4
votes
1answer
77 views

Model structure on sSet

Which is the model structure on $ \text{sSet} $ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $ \text{Cat} $ (of small categories) by the adjunction ...
4
votes
0answers
71 views

The two-sided simplicial bar construction is Reedy-cofibrant

Let $\mathcal{M}$ be a simplicial model category and let $\mathbb{C}$ be a small category. For a diagram $F : \mathbb{C} \to \mathcal{M}$ and a weight $G : \mathbb{C}^\mathrm{op} \to \mathbf{sSet}$, ...
4
votes
2answers
115 views

The empty set in homotopy theoretic terms (as a simplicial set/top. space)

I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my ...
3
votes
2answers
206 views

Kan fibrations and surjectivity

I have a basic question on the usual model structure on simplicial sets. What is the relation between being a Kan (trivial maybe ?) fibration and surjectivity ? Surjectivity here means either ...
2
votes
0answers
104 views

left inverse to trivial fibration is trivial cofibration

It is claimed that in the model category of simplicial sets (with usual model structure), a trivial fibration $X \to Y$ has a section, which is a trivial cofibration. Now, I see that there is a ...