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One natural setting where the delooping operation $\def\B{{\sf B}}\B$ can be defined very quickly is Γ-spaces, described by Segal in Categories and cohomology theories. A Γ-space is simply a functor $ …

Since the question uses semisimplicial sets, it makes sense to point out the following rather elegant model for the classifying space $\def\B{{\sf B}}\B G$ as a semisimplicial set: declare the set of …

As already pointed out in the comments, such a simplicial set is highly nonunique. For example, in addition to the simplicial set $S$ described in the second paragraph one could take the wedge of sim …

Given that the map $\def\E{{\sf E}}\def\B{{\sf B}}\E G→\B G$ is the geometric realization of a simplicial covering map (namely, the nerve of the functor $\def\sq{/\!/} G\sq G→*\sq G$), the canonical t …

This is an answer to the edited question. First, observe that the composition of functors $\def\N{{\rm N}}\def\Sing{{\rm Sing}}\N∘\Sing$ in the main post computes the homotopy colimit of the simplicia …

For $\def\B{{\rm B}} \def\bB{{\bf B}} \def\Spin{{\rm Spin}} \def\String{{\rm String}} \B\Spin(n)$, simply equip the $n$-planes with a spin structure, as originally proposed by Stolz and Teichner. For …

The easiest way to construct an explicit contracting homotopy is to observe that EG is the geometric realization of the nerve of the groupoid G//G, which has G as its set of objects and exactly one mo …

The nth Deligne cohomology is defined as cohomology with coefficients in a truncated chain complex of sheaves of U(1)-valued differential forms: U(1)→Ω^1→Ω^2→⋯→Ω^n for some n≥0. Thus starting with an …