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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.
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
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
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
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
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
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