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).

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

is a Quillen equivalence (Prop. below) 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.

The construction generalized from simplicial groups to simplicial groupoids, where the groupoidal $\overline{W}$ is accompanied by a corresponding left adjoint known as the Dwyer-Kan loop groupoid-construction, and together they constitute a Quillen equivalence between then model structure on simplicial groupoids and the classical model structure on simplicial sets, exhibiting simplicial groupoids as an equivalent presentation of classical homotopy theory.

## Definition

### In components

###### Definition

(standard universal principal simplicial complex)
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

(1)\begin{aligned} & 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. \end{aligned}
• degeneracy maps are given by

(2)\begin{aligned} & 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-i}), \, e, \, g_{n-i-1}, \, \cdots, \, g_0 \big) \,, \end{aligned}

where $e$ denotes the respective neutral element.

This carries a $G$-action by left multiplication on the top degree component:

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

###### Remark

It is the straightforward simplicial incarnation of the left $G$-action (3) that singles out the model $W G$ (Def. ) for the universal simplicial principal space. For another model with an alternative good property see at groupal model for universal principal simplicial complex.

###### Definition

(standard simplicial classifying complex)
For $G$ a simplicial group, its standard simplicial classifying complex is the quotient of $W G$ (Def. ) by its $G$-action (3)

(4)$\overline{W} G \;\coloneqq\; (W G) / G \,.$

The corresponding quotient coprojection, whose fiber is, manifestly, $G$

(5)$\array{ G &\xhookrightarrow{fib}& W G \\ && \big\downarrow \\ && \overline{W} G \mathrlap{\; \coloneqq (W G)/G} }$

is known as the standard model for the simplicial $G$-universal principal bundle (see below).

This means, under the isomorphism

$\big( \overline{W}G \big)_{n} \;\coloneqq\; (W G)_n/G_n \;\simeq\; G_n/G_n \times G_{n-1} \times \cdots \times G_0 \;\simeq\; G_{n-1} \times \cdots \times G_{0} \,,$

that the above face maps (1) and degeneracy maps (2) of $W G$ imply the following structure maps on the simplicial classifying complex:

$\overline{W}G \;\;\; \in \; SimplicialSets$
• underlying sets are

$(\overline{W}G)_n \;\coloneqq\; G_{n-1} \times \cdots \times G_1 \times G_0 \,;$
• face maps are given by:

(6)\begin{aligned} & d_i \big( g_{n-1}, \cdots, g_0 \big) \\ & \;\coloneqq\; \left\{ \array{ \big( g_{n - 2}, \, \cdots, \, g_0 \big) & \text{if} & i = 0 \\ \big( 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} & 0 \lt i \lt n \\ \big( d_{n-1}(g_{n-1}), \, \cdots, \, d_1(g_1) \big) & \text{if} & i = n \mathrlap{\,;} } \right. \end{aligned}
• degeneracy maps are given by:

(7)\begin{aligned} & s_i(g_{n-1}, \cdots, g_0) \\ & \;\coloneqq\; \left\{ \array{ \big( e, \, g_{n-1}, \, \cdots, \, g_0 \big) & \text{if} & i = 0 \\ \big( s_{i - 1}(g_{n-1}), \, \cdots, \, s_0(g_{n-i}), \, e, \, g_{n-i-1}, \, \cdots, \, g_0 \big) & \text{if} & 0 \lt i \mathrlap{\,.} } \right. \end{aligned}

(This goes back to MacLane 1954, p. 3, Kan 1958, Def. 10.3; the above follows Goerss & Jardine 1999/2009, p. 269.)

More generally:

###### Definition

For $\mathcal{G}$ a Dwyer-Kan simplicial groupoid, $\overline{W}\mathcal{G}$ is the simplicial set with

1. set of vertices (0-cells) equal to the set of objects of $\mathcal{G}$,

2. set of $n$-simplices for $n \geq 1$ equal to sequences of morphisms of the form

$x_n \overset{f_{n-1}}{\longrightarrow} x_{n-1} \overset{f_{n-2}}{\longrightarrow} x_{n-2} \to \cdots \to x_1 \overset{f_{0}}{\longrightarrow} x_0$

where

• $x_i \in Obj(\mathcal{G})$

• $f_k \in Mor(\mathcal{G}_k)$

[Dwyer & Kan 1984, §3.2]

###### Remark

(décalage)
Conversely, comparison of Def. with Def. shows that $W G$ is obtained from $\overline{W} G$ by shifting down in degree and discarding the 0th face- and degeneracy maps:

$(W G)_{n} \;\simeq\; (\overline{W}G)_{n+1}$
\begin{aligned} d^{W G}_i & \;=\; d^{ \overline{W}G }_{i+1} \\ s^{W G}_i & \;=\; s^{\overline{W}G}_{i + 1} \mathrlap{\,.} \end{aligned}

One refers to this relation as saying that $W G$ is the décalage of $\overline{W}G$, in the form

$W G \;=\; Dec^0\big( \overline{W}G \big) \,.$

See the example there.

###### Example

(low-dimension cells of $W G$)
Unwinding the definition of the face maps (1), one finds that the generic 1-simplex in $W G$ (Def. ) looks as follows:

while the generic 2-simplex in $W G$ looks as follows:

###### Example

(simplicial classifying space of an ordinary group)
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

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

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 (4) 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

(simplicial classifying spaces are Kan complexes)
The underlying simplicial set of any simplicial classifying $\overline{W}G$ (Def. ) is a Kan complex.

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

This follows as the combination of the following facts:

1. every simplicial group is fibrant in the projective model structure on simplicial sets (this Prop.);

2. $\overline{W}(-)$ is a right Quillen functor from there to the injective model structure on reduced simplicial sets (Prop. );

3. every injectively fibrant reduced simplicial set is a Kan complex (this Prop.).

###### Proposition

For a simplicial group action of $G$ on a Kan complex $X$, the canonical coprojection from the Borel construction to the simplicial classifying space is a Kan fibration:

$(W G \times X)/G \xrightarrow{ \;\in Fib \; } \overline{W}G \,.$

###### Proof

By the fact (this Prop.) that the Borel construction is a right Quillen functor from the model structure on simplicial group actions to the slice model structure of the classical model structure on simplicial sets over the simplicial classifying space; see this Example.

In particular:

###### Proposition

The coprojection $W G \overset{}{\longrightarrow} \overline{W}G$ (5) is a Kan fibration.

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

This is the special case of Prop. for $X = G$ equipped with the left multiplication action on itself, using again that the underlying simplicial set of any simplicial group is a Kan complex (this Prop.):

$\array{ G &\longrightarrow& W G \\ && \big\downarrow{}^{\mathrlap{\in Fib}} \\ && \overline{W}G } {\phantom{AAAAAA}} \simeq {\phantom{AAAA}} \left( W G \times \left( \array{ G \\ \downarrow \\ \ast } \right) \right) \big/ G \,.$

###### Proposition

The simplicial set $W G$ is contractible.

(e.g Goerss & Jardine 09, Sec. V Lemma 4.6 (p. 270), see also the discussion at décalage)

###### Proposition

The simplicial homotopy groups of $\overline{W} G$ are those of $G$, shifted up in degree by one:

$\pi_n(G) \;\simeq\; \pi_{n+1}\big(\overline{W}(G)\big) \,.$

###### Proof

By Prop. the universal simplicial principal bundle (5) is a Kan fibration between Kan complexes (by Prop. and this Prop.). Therefore this is a homotopy fiber sequence (by this Prop.)

$G \xrightarrow{hofib} W G \xrightarrow{\;\;q\;\;} \overline{W}G \,.$

This implies a long exact sequence of homotopy groups of the form

$\cdots \xrightarrow{\;} \pi_{n+1}(W G) \xrightarrow{\;\;} \pi_{n+1}(\overline{W}G) \xrightarrow{\;\;} \pi_n(G) \xrightarrow{\;\;} \pi_n(W G) \xrightarrow{\;} \cdots \,.$

But Prop. says that $\pi_n(W G)$ is trivial for all $n$, so that this collapses to short exact sequences:

$0 \xrightarrow{\;\;} \pi_{n+1}(\overline{W}G) \xrightarrow{\;\simeq\;} \pi_n(G) \xrightarrow{\;\;} 0$

which exhibit the claim to be proven.

###### Proposition

Let $\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2$ be a homomorphism of simplicial groups which is a Kan fibration. Then the induced morphism of simplicial classifying spaces $\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2$ is a Kan fibration if and only if $\phi$ is a surjection on connected components: $\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1)$.

(Goerss & Jardine, Ch. V, Cor. 6.9, see the proof here)

### 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).

### Quillen equivalence between simplicial groups and reduced simplicial sets

###### Proposition

(Quillen equivalence between simplicial groups and reduced simplicial sets)
The simplicial classifying space-construction $\overline{W}(-)$ (Def. ) is the right adjoint in a Quillen equivalence between the projective model structure on simplicial groups and the injective model structure on reduced simplicial sets.

$Groups(sSet)_{proj} \underoverset {\underset{\overline{W}}{\longrightarrow}} {\overset{ \Omega }{\longleftarrow}} {\;\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;\;} (sSet_{\geq 0})_{inj} \,.$

The left adjoint $\Omega$ is the simplicial loop space-construction.

### 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.

The idea of constructing $\overline{W}$ using the bar construction is due

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 $\overline{W}$ is defined explicitly for simplicial groups and the adjunction between simplicial groups and reduced simplicial sets is explicitly spelled out:

• 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 groupoids, now right adjoint to the Dwyer-Kan loop groupoid-construction:

Generalization to simplicial presheaves: