49
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 ...
38
votes
Accepted
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 ...
- 53.9k
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 ...
- 11.8k
38
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 ...
- 24.8k
35
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
...
- 26.4k
30
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 ...
- 16.1k
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 ...
- 24.5k
25
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 ...
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}...
- 113k
23
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'...
- 5,739
20
votes
Accepted
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 ...
- 32.6k
20
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 ...
- 129k
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 ...
- 38.7k
17
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 ...
- 34.5k
16
votes
Accepted
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 ...
- 17.4k
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 ...
- 38.7k
15
votes
Accepted
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 ...
- 34.5k
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 ...
- 14.7k
14
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 ...
- 2,726
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 ...
- 13.1k
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 ...
- 5,215
13
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.
- 9,732
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 ...
- 11.6k
13
votes
Accepted
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 ...
- 50k
13
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 ...
- 32.9k
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 ...
- 38.7k
12
votes
The category theory of $(\infty, 1)$-categories
Browsing through the old unanswered questions, I've come across this one, which happily can be partially answered now by the work of Riehl and Verity (Zhen Lin will be aware of this, which is why I'll ...
12
votes
Accepted
Fibrant-cofibrant models of Eilenberg-MacLane spectra
The following four categories are models for spectra with Eilenberg-MacLane spectra of the desired form.
Kan's category of semisimplicial spectra [1]
The category $\mathbf{Sp}^\mathbb{N}(\mathbf{\...
- 136
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