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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
the model structure on reduced simplicial sets: is is cofibrantly generated? Does the projective model structure on functors with values in it exist?
Never mind, Danny kindly points out that it follows easily from prop. A.2.6.13 of HTT.
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)
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?
added (here) statement of Goerss&Jardine V Cor. 6.9:
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 $\pi_1(f) \colon \pi_1(X) \twoheadrightarrow \pi_1(Y)$.
added (here) the following statement:
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.
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.
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