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Results tagged with model-categories
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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.
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
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
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
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