The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor $sGrp\leftarrow sPtSet:\Omega\Sigma$. This can be proved by observing that
- looping and delooping $B:sGrp \rightleftarrows sSet_0:\Omega$ is a homotopy equivalence between reduced simplicial sets and simplicial groups.
- Looping and suspension $\Sigma:sPtSet \rightleftarrows sSet_0:\Omega$ are adjoints.
The content of this is that Milnor's construction $F[X]$ of the reduced free simplicial group on a pointed simplicial set $X$ is of homotopy type $\Omega\Sigma X$.
Similarly, the left adjoint to the inclusion $sMon\hookrightarrow sPtSet$ from the category of simplicial monoids is also homotopy equivalent to $\Omega\Sigma$. The content of this is that Jame's construction $J(X)$ of the reduced free simplicial monoid on a pointed simplicial set $X$ is also of homotopy type $\Omega\Sigma X$.
I would like to know the left adjoints, up to homotopy, of the following inclusion functors:
- $sGrp\hookrightarrow sGrpd$ where $sGrpd$ is the category of simplicial groupoids
- $sMon\hookrightarrow sCat$ where $sMon$ is the category of simplicial monoids
This information would provide alternative model theoretic proofs of the corresponding algebraic constructions of simplicial objects, as with Milnor and James' constructions.
