Let $X$ be a homotopy 1-type (a space with vanishing homotopy groups above degree one). It is a classical fact that $X$ can be recovered completely from its fundamental groupoid.

On the the other hand, there is an equivalence of categories between (unpointed) simplicial sets and simplicial groupoids which sends a simplicial set to its Dwyer-Kan loop groupoid: https://ncatlab.org/nlab/show/Dwyer-Kan+loop+groupoid

When $X$ is a homotopy 1-type, is the Dwyer-Kan loop groupoid equivalent to the image of the fundamental groupoid under the inclusion of groupoids in simpicial groupoids?


The answer is yes. This follows from checking that the Quillen equivalence $G: sSet ^\to_\leftarrow sGpd : \bar W$ corresponds when we pass to spaces to the expected equivalence between spaces and groupoids in spaces. Then because every $X \in sSet$ is cofibrant and $G$ is left Quillen, $GX$ is equivalent to what we expect, namely the path groupoid of $X$ (well, this is not strictly a groupoid, but there should be an equivalent thing which is a groupoid using Moore paths). Then you can reason as in spaces to conclude that $G$ restricts to an equivalence between 1-types and discrete groupoids.

To see that the equivalence is the expected one, consider the right adjoint $\bar W$. From the description in 3.2 of Dwyer and Kan's paper, one sees that $\bar W$ just takes the diagonal of a simplicial groupoid, regarded via the nerve as a bisimplicial set. This is equivalently the homotopy colimit of the corresponding functor $\Delta^{op} \to sSet$, which is indeed what we expect this adjunction to be doing.


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