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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 …

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 …

I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do …

There are many situations which arise where one might consider different Model categories with the same underlying category. For example in (left) Bousfield localization you start with a model categor …

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial set …

So in Julie Bergner's work on $(\infty, 1)$-categories arXiv:0610239, she considers several model categories which model $(\infty, 1)$-categories, which are known to be equivalent. I'm guessing that t …

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 …

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 …

If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the le …

Given a suitable nice symmetric monoidal category $C$, symmetric monoidally enriched, tensored, and cotensored over a symmetric monoidal category $S$, and an operad $\mathcal{O}$ in $S$, we can consid …

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 …