50 votes

What is modern algebraic topology(homotopy theory) about?

While I think that Andre is right in saying that homotopy theory (or algebraic topology) is ready to study everything that fits into the framework of abstract homotopy theory, some things have still ...
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 ...
David White's user avatar
  • 29.4k
38 votes
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The homotopy category is not complete nor cocomplete

I'll work in the based category, and consider $S^1$ as $\{z\in\mathbb{C}:|z|=1\}$. Consider the maps $$\text{point}\xleftarrow{}S^1\xrightarrow{f}S^1, $$ where $f(z)=z^2$. Suppose that there is a ...
Neil Strickland's user avatar
38 votes

Why do we need model categories?

Model categories capture the idea that in many cases you resolve an object by an equivalent object that is better behaved. The standard example is replacing a chain complex by a chain complex of ...
Lennart Meier's user avatar
37 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 ...
Charles Rezk's user avatar
  • 26.6k
31 votes
Accepted

Why is Voevodsky's motivic homotopy theory 'the right' approach?

(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in). I'm going to try with a very naive answer, although I'm not sure I understand ...
Denis Nardin's user avatar
  • 16.2k
26 votes

What is modern algebraic topology(homotopy theory) about?

Abstract homotopy theory allows one to use the tools of homotopy theory (e.g. inverting weak equivalences, computing homotopy colimits, doing Bousfield localization, taking fibrant and cofibrant ...
26 votes
Accepted

Counter-example to the existence of left Bousfield localization of combinatorial model category

A surprisingly effective way to construct counterexamples in model category theory is to just write down all the objects and morphisms involved and try to give the resulting (finite!) diagram the ...
Reid Barton's user avatar
  • 24.8k
24 votes

Why do we need model categories?

This answer is an elaboration on Dylan's comments. 1) Let us define a homotopy theory to be a pair $(C, W)$, where $C$ is a category and $W$ is some class of morphisms called weak equivalences. (Let'...
AAK's user avatar
  • 5,841
23 votes

The homotopy category is not complete nor cocomplete

Here's a class of counterexamples for the pointed homotopy category of connected CW complexes (so even this restriction does not save you). Let $hCW_{\ast}$ denote this category, and let $\pi_{\bullet}...
Qiaochu Yuan's user avatar
21 votes

Why is Voevodsky's motivic homotopy theory 'the right' approach?

I just want to point out, with regards to your second question, that the fact that the $\mathbb P^1$ is not equivalent to a simplicial complex homotopic to the sphere built out of affine spaces (or ...
Will Sawin's user avatar
  • 135k
20 votes
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Is the homotopy category of an abelian model category abelian?

No. The projective model structure on chain complexes of modules over a ring is an abelian model category, and the homotopy category is the derived category, which is never abelian unless the ring is ...
Jeremy Rickard's user avatar
18 votes

What is modern algebraic topology(homotopy theory) about?

In response to Ryan Budney's comment, let me try to say something about topological data analysis, and other recent applications of algebraic topology outside of traditional mathematics. Applied ...
18 votes

Why do we need model categories?

Today we understand that what we are really interested in when we talk about "homotopy theory" are in the end "$\infty$-categories". In fact I have even heard some peoples claim that maybe in the ...
Simon Henry's user avatar
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17 votes
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When is a functor a right derived functor?

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories with enough injective objects. Let me use the notation $D^+(\mathcal{A})$ and $D^+(\mathcal{B})$ to denote the stable $\infty$-categories ...
Jacob Lurie's user avatar
  • 17.5k
17 votes
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When did the Joyal model structure on simplicial sets originate?

Here is what André Joyal wrote in an email to me: No, I have not discovered the model structure for quasi-categories in the 1980's. I became interested in quasi-categories (without the name) around ...
Dmitri Pavlov's user avatar
16 votes

Why do we need model categories?

Model categories provide a powerful framework for commuting (homotopy) limits and colimits, and, more generally, for commuting left adjoint functors and (homotopy) limits, as well as right adjoint ...
Dmitri Pavlov's user avatar
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 ...
Dan Petersen's user avatar
  • 39.2k
15 votes

Non-small objects in categories

In the opposite category of the category of sets, and of many algebraic categories, the only small objects are the empty set and the singleton. A conceptual reason for this is Freyd's (or Gabriel and ...
Kevin Arlin's user avatar
  • 2,964
14 votes

On model categories where every object is bifibrant

Another example is given by Strom's model structure on topological spaces where Fibrations: Hurewicz fibrations, Weak equivalences : (strong) homotopy equivalences.
David C's user avatar
  • 9,792
14 votes
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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 ...
Dylan Wilson's user avatar
  • 13.2k
14 votes

When did the Joyal model structure on simplicial sets originate?

My suspicion is now that it was some time between 2004 and 2006. I have a lot more citations in this blog post, but I note three points, in reverse chronological order: Multiple experts are referring ...
David Roberts's user avatar
  • 33.8k
13 votes

What is modern algebraic topology(homotopy theory) about?

I'm going to give an algebraist's perspective. First let's discuss homological algebra (which has roots in topology). There's a quote (attributed, I think, to Connes) that a great mystery of ...
13 votes
Accepted

Non-Cartesian Monoidal Model Structure on a Slice Category

This construction came up in an Australian Category Seminar talk given by Ross Street last month, from which I will copy for 1. and 2. below. I'm afraid I don't know a reference. 1. (monoidal ...
Alexander Campbell's user avatar
13 votes

Non-small objects in categories

In the category $\mathsf{Top}$ of topological spaces and continuous maps the only $\lambda$-presentable objects are discrete spaces. This appears 1.14(6) in Locally presentable and Accessible ...
Ivan Di Liberti's user avatar
13 votes

Correspondence between classes of model categories and classes of $\infty$-categories

Regarding (1) : A) Every model category has an associated $\infty$-category, obtained for example by taking the Dwyer-Kan localization at the class of all equivalence, (But there are other more ...
Simon Henry's user avatar
  • 39.9k
13 votes
Accepted

Homotopy coherent colimits in chain complexes

The result is not only true for simplicial model categories, but for plain combinatorial model categories too - this is Higher Algebra 1.3.4.25.. In fact, for this you can reduce to the case of ...
Maxime Ramzi's user avatar
  • 13.3k
13 votes
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Two $\infty$-categories of chain complexes

The two categories you describe are not equivalent in the fashion that you hope. No matter what kind of simplicial category $C$ is, the quasicategory $N_\Delta(C)$ has an explicit description of its ...
Tyler Lawson's user avatar
  • 51.1k
13 votes

sSet-enriched categories, quasi-categories and the model-independent theory

This has not been done, and there are good reasons for it: While $sSet$-enriched categories are indeed very good to easily get examples of $\infty$-categories, they are very bad at understanding what ...
Simon Henry's user avatar
  • 39.9k

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