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Results tagged with model-categories
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user 184
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.
73
votes
Do we still need model categories?
Here are some rough analogies:
Model Category :: $(\infty, 1)$-category
Basis :: Vector space
Local coordinates :: Manifold
I especially like the last one. When you do, say, differential geometry …
- 27.5k
49
votes
What are surprising examples of Model Categories?
The category of sets admits precisely nine model category structures, no more no less.
I learned this fact from Tom Goodwillie's comments on a different MO question. It always shocks people when I m …
- 27.5k
16
votes
Accepted
Derived categories and homotopy categories
Yes. The former is a special case of the latter. There is a model category structure on the category of (say bounded) chain complexes of objects in your given abelian category. The weak equivalences a …
- 27.5k
11
votes
Accepted
Is the simplicial nerve a localization?
This is not true. Here is a counter example. We let $\mathcal{C}_*$ be the following simplicial category. It has two objects 0 and 1. Their only endomorphisms are the identity. There are no morphisms …
- 27.5k
4
votes
Accepted
Is a left Bousfield localization of simplicial presheaves a locally cartesian closed model c...
Section 2 of this paper of Rezk addresses exactly the question of when the localization by S yields a Cartesian model category. For that the relevant property is that that if you take the product of a …
- 27.5k