# Equivalence between the Dwyer-Kan loop groupoid and the fundamental groupoid

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?

## 1 Answer

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.