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For $G$ a simplicial group, there is a reduced simplicial set, traditionally denoted $\overline W G$ and called the classifying space or classifying complex of $G$, which is a model for the delooping of $G$ and such that the functor $\overline{W}(-)$ is right adjoint to the standard simplicial loop space-construction $G$ (here denoted by $L$ to avoid a clash of notations).
This pair of adjoint functors
is a Quillen equivalence between the model structure on simplicial groups and the model structure on reduced simplicial sets, modelling looping and delooping of homotopy types in simplicial homotopy theory.
For $G$ a simplicial group, one writes
for the the simplicial set whose
underlying sets are
face maps are given by
degeneracy maps are given by
where $e$ denotes the respective neutral element.
This carries a $G$-action by left multiplication on the first factor:
The quotient of $W G$ by this $G$-action (1) is denoted
and the quotient coprojection
is known as the standard model for the simplicial $G$-universal principal bundle (see below).
(due to MacLane 54, p. 3, Kan 58, Def. 10.3, see also Goerss & Jardine 09, p. 269)
In the special case that
is an ordinary discrete group regarded as a simplicial group (which is constant as a functor on the opposite simplex category) the definitions in Def. reduce as follows:
The simplicial set $W G$ is that whose
underlying sets are
face maps are given by
degeneracy maps are given by
This identifies
with the nerve of the action groupoid of $G$ acting on itself by right multiplication (isomorphic to the pair groupoid on the underlying set of $G$):
Finally this means that the simplicial classifying complex (2) of an ordinary group is isomorphic to the nerve of its delooping groupoid:
Equivalently, $\overline{W}(-)$ is the following composite functor:
(Stevenson 11, Lemma 15 following John Duskin, see also NSS 12, Def. 3.26)
Here:
$\mathbf{B} \;\colon\; Groups \to Groupoids$ forms the one-object groupoid with hom-set the given group (the delooping groupoid);
$N \;\colon\; Groupoids \longrightarrow SimplicialSets$ is the nerve construction;
$\sigma_\ast$ is the total simplicial set-functor (right adjoint to pre-composition with ordinal sum).
In all of the following, $G$ is any simplicial group.
The simplicial set $\overline{W}G$ is a Kan complex.
The coprojection $W G \overset{}{\longrightarrow} \overline{W}G$ is a Kan fibration.
The simplicial set $W G$ is contractible.
The object $\overline{W}G$ serves as the classifying space for simplicial principal bundles (May 67, §21, Goerss & Jardine 09, Section V, Thm. 3.9, see also NSS 12, Section 4.1).
The slice model category of the classical model structure on simplicial sets over the simplicial classifying complex $\overline{W}G$ is Quillen equivalent to the Borel model structure for $G$-equivariant homotopy theory:
(Dror, Dwyer & Kan 1980) See there for details. This is a model for the general abstract situation discussed at ∞-action.
The idea of constructing $\overline{W}$ using the bar construction is due to Eilenberg and MacLane, who apply it to simplicial rings with the usual tensor product operation:
This was also later discussed in
The first reference where $\bar W$ is defined explicitly for simplicial groups and the adjunction between simplicial groups and reduced simplicial sets is explicitly spelled out is
The left adjoint simplicial loop space functor $L$ is also discussed by Kan (there denoted “$G$”) in
The Quillen equivalence was established in
Textbook accounts:
Peter May, p. 87-88 in: Simplicial objects in algebraic topology, University of Chicago Press 1967 (ISBN:9780226511818, djvu, pdf)
Paul Goerss, J. F. Jardine, Section V.4 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)
Streamlining:
Identification of the slice model structure over $\overline{W}G$ with the Borel model structure:
Generalization to simplicial presheaves:
Last revised on June 6, 2021 at 04:24:55. See the history of this page for a list of all contributions to it.