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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.
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}^+ …
- 5,539
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 …
- 5,539
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 …
- 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" …
- 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
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 …
- 5,539
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.)
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