41
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 ...

- 15.8k

31
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
...

- 26k

28
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.

- 19k

25
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/...

- 15.8k

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 ...

- 3,397

23
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'...

- 60.9k

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

### 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) ...

- 122k

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 ...

- 15.8k

20
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 $\mathcal{C}$ ...

- 17.1k

19
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 ...

- 29.3k

18
votes

### Model independent proof of colimit formula for left Kan extensions

Dominic Verity and I have been working to develop model independent foundations of $(\infty,1)$-category theory. Our aim at present isn't to cover all models of $(\infty,1)$-categories but only the "...

- 1,529

17
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

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 $(\...

- 26.3k

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....

- 5,649

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

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 ...

- 6,596

16
votes

### 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 ...

- 10.2k

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 ...

- 17.1k

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 ...

- 37.9k

15
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 ...

- 26.3k

14
votes

Accepted

### Is there an $(\infty,2)$-category with morphisms given by $D^b\text{Coh}$?

For smooth projective varieties it is known that
$$D^b_\infty Coh(X\times Y)\simeq Hom(D^b_\infty Coh(X), D^b_\infty Coh(Y))$$ compatibly with the composition you describe
(note here $D^b_\infty Coh$...

- 22.2k

14
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 ...

- 26k

14
votes

Accepted

### Quasicategories for non-simplicial model categories

It's not quite in the literature, but there is a fully explicit construction that avoids hammock localisation or any kind of fibrant replacement: by a recent result of Lennart Meier, a certain "double ...

- 13.6k

14
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 ...

- 8,886

14
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.
...

- 1,891

14
votes

Accepted

### Deformation of a diagram preserve the homotopy limit

This is false.
Consider the two $C_2$-spaces $S^{2\sigma}$ and $S^2$, where $\sigma$ is the sign representation and $S^V$ denotes the one-point compactification. Then the two underlying spaces are the ...

- 12.8k

13
votes

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

Besides Gaitsgory-Rozenblyum (http://www.math.harvard.edu/~gaitsgde/GL/), you might try looking at Lee Cohn's work (http://arxiv.org/abs/1308.2587), which establishes some equivalence between the ...

- 1,126

13
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^∞...

Community wiki

13
votes

### What does the homotopy coherent nerve do to spaces of enriched functors?

Have you looked in the paper by Jean-Marc Cordier and myself:
Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1 - 54?
We defined a simplicial set of coherent natural ...

- 8,667

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