47
votes

Accepted

### Why stable $\infty$-categories?

I already answered some version of this question in this answer, but let me try to expand a bit on the concrete advantages in mathematical practice. For understanding the following you need to take on ...

36
votes

Accepted

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

Presentable $\infty$-categories can be understood without every having to think about cardinals. An $\infty$-category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$, where
...

28
votes

Accepted

### Condensed criterion for sheafiness of adic spaces

Thanks for the question! One interpretation of the conjecture is true. Let me elaborate. The following results are kind of implicit in some discussion towards the end of www.math.uni-bonn.de/people/...

28
votes

### What is the motivation for infinity category theory?

There are many motivations, but the short answer is that many desirable properties are only available in the world of $\infty$-categories. This is a wonderful miracle.
This is particularly visible ...

27
votes

### Why not a Stacks project for Homotopy Theory?

Easy. Who's gonna write it?
JDJ (Johan de Jong) has written almost the entire stacks project himself.

25
votes

Accepted

### Grothendieck derivators vs $\infty$-categories

The short answer is that $(\infty,1)$-categories are the "real" object of interest. Derivators are a tool for working with them that is sometimes (for some people) easier to use, but doesn'...

23
votes

Accepted

### DG categories in algebraic geometry - guide to the literature?

Let me try to address the bulleted questions and simultaneously advertise the G-R book everyone has mentioned. Since the main question was about literature, I could also mention Drinfeld's article "DG ...

23
votes

Accepted

### Functorial kernel in derived category

Let $\mathcal{C}$ be a stable $\infty$-category. Then $\mathcal{C}$ has a homotopy category $h \mathcal{C}$, which is triangulated. The collection of morphisms $f: X \rightarrow Y$ of $\mathcal{C}$ ...

22
votes

### Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

The question used the phrase "still needed." This is a very loaded term, and the answer that you give will depend very strongly on how you interpret it.
If, as Dylan does, we interpret this as asking ...

Community wiki

20
votes

Accepted

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

My colleague Dylan answered first (I keep telling him not to spend too much time on this toy :) but I both agree and disagree with his "Yes of course". The same words are used with different meanings ...

20
votes

### Why not a Stacks project for Homotopy Theory?

According to the about page of Kerodon:
Kerodon is an online textbook on categorical homotopy theory and related mathematics. It currently consists of a single chapter, but should grow (slowly) ...

20
votes

Accepted

### Describing fiber products in stable $\infty$-categories

In fact what you need is that your ∞-category is additive (i.e. that it has direct sums and that the canonical commutative monoid structure on the mapping spaces is group-like). All stable categories ...

19
votes

### Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

At the risk of starting some kind of (un?)civil war, let me expand on my comments.
First and foremost, let's address the interpretation of the question. The OP asks "do we need a model category of ...

Community wiki

18
votes

Accepted

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

Topological field theory (TFT) is a major client of higher-dimensional category theory. For $(\infty,
2)$-categories specifically, this specializes to two-dimensional TFT. One significant research ...

17
votes

Accepted

### Are n-truncated quasicategories a model for n-categories?

Let $C$ be an $\infty$-category, and $n\geq -1$. The following are equivalent:
$C$ is $n$-truncated.
The $\infty$-groupoids $\def\Map{\operatorname{Map}}\Map(\Delta^0,C)$ and $\Map(\Delta^1,C)$ are ...

17
votes

Accepted

### Proj construction in derived algebraic geometry

It is instructive to look at the simplest case of Proj: that of a free module, i.e. the projective space. Lurie works these out for us quite carefully in his Spectral Algebraic Geometry tome.
...

17
votes

Accepted

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

The completeness condition is not really about making things invertible which weren't already. It is about where the information about invertible morphisms is stored.
We can already see this with $(\...

17
votes

Accepted

### What is the free symmetric monoidal $\infty$-category on one object?

Yes, it is the same as $\mathbb{F}$.
As John Baez points out, it is the same as the free symmetric monoidal $\infty$-groupoid on one object. (This can also be seen by playing around with the adjoints ...

17
votes

Accepted

### Projective objects in the derived category of chain complexes

In a stable $\infty$-category, there are no nontrivial projectives. Of course, $0$ is always projective.
Now let $X$ be an arbitrary projective in some stable $C$, $X\simeq\Sigma \Omega X$ is a ...

16
votes

### DG categories in algebraic geometry - guide to the literature?

There are plenty of interesting dg-categories one can associate to a scheme. From the point of view of six functor yoga, these should be viewed as "categories of coefficients" for cohomology theories....

16
votes

### Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

The original question has been answered in the sense that there are people who are confident to prove every statement about spectra they care about without recourse to models or model categories of ...

Community wiki

16
votes

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

$\newcommand{\Vect}{\mathrm{Vect}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\cc}{\mathbf{C}}$Here are three (related) examples. The first one is simple (although not really related to physics): an $...

Community wiki

16
votes

Accepted

### When did the Joyal model structure on simplicial sets originate?

Here is what André Joyal wrote in an email to me:
No, I have not discovered the model structure for quasi-categories in the 1980's.
I became interested in quasi-categories (without the name) around ...

15
votes

### Why Grothendieck's Homotopy Hypothesis is so difficult?

First, let me remark that your question does not seem to be about the homotopy hypothesis, but about rectification. More specifically, the homotopy hypothesis concerns the question of whether (some ...

15
votes

Accepted

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

Let $\mathcal{X}$ denote the $\infty$-topos $\mathcal{S}_{/S^1}$, whose objects are spaces $X$ with a map $X \rightarrow S^1$. Then $\mathcal{X}$ is generated under colimits by the object given by the ...

15
votes

Accepted

### Homotopy theories of operads

The answer is yes: see the paper of Chu-Haugseng-Heuts, "Two models for the homotopy theory of ∞-operads", arXiv:1606.03826.
In brief, already Cisinski and Moerdijk ("Dendroidal sets and simplicial ...

15
votes

### What is the motivation for infinity category theory?

Let me try to give an algebraist's answer. I struggled with exactly this question for many months in the context of the stable module category, and it took a good many conversations with some of the ...

15
votes

### What is the motivation for infinity category theory?

"Why should one, for example, look at morphisms and so on?"
In some sense people have been looking at morphisms between morphisms as long as they've been looking at morphisms. At least, they'...

14
votes

### A "universally non Hypercomplete" $\infty$-topos via Goodwillie calculus?

As discussed in the comments, I'm writing here the proof of the following fact:
Let $\mathscr{X}_∞$ be the ∞-topos of sheaves on $\mathrm{FinTop}^{op}$ under the atomic topology (the topology where ...

14
votes

### Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

Given that Sp is better behaved than all other existing models of spectra
No, Sp is not better behaved than other models.
The reason that it seems to be is because all operations in Sp
(e.g., Ω^∞, Σ^∞...

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