact Hausdorff

Recently I became through this mathoverflow question aware of the article Codensity and the ultrafilter monad by Tom Leinster. There he shows that the ultrafilter monad on the category $\mathrm{Set}$ ...
Lennart Meier's user avatar
25 votes
0 answers
629 views

Chromatic Spectra and Cobordism

I apologize in advance, if some of the things I've written are incorrect. The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...
Nerses Aramian's user avatar
25 votes
0 answers
1k views

$\infty$-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 ...
Simon Henry's user avatar
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24 votes
1 answer
2k views

Condensed criterion for sheafiness of adic spaces

Multiple times in talks about condensed mathematics (e.g. the Masterclass talks, Clausen's RAMpAGe talk), it is stated that the derived structure sheaf given by the condensed formalism "fixes&...
Jack J. Garzella's user avatar
23 votes
3 answers
3k views

how to make the category of chain complexes into an $\infty$-category

I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes. Has anyone ever written down ...
Yosemite Sam's user avatar
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23 votes
1 answer
857 views

What is higher equivariant homotopy?

In Lurie's "Survey of elliptic cohomology" it is claimed that there exists some mystical "2-equivariant homotopy theory" for elliptic cohomology. The classical equivariant elliptic cohomology is ...
Anton Fetisov's user avatar
21 votes
2 answers
3k views

Stable infinity categories vs dg-categories

What is the relation between dg-categories and stable $\infty$-categories? Given a dg-category one can form its dg-nerve and get a $\infty$-category (which will be stable if the dg-category is?). ...
Jan Weidner's user avatar
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21 votes
2 answers
1k views

$\infty$-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 ...
Simon Henry's user avatar
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21 votes
1 answer
1k views

Grothendieck derivators vs $\infty$-categories

I have some questions on derivators and $(\infty,1)$-categories, I would be grateful if someone could help me. Is there some problems that $(\infty,1)$-categories/derivators can resolve but ...
Amos Kaminski's user avatar
21 votes
1 answer
843 views

Natural examples of $(\infty,n)$-categories for large $n$

In Higher Topos Theory, Lurie argues that the coherence diagrams for fully weak $n$-categories with $n>2$ are 'so complicated as to be essentially unusable', before mentioning that several dramatic ...
21 votes
1 answer
700 views

The derived category does not satisfy descent - example

One motivation for studying infinity categories is that the derived category does not satisfy Zariski descent, although the infinity categorical version does. I would like to see an example of Zariski ...
Mathmop's user avatar
  • 313
20 votes
2 answers
3k views

What is a symmetric monoidal $(\infty,n)$-category?

This question arose from reading Jacob Lurie's "Classification of topological field theories" paper. In that paper, he uses complete $n$-fold Segal spaces as a model for $(\infty,n)$-categories, but ...
Johannes Ebert's user avatar
20 votes
2 answers
733 views

Limitations on model-categorical presentations

In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is ...
Mike Shulman's user avatar
20 votes
0 answers
344 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 ...
Mathemologist's user avatar
19 votes
2 answers
1k views

Internal categories in simplicial sets

Is there a model structure (or more generally a homotopy theory) on the category of internal categories in simplicial sets, which presents the theory of $(\infty,1)$-categories? Note that this ...
Mike Shulman's user avatar
19 votes
2 answers
1k 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 ...
Simon Henry's user avatar
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19 votes
1 answer
1k views

On a question motivating Lurie's treatment of formal moduli problems

Lurie, in his ICM 2010 proceedings paper Moduli Problems for Ring Spectra (pdf), says that one motivating problem (for him, I presume, and possibly others) for thinking about formal moduli problems is ...
David Roberts's user avatar
  • 33.8k
19 votes
0 answers
833 views

What is the status of the cobordism hypothesis?

Let $\mathscr{C}$ be a symmetric monoidal (weak) $n$-category. A framed extended TQFT of dimension $n$ with values in $\mathscr{C}$ is a symmetric monoidal functor from the framed bordism $n$-category ...
Miguel I. Solano's user avatar
19 votes
0 answers
445 views

monoidal (∞,1)-categories from weakly monoidal model categories

In Higher Algebra section 4.1.7, Jacob Lurie proves that the underlying $(\infty,1)$-category of a monoidal model category is a monoidal $(\infty,1)$-category. Dominic Verity and Yuki Maehara have (...
Emily Riehl's user avatar
  • 1,559
19 votes
0 answers
883 views

Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ...
Dmitri Pavlov's user avatar
18 votes
5 answers
3k views

Categories which are not compactly generated

Do you know natural examples of triangulated categories (or [presentable] stable $\infty$-categories) which are not compactly generated? (ideally they'd be defined algebraically, but curious to hear ...
David Ben-Zvi's user avatar
18 votes
4 answers
2k views

Proof that the homotopy category of a stable $\infty$-category is triangulated

I've been looking at various general strategies for proving that some category is triangulated, and Lurie manages to prove that a huge class of interesting examples of categories that we know about ...
Dylan Wilson's user avatar
  • 13.2k
18 votes
1 answer
936 views

$(\infty,2)$-categories: current applications and future prospects

Lately there has been a lot of progress on the foundations of $(\infty,2)$-categories (for example, all currently-known models for them were shown to be equivalent and finally we have a construction ...
Emily's user avatar
  • 10.3k
18 votes
1 answer
496 views

How aggressive is the fibrant replacement of $\mathrm{Bord}_n$?

Lurie (On the Classification of Topological Field Theories), with some corrections by Calaque and Scheimbauer (A note on the $(\infty,n)$-category of cobordisms), famously constructed a symmetric ...
Theo Johnson-Freyd's user avatar
18 votes
2 answers
2k views

What is a homotopy between $L_\infty$-algebra morphisms

A $L_\infty$-algebra can be defined in many different ways. One common way, that gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...
Mark.Neuhaus's user avatar
  • 2,004
18 votes
1 answer
611 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 ...
Simon Henry's user avatar
  • 39.9k
18 votes
1 answer
541 views

Is there a cotangent bundle of a stable $\infty$-category?

Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following? When $C$ is the ...
David Treumann's user avatar
18 votes
0 answers
403 views

Constructible derived category and fundamental category

Introduction (may be skipped) Given a nice topological space $X$, the category of local systems (say over a field $k$) on it is equivalent to the category of representations of its fundamental ...
Jan Weidner's user avatar
  • 12.8k
17 votes
1 answer
2k views

Is the $\infty$-category of spectra “convenient”?

A 1991 paper of Lewis, titled “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$: There is a symmetric monoidal smash ...
Emily's user avatar
  • 10.3k
17 votes
2 answers
679 views

Homotopy theories of operads

I know of three homotopy theories of colored operads. The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak ...
Dmitry Vaintrob's user avatar
17 votes
1 answer
1k views

The category theory of $(\infty, 1)$-categories

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent ...
Zhen Lin's user avatar
  • 14.9k
16 votes
2 answers
3k views

Semi-simplicial versus simplicial sets (and simplicial categories)

Hi, Let's denote by "semi-simplicial set" a simplicial set without degeneracies, i.e. a contravariant functor from the category $\Delta_{inj}$ of finite linearly ordered sets and order preserving ...
Peter Arndt's user avatar
16 votes
2 answers
542 views

How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?

If $\mathcal{C}$ is a symmetric monoidal $(\infty,1)$-category with duals, then there should be a functor $$ d: \mathcal{C} \longrightarrow \mathcal{C}^{op} $$ such that $d(x)$ is dual to $x$ for ...
Jan Steinebrunner's user avatar
15 votes
3 answers
2k views

Does the classification diagram localize a category with weak equivalences?

Let $(C,W)$ be a category equipped with a subcategory of weak equivalences. Its "classification diagram" or "bisimplicial nerve" $N(C,W)$ is a bisimplicial set, for which $N(C,W)_n$ is the nerve of ...
Mike Shulman's user avatar
15 votes
1 answer
1k 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 ...
Daniel Grady's user avatar
15 votes
1 answer
626 views

Adjoint functor theorem for infinity categories

In HTT, a version of the adjoint functor theorem for (locally) presentable infinity categories is proven (Corollary 5.5.2.9). Is there a more refined version of this somewhere, which more closely ...
David Carchedi's user avatar
15 votes
2 answers
913 views

Are strict $\infty$-categories localized at weak equivalences a full subcategory of weak $\infty$-categories?

One has a nice "folk" model structure on strict $\infty$-categories due to Yves Lafont, Francois Metayer and Krzysztof Worytkiewicz whose notion of weak equivalences seem to be the notion of weak ...
Simon Henry's user avatar
  • 39.9k
15 votes
2 answers
976 views

Non-unique splittings of homotopy idempotents

By a homotopy idempotent I mean a map $f:X\to X$, where $X$ is a space, equipped with a homotopy $f\circ f \sim f$. In contrast to the situation in stable homotopy theory (where $X$ would be a ...
Mike Shulman's user avatar
15 votes
1 answer
2k views

$\infty$-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 $ExDisc_\kappa \to \mathsf{RTop}$ ...
Maxime Ramzi's user avatar
  • 13.3k
15 votes
1 answer
473 views

Well pointed endofunctors on $\infty$-categories

In $1$-category theory, a well pointed endofunctor of a category $C$, is an endofunctor $F:C \rightarrow C$ endowed with a natural transformation $\sigma : Id \rightarrow F$ such that the two natural ...
Simon Henry's user avatar
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