# Kan's loop group construction

I'm looking for a good place to read about the loop group construction $G : \mathbf{sSet_0} \to \mathbf{sGrp}$ taking a reduced simplicial set $X$ and producing a simplicial group $GX$.

I would also like a good reference for the functor $\overline W :\mathbf{sGrp} \to \mathbf{sSet_0}$, right adjoint to $G$.

$G$ and $\overline W$ are described by Kan in "On Homotopy and c.s.s. Groups" (1958) and in Goerss and Jardine's "Simplicial Homotopy Theory" Chap. 5. However, Kan's article uses a slightly different definition from Goerss-Jardine, as well as pre-model-category-theoretic language. The description in Goerss-Jardine seems quite messy to me and seems to require more/better appreciation of cocycles in simplicial groups than I have.

Also, what is a good name for $\overline W (-)$? "Simplicial classifying space"? Would $B(-)$ then be a better notation? Similarly would $\Omega(-)$ be a better notation for $G(-)$?

• I like Stevenson's "Décalage and Kan's simplicial loop group functor" and some references therein. You can also look at May's "simplicial objects in algebraic topology". I also believe they're discussed in Curtis' "simplicial homotopy theory" survey paper. – Bruno Stonek Mar 19 '16 at 9:28