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Questions tagged [higher-category-theory]

For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

8
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3answers
287 views

Cofiber of the inclusion of an $E_0$-algebra $M$ into the free $E_k$-algebra generated by it

Let $\mathcal{C}$ be the $E_k$-monoidal $\infty$-category of left modules over a fixed connective $E_{k+1}$-ring spectrum $A$. Suppose that $M$ is an object of $\mathcal{C}$ which is an $E_0$-algebra, ...
10
votes
1answer
519 views

On HTT's Lemma 3.3.4.1

While studying the book Higher Topos Theory I have encountered some difficulty with Lemma 3.3.4.1, which says that the pullback along a cartesian fibration of a map q such that $q^{op}$ is cofinal is ...
7
votes
1answer
353 views

Is the derived category of $A$-dg-modules as a dg-category coincide with the ordinary definition of derived category?

Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We ...
3
votes
1answer
269 views

Ge-categories and applications

Ge-categories, i.e., categores enriched over groupoids (these are 2-categories where the set of morhisms $HOM(a,b)$ has a groupoid structure) seem to be useful in homotopy theory. Question: What are ...
8
votes
0answers
241 views

Bar construction and the $\infty$-categorical Barr-Beck theorem

I am studying the proof of the $\infty$-categorical version of the Barr-Beck theorem in Lurie's Higher Algebra, but there is a step of the proof that is puzzling me. In Lemma 4.7.3.13, a simplicial ...
3
votes
1answer
262 views

Dual objects in the $\infty$-category of spectra

We say (according to https://ncatlab.org/nlab/show/%28infinity%2Cn%29-category+with+duals) that a symmetric monoidal $(\infty,1)$ category $\mathcal{C}$ has duals if its homotopy category $h\mathcal{C}...
9
votes
3answers
583 views

What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?

The $(\infty, 1)$ category $Sp$ of spectra as defined by Lurie in Higher Algebra has the structure of a symmetric monoidal category. Although I know the definition of symmetric monoidal category in ...
15
votes
0answers
962 views

What to expect from spectral algebraic geometry

So I've been trying to learn some derived algebraic geometry, and I've chosen to approach the subject from the perspective of spectral or "brave new" algebraic geometry. Without having to go through ...
5
votes
1answer
123 views

Characterizing freely adjoining K-filtered colimits as K-continuous presheaves

Note: This question has a 1-categorical and an $\infty$-categorical versions. I am interested in the $\infty$-categorical one so this is the version that I write below, but an answer for the 1-...
12
votes
2answers
855 views

$K$-theory backwards

Let $R$ be a ring. The $K$-theory of $(Mod(R)^{f.g.proj},\oplus)$ is obtained by first throwing out non-isomorphisms and then group completing. What happens if these steps are reversed? That is, ...
3
votes
0answers
128 views

Can the cobordism hypothesis be formulated as a statement about adjoint functors?

I would like to formulate the cobordism hypothesis for general tangential structure as a statement about adjoint $(\infty,1)$-functors. For a space $Y$ with an action of $O(n)$ let $X=Y\times_{O(n)} ...
4
votes
0answers
333 views

The lisse-etale site and derived algebraic geometry

If one reads say Olsson's book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff's of the ...
5
votes
3answers
304 views

(Co)limits of locally cartesian closed categories

Do locally cartesian closed $\infty$-categories form a presentable $\infty$-category? It seems like they should, and that the inclusion $\text{LCC}\rightarrow\text{Cat}$ preserves limits and maybe ...
10
votes
1answer
219 views

Criterion for homotopy pullback square of simplicial categories

Assume given a pullback square of simplicial categories $$\begin{array}[c]{ccc} A&{\rightarrow}&B\\ \downarrow&&\downarrow\\ C&{\rightarrow}&D. \end{array}$$ Suppose further ...
10
votes
2answers
351 views

The cofibration/fibration $\leftrightarrow$ epi/mono confusion

A recent question Why do we need model categories? reminded me of this long-standing confusion of mine -- I mentioned it in an answer there, and then decided to ask a separate question about it. I ...
38
votes
4answers
4k views

What's there to do in category theory?

Disclaimer: I posted this question on MSE only a few days ago; and received very few comments. I know that the etiquette is to wait a bit more than that before moving a post from MSE to MO, but I ...
46
votes
0answers
1k views

What did Gelfand mean by suggesting to study “Heredity Principle” structures instead of categories?

Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the following ...
3
votes
1answer
94 views

Reedy-indexed diagrams in higher categories

It is well known that for a functor $F: \mathcal R \to \mathcal M$, where $\mathcal R$ is a Reedy category and $\mathcal M$ is suitably bicomplete, the following decomposition determines the structure ...
2
votes
1answer
97 views

Pseudo or lax algebras for a 2-monad, reference request

I would like to find explicit definitions of pseudo, or even lax, algebras for a 2-monad, and their lax morphisms, with all the coherence diagrams included. Alternatively, coherent lax algebras for ...
4
votes
1answer
181 views

The conceptual difference in notations of Cat

There have been some places in which a conceptual difference (that of criteria of identification, etc) is avoided in the categorical notation, namely Let $Cat$ be at the same time the 2-category of ...
8
votes
1answer
565 views

The universal property of the unseparated derived category

In Appendix C of his book in progress Spectral Algebraic Geometry, Lurie defines the unseparated derived category $\check{{\cal D}}({\cal A})$ (see Definition C.5.8.2 loc.cit) associated to a ...
5
votes
1answer
225 views

Monadic interpretation of coalgebras over operads

The structure of an algebra $A$ over a operad $O$ is encoded by an operad morphisms from $O$ to $\{Hom(A^{\otimes k},\, A)\}_{k}$. The same structure can be stored using the structure $M_OA\to A$ of ...
1
vote
0answers
99 views

Synthetic type theory for virtual double category and its higher categories

For some monad T on a virtual equipment, the paper A unified framework for generalized multicategories by Cruttwell and Shulman (arXiv:0907.2460) proposes the normalized T-monoid. Another paper, by ...
2
votes
1answer
202 views

Coproducts of weak equivalences

In a model category $\mathcal{C}$ admitting a forgetful functor to simplicial sets, is the coproduct of weak equivalences a weak equivalence? Say even just coproducts indexed by $\mathbf{N}$. A ...
5
votes
0answers
90 views

Cartesian liftings in double categories

The question: I wonder whether the following definition, or something similar, has appeared somewhere (see below for motivations). Any reference or pointer is welcome! (In what follows, I denote ...
4
votes
0answers
237 views

Categorical formalism for higher non-abelian group cohomology / obstruction theory for gerbes?

I'm sure this is very well known but I haven't found any references for this searching the internet so hence the question: What's the neat abstract framework for obstruction theory for non-abelian ...
9
votes
0answers
300 views

Is every weak $\infty$-bicategory (à la Lurie) an $\infty$-bicategory?

In Definition 4.1.1 of $(\infty,2)$-Categories and the Goodwillie Calculus I, Lurie defines a weak $\infty$-bicategory to be a scaled simplicial set that has the extension property with respect to ...
2
votes
1answer
312 views

Criteria for being an $\infty$-category?

Let $\mathcal{C}$ be a simplicial category, such that for any two objects $X, Y\in\mathcal{C}$, $\text{Hom}_{\mathcal{C}}(X,Y)$ is a simplicial commutative monoid. Is the simplicial nerve $\text{N}(\...
19
votes
1answer
674 views

What's with equivariant homotopy theory over a compact Lie group?

For some reason -- I'm not quite sure why -- I've developed the impression that I'm supposed to "tiptoe" around equivariant homotopy theory over groups that are not finite. Should I? Let me explain. ...
5
votes
2answers
285 views

Type Theory to Study $(\infty,n)$-Categories and $(r,n)$-Categories

The recent paper "A Type Theory for Synthetic $\infty$-Categories" proposes the syntax as the theory of the strict interval. In principle, any other suitable theory could be used instead. For instance,...
10
votes
1answer
211 views

The category theory of Span-enriched categories / 2-Segal spaces

The category $\mathsf{Span}$ of spans of sets is symmetric monoidal closed under $\times$ (the cartesian product from $\mathsf{Set}$, which is not the categorical product in $\mathsf{Span}$), ...
9
votes
1answer
411 views

Relationship between synthetic differential geometry and differential cohesion?

I'm a big fan of synthetic differential geometry (or smooth infinitesimal analysis), as developed by Anders Kock and Bill Lawvere. It's a beautiful and intuitive geometric theory, which gives ...
12
votes
1answer
556 views

Did Lurie's model of a homotopy coherent idempotent change?

In the published version of HTT (Def 4.4.5.2) and on the nlab one finds one definition of a split homotopy coherent idempotent corepresented by a quasicategory $Idem^+_\mathrm{old}$. A few months ago, ...
7
votes
0answers
141 views

Generalized (co)-presheaves for Generalized Multicategories?

A $T$-multicategory in the sense of Crutwell-Shulman, where $T$ is a monad on a virtual double category $C$ is a monoid in the "horizontal kleisli category", i.e., an object of objects $O$, a ...
5
votes
1answer
244 views

Maximal Cisinski model structure on simplicial sets

This is a very simple question coming from the observation that every (pre)sheaf category has the maximal Cisinski model structure on it. This is the Cisinski model structure with the smallest class ...
61
votes
2answers
4k views

What is Homology anyway?

Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
8
votes
0answers
163 views

Pushouts of commutative pseudomonoids

Let $(\mathcal{C},\otimes)$ be a symmetric monoidal bicategory. Assume that $\mathcal{C}$ has bicategorical coequalizers which are preserved by $\otimes$ in each variable. My question is if then the ...
3
votes
0answers
47 views

Can one relate $K_0$ of an $A_\infty$-category $\mathcal A$ to $K_0(Fun_{A_\infty}(\mathcal A, \mathcal A))$?

For an $A_\infty$-category $\mathcal A$, one defines the group $K_0(\mathcal A)$ by $$K_0(\mathcal A) := \mathbb Z \operatorname{Ob} \operatorname{Tw} \mathcal A / \left<[A]+[B]-[C]\right>$$ ...
19
votes
1answer
534 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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vote
0answers
92 views

Localization of a 2-category

I am looking for a basic reference about localization of 2-categories, possibly avoiding the full formalism of n-categories.
3
votes
0answers
245 views

Linear $\infty$-categories $\mathrm{QC(X)}$ and $\mathrm{Perf(X)}$ of a “derived” stack $\mathrm{X}$

For each scheme or algebraic stack their $\infty$-category of quasicoherent sheaves (resp., perfect complexes) on it is $k$-linear for a commutative ring $k$. That is (by a recent result of L.Cohn), ...
4
votes
1answer
112 views

Quasi-categorical factorization system induced on $X^S$

Let $X$ be an $\infty$-category (I am happy to assume it is bicomplete and stable, but this should not be necessary) and consider a factorization system $F=(E,M)$ on $X$ (this is defined in Section 24 ...
5
votes
1answer
186 views

Geometric realization of the mapping stack

Some background and notation Let $Sh_{\infty}(Cartsp)$ be the infinity category of smooth simplicial sheaves on the site of cartesian spaces (convex open subsets of $\mathbb{R}^n$ and smooth maps ...
6
votes
2answers
312 views

Classification of weak 3-groups

Weak 2-groups can be classified by the data $(\pi_1,\pi_2, t, \omega)$, where $\pi_1$ is a group, $\pi_2$ an Abelian group, $t: \pi_1 \to Aut(\pi_2)$, and $\omega \in H^3(B\pi_1,\pi_2)$. I wonder do ...
8
votes
2answers
679 views

Grothendieck toposes and logic

I am searching results in which one can extract logic information from a topological (Grothendieck topos) perspective (such as Gödel's Completeness Theorem and Deligne's Theorem ("theorem by P. ...
8
votes
0answers
236 views

Practice of higher categories - giving rigorous constructions

1) Let $\mathcal{C}$ be a monoidal $\infty$-category, $A$ an algebra object in $\mathcal{C}$, and $M$ a left $A$-module in $\mathcal{C}$. So, in Lurie's formalism, $\mathcal{C}$ is encoded by some ...
11
votes
2answers
393 views

n-categorical pasting diagram overview

There has been a lots of approaches to the notion of n-categorical diagram and n-categorical pasting diagram: Street : "Parity complexes" Power : " An n-categorical pasting theorem" Johnson : "The ...
5
votes
0answers
325 views

Question about $(\infty, 1)$-category of $(\infty, 1)$-categories

I would like to realize $QCat:=(sSet)_{\text{Joyal}}$ as an $(∞,1)-category$ in the sense of Lurie (i.e. weak Kan complex). I would like to do this in the following way: 1) First since $QCat$ is a ...
8
votes
1answer
506 views

Derived noncommutative geometry includes derived, or spectral algebraic geometry?

Let $k$ be a commutative ring. In derived noncommutative (algebraic) geometry a "noncommutative space over $k$" is a $k$-linear $\mathrm{DG}$-category. This is motivated by the fact that homological ...
27
votes
2answers
1k views

What is the relationship between connective and nonconnective derived algebraic geometry?

"Derived algebraic geometry" usually means the study of geometry locally modeled on "$Spec R$" where $R$ is a connective $E_\infty$ ring spectrum (perhaps with further restrictions). Why "connective", ...