Let $\mathcal{G}$ be a strict topological $2$-group, i.e. a strict $2$-category with a single object, a space of invertible $1$-morphisms, a space of invertible $2$-morphisms and continuous structure maps. This corresponds to a topological crossed module. I would like a reference for the comparison of the following two ways of defining the classifying space of $\mathcal{G}$:
- The classifying space $\lvert \mathcal{G}(\ast,\ast)\rvert$ of the category $\mathcal{G}(\ast,\ast)$ of $1$-morphisms can be obtained as a monoid and $B\lvert \mathcal{G}(\ast,\ast)\rvert$ is a classifying space for $\mathcal{G}$.
- There is also a topological version of the Duskin nerve of $\mathcal{G}$, which is defined as a simplicial space $D_{\ast}\mathcal{G}$ with $D_0\mathcal{G}$ the single object of $\mathcal{G}$, $D_1\mathcal{G}$ the space of $1$-morphisms, $D_2\mathcal{G}$ the space of commutative triangles filled by a $2$-morphism, $D_3\mathcal{G}$ commutative tetrahedral diagrams and $3$-coskeletal from degree $3$ onwards. The geometric realisation $\lvert D_{\ast}\mathcal{G}\rvert$ of this simplicial space is the second construction I would like to consider.
The paper The classifying space of a topological $2$-group hints on page 19 at the fact that these two constructions are homotopy equivalent and that this result seems to be well-known among experts. I am wondering
Is there a reference for the homotopy equivalence of the above constructions somewhere in the literature?
The closest statements I could find are the corresponding statement for simplicial sets in On the geometry of 2-categories and their classifying spaces and the proof of Lemma 5.8 in Two-Categorial Bundles and Their Classifying Spaces, where the topological case is reduced to the discrete case using the singular simplex functor.
