# Tag Info

### 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 ...
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

### 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
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
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
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
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