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 ...

Community wiki

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 ...

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 ...

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 ...

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
...

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 ...

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 ...

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 ...

Community wiki

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}...

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'...

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 ...

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 ...

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 ...

Community wiki

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 ...

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 ...

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 ...

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 ...

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 ...

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
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 ...

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
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 ...

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.

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 ...

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 ...

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 ...

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 ...

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 ...

Community wiki

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{\...

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