29 votes
Accepted

Higher Topos Theory- what's the moral?

It seems there are really two questions here: Why higher category theory? What questions can you pose without the language of higher category theory which are best answered using higher category ...
  • 52.6k
25 votes

Higher Topos Theory- what's the moral?

I'm going to give a general answer first, and a specific answer below. It is my opinion that when Jacob Lurie wrote Higher Topos Theory, he was channeling Grothendieck. When Grothendieck ...
  • 24.1k
20 votes

Higher Topos Theory- what's the moral?

Here is a belated addendum to the other answers. In short, I think the comparison with Grothendieck is on point. More specifically I want to argue that HTT accomplished for higher category theory what ...
  • 37.9k
7 votes
Accepted

Are free $E_\infty$-algebras in spaces closed under pullbacks? (as a full subcategory of all $E_\infty$-algebras in spaces)

Yes, the full subcategory $\mathrm{Mon}_{\mathbb{E}_\infty}(\mathcal{S})^{\rm free} \subset \mathrm{Mon}_{\mathbb{E}_\infty}(\mathcal{S})$ on those $\mathbb{E}_\infty$-monoids that are equivalent to a ...
7 votes
Accepted

Do the various notions of morphism spaces of simplicial sets agree on the underived level?

There is no hope of comparisons between $\pi_0\operatorname{Hom}^L_X(x,y)$ and $\pi_0\operatorname{Hom}^R_X(x,y)$ in general, even when $x=y$. (Note that the statement you cited says $\operatorname{...
6 votes
Accepted

Intuition for isofibrations in $\infty$-categories

As has been mentioned, there's no homotopically meaningful content to the notion of an isofibration, since every map of $\infty$-categories is equivalent to an isofibration. So the point is really ...
  • 2,172
6 votes
Accepted

limits and products stable $\infty$-category

In the case of an $\mathbb N^{op}$-indexed system specifically, the answer is yes (note that this is implicit in the Stacks project link you gave); in fact if you replace "fiber sequence" by ...
  • 10.2k
5 votes
Accepted

Does the forgetful functor from an over-$(\infty,1)$-category create weakly contractible limits?

You can also use Quillen's theorem A : to prove that $C \to D$ is initial, it suffices to show that for every $d\in D$, $C \times_D D_{/d}$ is weakly contractible. In the case where $C\to D$ is fully ...
  • 10.2k
4 votes

Is there any elementary text unravelling the definitions of 2-category, lax functor and lax transformation, allowing people who do not know in the first place what these things are to really understand the definitions?

The book 2-dimensional categories by Johnson and Yau does seem to satisfy all of the required conditions: it unravels all definitions in full detail, spelling out the details for 2-categories and ...
3 votes

When is a stable $\infty$-category the stabilization of an $\infty$-topos?

Here is another sort of constraint. I'll write $Sp(\mathcal C)$ instead of $Stab(\mathcal C)$. Claim: If $\mathcal A \simeq Sp(\mathcal X)$ for a nontrivial [1] $\infty$-topos $\mathcal X$, then for ...
  • 52.6k
3 votes

references to learn the general theory Lie $\infty$-groupoids and Lie $\infty$-algebroids

There is no introductory book on Lie ∞-groupoids and ∞-algebroids analogous to Mackenzie's book. The only book-length treatment that covers these subjects is Urs Schreiber's Differential cohomology in ...
3 votes
Accepted

Are there strictly connective smooth proper algebras over $\mathbb{F}_p$?

Here's an example: consider the $\infty$-category $Fun(\Delta^1,Perf(\mathbb F_p))$. It has two canonical generators $A= \mathbb F_p\to \mathbb F_p$ and $B=\mathbb F_p\to 0$; and I claim that $R= End(...
  • 10.2k
3 votes

Domains that may require a good categorical background

The blog by Bartosz Milewski comes to my mind. It focusses on the interplay between Haskell and category theory.
2 votes

Domains that may require a good categorical background

Besides the CT-functional programming connection, in recent years a field of "applied category theory" (ACT) has emerged that seeks to apply category-theoretic ideas to fields beyond the ...
2 votes
Accepted

Rotation axiom for the triangulation on a derivator

The $-\Sigma f$ bit follows from the following claims. Let me know if the last two seem to need further fleshing out. Keeping track of the differences in things that are all called $0$ is the ...
  • 2,172
1 vote

Density Theorem for $\infty$-Categories (HTT, Lemma 5.1.5.3)

I give different arguments for this kind of things here. In short, the main case is the one of the terminal presheaf: one proves by cofinality arguments that the colimit of the Yoneda embedding $S\to\...
1 vote

Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The paper The enriched Thomason model structure on 2-categories answers the question in the affirmative. Specifically, it constructs a model structure on the category of 2-categories and proves that ...

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