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Results tagged with model-categories
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user 57405
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.
28
votes
Accepted
Joyal's letter to Grothendieck
The letter may be found on Georges Maltsiniotis' webpage containing material related to Pursuing Stacks. (A direct link to the pdf.)
- 5,539
13
votes
Accepted
Non-Cartesian Monoidal Model Structure on a Slice Category
This construction came up in an Australian Category Seminar talk given by Ross Street last month, from which I will copy for 1. and 2. below. I'm afraid I don't know a reference.
1. (monoidal struct …
- 5,539
12
votes
Accepted
Weak complicial sets: Are the morphisms too strict?
Indeed there is no such coherence result: it is false already for $2$-categories (see for instance Lemma 2 of this paper of Steve Lack). The solution to your troubling corollary is that the "correct" …
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11
votes
Accepted
Proof of existence of Joyal model structure via Cisinski theory?
Such a proof is given in Chapter 3 of Cisinski's book Higher categories and homotopical algebra, see Definition 3.3.7 and Theorem 3.6.1. (Note that Cisinski's proof uses as the interval object not the …
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8
votes
Accepted
Does the Dwyer-Kan model structure make dgCat a model $2$-category?
No. If dgCat were a model 2-category, then the 2-functor from dgCat to Cat that sends a dg-category $A$ to its underlying category (which has the same objects as $A$, and whose morphisms are the $0$-c …
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4
votes
A model category of abelian categories?
Expanding on my comment, there ought to be a finitary 2-monad $T$ on Cat with $\mathfrak{M} = T\text{-Alg}_s$ and $\mathfrak{A} = T\text{-Alg}$. If this is so, then all of your questions are answered …
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3
votes
Accepted
Fibrations of fibrant marked simplicial sets
Yes, this is true. There are various ways to prove this. Here's the shortest argument I can think of. One direction is easy to prove, so let's prove the other direction.
Let $U \colon \mathbf{sSet}^+ …
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