# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 21st 2011
• (edited Feb 21st 2011)

have created model structure on reduced simplicial sets

(I thought I had a vague memory that this or something similarly titled already existed, but apparently it didn’t).

Also added a little bit more detail on the Quillen equivalence with simplicial groups here and there, notably in the last section at groupoid object in an (infinity,1)-category

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 13th 2012

the model structure on reduced simplicial sets: is is cofibrantly generated? Does the projective model structure on functors with values in it exist?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 13th 2012

Never mind, Danny kindly points out that it follows easily from prop. A.2.6.13 of HTT.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 28th 2017

I have added to model structure on reduced simplicial sets statement and (immediate) proof that reduced suspension/looping (co-)restricts to a Quillen endo-adjunction on reduced simplicial sets (here)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 1st 2017

Some questions:

Is the model structure on reduced simplicial sets still right proper?

And how about 2-reduced simplicial sets and $n$-reduced simplicial sets? Will they still carry the analogous model structures? And will they be right proper?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJul 4th 2021
• (edited Jul 4th 2021)

Let $f \colon X \longrightarrow Y$ be a fibration in the model structure on reduced simplicial sets such that both $X$ and $Y$ are Kan complexes. Then $f$ is a Kan fibration precisely if it induces a surjection on the first simplicial homotopy group $\pi_1(f) \colon \pi_1(X) \twoheadrightarrow \pi_1(Y)$.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJul 4th 2021

Let $\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2$ be a homomorphism of simplicial groups which is a Kan fibration. Then the induced morphism of simplicial classifying spaces $\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2$ is a Kan fibration if and only if $\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1)$ is a surjection.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 4th 2021

added (here) the following statement and its proof:

Let $\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2$ be a homomorphism of simplicial groups which is a Kan fibration. Then the induced morphism of simplicial classifying spaces $\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2$ is a Kan fibration if and only if $\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1)$ is a surjection.