Contents

group theory

# Contents

## Idea

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).

$SimplicialGroups \underoverset {\;\;\;\underset{\overline{W}}{\longrightarrow}\;\;\;} {\;\;\;\overset{L}{\longleftarrow}\;\;\;} {\bot} SimplicialSets_{red}$

## Definition

### In components

###### Definition

For $G$ a simplicial group, one writes

$W G \;\in\; SimplicialSets$

for the the simplicial set whose

• underlying sets are

$(W G)_n \;\coloneqq\; G_{n} \times G_{n-1} \times \cdots \times G_0$
• face maps are given by

$d_i \big( g_n, g_{n-1}, \cdots, g_0 \big) \;\coloneqq\; \left\{ \array{ \big( d_i(g_n), \, d_{i-1}(g_{n-1}), \, \cdots ,\, d_0(g_{n-i}) \cdot g_{n-i-1}, \, g_{n - i - 2}, \, \cdots, \, g_0 \big) & \text{if} & i \lt n \\ \big( d_n(g_n), \, d_{n-1}(g_{n-1}), \, \cdots, \, d_1(g_1) \big) & \text{if} & i = n } \right.$
• degeneracy maps are given by

$s_i(g_n, g_{n-1}, \cdots, g_0) \;\coloneqq\; \big( s_i(g_n), \, s_{i - 1}(g_{n-1}), \, \cdots, \, s_0(g_{n-1}), \, e, \, g_{n-i-1}, \, \cdots, \, g_0 \big) \,,$

where $e$ denotes the respective neutral element.

This carries a $G$-action by left multiplication on the first factor:

(1)$\array{ G \times W G &\overset{}{\longrightarrow}& W G \\ \big(h_n, (g_n, g_{n-1}, \cdots, g_0)\big) &\mapsto& (h_n \cdot g_n, \, g_{n-1}, \cdots, g_0) \mathrlap{\,.} }$

The quotient of $W G$ by this $G$-action (1) is denoted

(2)$\overline{W} G \;\coloneqq\; W G / G$

and the quotient coprojection

$W G \longrightarrow \overline{W} G$

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)

###### Example

In the special case that

$G \in Groups \overset{const}{\hookrightarrow} SimplicialGroups$

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

$(W G)_n \;\coloneqq\; G^{\times_{n + 1}}$
• face maps are given by

$d_i \big( g_n, g_{n-1}, \cdots, g_0 \big) \;\coloneqq\; \left\{ \array{ \big( g_n, \, g_{n-1}, \, \cdots ,\, g_{n-i} \cdot g_{n-i-1}, \, g_{n - i - 2}, \, \cdots, \, g_0 \big) & \text{if} & i \lt n \\ \big( g_n, \, g_{n-1}, \, \cdots, \, g_1 \big) & \text{if} & i = n } \right.$
• degeneracy maps are given by

$s_i(g_n, g_{n-1}, \cdots, g_1) \;\coloneqq\; \big( g_n, \, g_{n-1}, \, \cdots, \, g_{n-1}, \, e, \, g_{n-i-1}, \, \cdots, \, g_0 \big) \,,$

This identifies

$W G \;=\; N \big( G \times G \rightrightarrows G \big)$

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:

$\overline{W}G \;\simeq\; N \big( G \rightrightarrows \ast \big) \,.$

### Via total simplicial sets

Equivalently, $\overline{W}(-)$ is the following composite functor:

$\overline{W} \;\colon\; [\Delta^{op}, Groups] \overset {\;\;[\Delta^{op}, \mathbf{B}]\;\;} {\longrightarrow} [\Delta^{op}, Groupoids] \overset {\;\;[\Delta^{op}, N]\;\;} {\longrightarrow} [\Delta^{op}, SimplicialSets] \overset {\;\;\sigma_\ast\;\;} {\longrightarrow} SimplicialSets \,.$

(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).

## Properties

In all of the following, $G$ is any simplicial group.

### Basic properties

###### Proposition

The simplicial set $\overline{W}G$ is a Kan complex.

(e.g. Goerss & Jardine 09, Sec. V Cor. 6.8 (p. 287))

###### Proposition

The coprojection $W G \overset{}{\longrightarrow} \overline{W}G$ is a Kan fibration.

(e.g Goerss & Jardine 09, Sec. V Lemma 4.1 (p. 270))

###### Proposition

The simplicial set $W G$ is contractible.

(e.g Goerss & Jardine 09, Sec. V Lemma 4.6 (p. 270))

### Classification of simplicial principal bundles

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).

### Slice model structure

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:

$\big( SimplicialSets_{Qu} \big)_{/\overline{W} G} \;\simeq_{Qu}\; G Actions(SimplicialSets)_{proj} \,.$

(Dror, Dwyer & Kan 1980) See there for details. This is a model for the general abstract situation discussed at ∞-action.

## References

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

• Saunders MacLane, Constructions simpliciales acycliques, Colloque Henri Poincaré 1954 (pdf) (See, in particular, §3.)

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

• Daniel Kan, Sections 10-11 in: On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53 (jstor:1970042)

The left adjoint simplicial loop space functor $L$ is also discussed by Kan (there denoted “$G$”) in

• Daniel M. Kan, §7 of: A combinatorial definition of homotopy groups, Annals of Mathematics 67:2 (1958), 282–312. doi.

The Quillen equivalence was established in

• Dan Quillen, Section 2 of: Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (jstor:1970725)

Textbook accounts:

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.