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Results tagged with model-categories
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user 43054
A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.
33
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 understan …
- 16.5k
3
votes
Accepted
Kan complexes and semigroups
(2) is true (and so (1) is false).
To see it, note that every horn $\Lambda^n_i\to S$ to a constant simplicial set must be constant, and so it can be filled by the constant horn $\Delta^n\to S$. Equi …
- 16.5k
4
votes
Accepted
Does the existence of a derived functor imply existence of model structure?
This is not quite the answer to your question as you pose it. I hope it will be useful anyway. By and large I am just expanding user337830 comments. Everything will use homological grading (what can I …
- 16.5k
6
votes
Geometric Meaning of Different K-theories
Fibrant replacement is essentially sheavification with respect to the corresponding topology. So étale K-theory is nothing more than the part of K-theory that satisfies étale descent. Concretely (and …
- 16.5k
8
votes
Accepted
Homotopy function complex for quasi-categories
Yes, you can compute the mapping spaces in ∞-categories by taking the biggest Kan subcomplex of the internal hom.
The trick is not to use the Joyal model structure, but instead the model structure on …
- 16.5k
3
votes
Accepted
A distinguished triangle of mapping spectra arising from recollement
I'm going to do a proof assuming we are in a stable $\infty$-category (I'm pretty sure this is almost equivalent to your "sufficiently rich" situation anyway). In your case $F=j_!j^!$ and $G=i_*i^*$.
…
- 16.5k