62
votes
Accepted
What is homology anyway?
Let's take coefficients in a field $k$, for simplicity.
On 2): the singular cohomology of a topological space $X$ is the dual of its singular homology, almost by definition. But if $X$ is a space for ...
- 17.6k
59
votes
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
The MathSciNet review (by Julie Bergner) of Simpson's book:
Homotopy theory of higher categories,
New Mathematical Monographs, 19. Cambridge University Press, Cambridge, 2012,
has a note about the ...
- 2,443
52
votes
Accepted
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
Here is my guess. To compare spaces with their notion of strict $\infty$-groupoids (in which everything is strict except inverses) Kapranov and Voevodsky use an intermediate category of Kan ...
- 9,246
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 ...
- 16.2k
40
votes
Accepted
Mark Hovey's open problems in the theory of model categories
I am a former student of Mark Hovey's, and during grad school, I wrote a document giving an update on the status of the 13 problems (as of 2012 or 2013, I guess). I just briefly went through it a ...
- 29.8k
39
votes
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
It's been more than a year and a half since I asked this question and I had a lot of thought about it so I decided I will post my own answer.
First I entirely agree with Yonatan that the main problem ...
- 40.2k
38
votes
What's there to do in category theory?
There is a majestic paper by Mac Lane
MacLane, Saunders. "Possible programs for categorists." Category Theory, Homology Theory and their Applications I. Springer, Berlin, Heidelberg, 1969. 123-131.
...
38
votes
What parts of the theory of quasicategories have been simplified since the publication of HTT?
In Higher Topos Theory, Lurie accomplishes many things. Let me highlight a few:
A study of the Joyal model structure and comparison to the Bergner model structure.
A study of cartesian fibrations and ...
- 61.6k
38
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
...
- 26.7k
38
votes
Should every modern day mathematician care about category theory?
When I was young I didn’t like sheaves or cohomology, so wanted to find something that was algebraic but didn’t involve too much sheaves or cohomology. I didn’t really need to know much about either ...
36
votes
Accepted
Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme
The problem is that the question is highly dependent on the definition of $n$-groupoids. The notion of strict $n$-groupoid is very clear and precise but we know very well (and Grothendieck knew that) ...
- 40.2k
36
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 ...
- 61.6k
33
votes
Accepted
What is the relationship between connective and nonconnective derived algebraic geometry?
Here is an example of a nonconnective $E_\infty$ ring spectrum which, I think, illustrates a key problem. (A more extensive discussion of this phenomenon occurs in Lurie's DAG VIII and in a paper by ...
- 51.5k
33
votes
Accepted
What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?
Lurie characterizes the symmetric monoidal structure on $\mathsf{Sp}$ by a universal property (HA.4.8.2.19): it is uniquely determined up to a contractible space of choices by the property that $S^0$ ...
- 13.2k
31
votes
Accepted
Spectral algebraic geometry vs derived algebraic geometry in positive characteristic?
I'll try to answer this question from the topological viewpoint. The short summary is that structured objects in the spectral setting have cohomology operations and power operations, which forces ...
- 5,550
31
votes
Accepted
Why are quasi-categories better than simplicial categories?
As a preface, I think that this question should be viewed as analogous to "what are the advantages of ZFC over type theory" or vice versa. We're talking about foundations -- in principle, it ...
- 61.6k
30
votes
Accepted
Errata on Rezk's paper
It looks like I completely missed this.
Here's what I guess happens: although the original 2.19 was wrong, there is a weaker version that is true (I'll just state it for simplicial sets): If $X$ ...
29
votes
Accepted
Derived categories and $\infty$-categories necessary for condensed mathematics
There are several questions (implicit) here.
In the texts as they are written, how much knowledge on derived categories (as triangulated categories, or as stable $\infty$-categories) is assumed?
...
- 19.9k
29
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 ...
- 29.8k
28
votes
Should every modern day mathematician care about category theory?
I say many (most?) mathematicians with thriving research careers completely ignore large parts of mathematics in their work. Probably, they don't even remember what they learned in some of their ...
27
votes
What is homology anyway?
For a long time (and still today), I very much shared the confusion of the OP. I think Jacob Lurie gives a very clear take on the standard perspective, but Mike Shulman does have a very valid ...
- 19.9k
26
votes
Is Mac Lane still the best place to learn category theory?
I just reviewed what I firmly believe will be the book that will replace MacLane as The Gold Standard for introductions to category theory for graduate students: Category Theory in Context by Emily ...
26
votes
What is homology anyway?
I generally think about the relationship differently than Jacob, probably because I'm coming from an algebraic topology background rather than an algebraic geometry one. I would say that if $\mathcal{...
- 65.1k
26
votes
Should every modern day mathematician care about category theory?
You can look at the edit history of this post to see previous versions, which took a different tack whose thread I have honestly lost. I want to take a different tack, though.
What makes this question ...
25
votes
Accepted
What are _all_ of the exactness properties enjoyed by stable $\infty$-categories?
I don't think this "finite limits and finite colimits coincide" business can be taken very far. If you take any small category $S_0$ you can add an initial and a terminal object to form $S = \mathrm{...
- 5,219
25
votes
Accepted
Deligne's doubt about Voevodsky's Univalent Foundations
It is a bit difficult to understand what he is asking. The already-linked nForum discussion includes some clarification about his example, which at the meeting took us a while to figure out.
More ...
- 65.1k
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'...
- 65.1k
25
votes
Accepted
Is every category a localization of a poset?
Yes, this is true. It follows form the work of Barwick and Kan on relative categories as a model for $\infty$-categories.
The idea is similar to how Thomason's work shows that every homotopy type can ...
- 27.2k
24
votes
Accepted
Is it always possible to write a scheme as a colimit of affine schemes?
Yes, this is just a basic fact in category theory, if interpreted correctly. For $C$ any category, and $F$ any preheaf on $C,$ $F$ is the colimit in presheaves of the diagram $C/F \to C \stackrel{y}{\...
- 15.3k
24
votes
Accepted
How do $\infty$-categories allow us to do descent on the derived level?
Let $X$ be a topological space covered by open sets $U$ and $V$.
Let $\mathscr{F}$ and $\mathscr{G}$ be complexes of sheaves defined on $U$ and $V$, respectively. Suppose you are given an isomorphism $...
- 17.6k
Only top scored, non community-wiki answers of a minimum length are eligible