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 ...
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.
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/...
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

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 ...
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 ...
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 $(\...
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 ...
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 ...
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 $D^b_\infty Coh$...
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 ...
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 ...
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. ...
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^∞...
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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