# Questions tagged [higher-category-theory]

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

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votes

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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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**2**answers

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

**4**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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", ...