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 ...

- 1,042

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{...

- 20.9k

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 ...

- 32k

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 ...

- 32k

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.

- 57.9k

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\...

- 12.5k

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 ...

- 32k

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