I am thinking about the classifying space construction.
The loop space functor Ω : Grpd₀ ⭢ Grpd sends a space to an internal pseudo group.
Elsewhere I have seen people write BΩX for a based connected CW-complex.
I am trying to think about how the construction of E(ΩX) should go, and then form B(ΩX) as a quotient of it by an action. ΩX is an internal pseudo group; it is associative and unitial up to homotopy.
I can't figure out how to describe a homotopy colimit of a pseudo group which will produce a contractible space. I think it should involve a geometric realization
$\text{coeq}$ $\amalg$ ΩX${}^{n}$ × Δ${}^{m}$ $\Rightarrow$ $\amalg$ ΩX${}^{n}$ × Δ${}^{n}$.
My question is whether this defines a pseudo functor which can play the role of E.