nLab
simplicial group

Idea

Recall that a simplicial set is a combinatorial model for a topological space. This relation is most immediate when the simplicial set is in fact a Kan complex (an ∞-groupoid).

A simplicial group is a simplicial set with the structure of a group on it. It turns out that this necessarily means that it is also a Kan complex. Therefore a simplicial group is

• an ∞-groupoid with an extra group structure on it;

• a model for a topological space with a group structure.

Accordingly (as discussed at group) a simplicial group $G$ gives rise to

• a one-object $\mathrm{\infty }$-groupoid $BG$ whose explicit standard realization as a simplicial set is denoted $\overline{W}G$

• an $\mathrm{\infty }$-groupoid $EG$ whose explicit standatd realization as a simplicial set (even a simplicial group, again) is denoted $WG$

• such that there is a fibration

  $$\begin{array}{ccc}EG& :=& WG\\ \downarrow & & \downarrow \\ BG& :=& \overline{W}G\end{array}$$\array{ \mathbf{E} G &:=& W G \\ \downarrow && \downarrow \\ \mathbf{B} G &:=& \bar W G }

  which is the universal G-bundle.

Definition

A simplicial group, $G$, is a simplicial object in the category of groups.

Notation

The category of simplicial groups is the category of functors from ${\Delta }^{\mathrm{op}}$ to Grp. It will be denoted $SimpGrp$.

Properties

Theorem (J. C. Moore)

The simplicial set underlying any simplicial group (by forgetting the group structure) is a Kan complex.

In fact, not only are the horn fillers guaranteed to exist, but there is an algorithm that provide explicit fillers. This implies that constructions on a simplicial group that use fillers of horns can often be adjusted to be functorial by using the algorithmically defined fillers. An argument that just uses ‘existence’ of fillers can be refined to give something more ‘coherent’.

Proof

Let $G$ be a simplicial group.

Here is the explicit algorithm that computes the horn fillers:

Let $(y_0,\dots ,y_{k-1},-,y_{k+1},\dots ,y_n)$ give a horn in $G_{n-1}$, so the $y_i$s are $(n-1)$ simplices that fit together as if they were all but one, the $k^{\mathrm{th}}$ one, of the faces of an $n$-simplex. There are three cases:

1. if $k=0$:

   • Let $w_n=s_{n-1}{y}_n$ and then $w_i=w_{i+1}(s_{i-1}{d}_i{w}_{i+1}{)}^{-1}{s}_{i-1}{y}_i$ for $i=n,\dots ,1$, then $w_1$ satisfies $d_i{w}_1=y_i$, $i\ne 0$;

2. if $0<k<n$:

   • Let $w_0=s_0{y}_0$ and $w_i=w_{i-1}(s_i{d}_i{w}_{i-1}{)}^{-1}{s}_i{y}_i$ for $i=0,\dots ,k-1$, then take $w_n=w_{k-1}(s_{n-1}{d}_{\mathrm{nw}}{}_{k-1}{)}^{-1}{s}_{n-1}{y}_n$, and finally a downwards induction given by $w_i=w_{i+1}(s_{i-1}{d}_i{w}_{i+1}{)}^{-1}{s}_{i-1}{y}_i$, for $i=n,\dots ,k+1$, then $w_{k+1}$ gives $d_i{w}_{k+1}=y_i$ for $i\ne k$;

3. if $k=n$:

   • use $w_0=s_0{y}_0$ and $w_i=w_{i-1}(s_i{d}_i{w}_{i-1}{)}^{-1}{s}_i{y}_i$ for $i=0,\dots ,n-1$, then $w_{n-1}$ satisfies $d_i{w}_{n-1}=y_i$, $i\ne n$.

Remark

• The filler for any horn can be chosen to be a product of degenerate elements.

• The simplicial homotopy groups of a simplicial group, $G$, can be calculated as the homology groups of the Moore complex of $G$. This is, in general, a non-Abelian chain complex.

• A simplicial group can be considered as a simplicial groupoid having exactly one object. If $G$ is a simplicial group, the suggested notation for the corresponding simplicially enriched groupoid would be $BG$ according to notational conventions suggested elsewhere in the nLab.

• There is a functor due to Dwyer and Kan, called the Dwyer-Kan loop groupoid that takes a simplicial set to a simplcial groupoid. This has a left adjoint $\overline{W}$ (see below), called the classifying space functor, and together they give an equivalence of categories between the homotopy category of simplical sets and that of simplicial groupoids. We thus have that all homotopy types are modelled by simplicial groupoids … and for connected homotopy types by simplicial groups. One important fact to note in this equivalence is that it shifts dimension by 1, so if $G(K)$ is the simplicial group corresponding to the connected simplicial set $K$ then ${\pi }_k(K)$ is the same as ${\pi }_{k-1}(G(K))$. This is important when considering algebraic models for a homotopy n-type.

The adjunction between simplicial groups and simplicial sets

Lemma

The forgetful functor

$$F:\mathrm{AbSGrpg}\to \mathrm{SSet}$$F : AbSGrpg \to SSet

from simplicial abelian group to the underlying simplicial sets has a left adjoint

$$\mathbb{Z}:\mathrm{SSet}\to \mathrm{AbSimpGrp}$$\mathbb{Z} : SSet \to AbSimpGrp

from simplicial sets to abelian simplicial groups, the free simplicial abelian group functor that sends the set $X_n$ of $n$-simplices to the free abelian group $(\mathbb{Z}X{)}_n=\mathbb{Z}{X}_n$ over it.

This functor $\mathbb{Z}$ has the following properties:

• it preserves weak equivalences

• $\mathbb{Z}X$ is a cofibrant simplicial group

Classifying space and universal bundle of a simplicial group.

Every simplicial group $H$ gives rise to a one-object ∞-groupoid $BG$ whose explicit realization as a Kan complex is traditionally denoted $\overline{W}H$.

We will give this in the more general form needed for a simplicial groupoid.

For a general discussion on classifying spaces go to that entry.

Let $H$ be a simplicial groupoid, then $\overline{W}H$ is the simplicial set described by

• $(\overline{W}H{)}_0=\mathrm{ob}(H_0)$, the set of objects of the groupoid of 0-simplices (and hence of the groupoid at each level);

• $(\overline{W}H{)}_1=\mathrm{arr}(H_0)$, the set of arrows of the groupoid $H_0$:

and for $n\ge 2$,

• $(\overline{W}H{)}_n=\{(h_{n-1},\dots ,h_0)\mid h_i\in \mathrm{arr}(H_i)$ and $s(h_{i-1})=t(h_i),0<i<n\}$.

Here $s$ and $t$ are generic symbols for the domain and codomain mappings of all the groupoids involved. The face and degeneracy mappings between $\overline{W}(H{)}_1$ and $\overline{W}(H{)}_0$ are the source and target maps and the identity maps of $H_0$, respectively; whilst the face and degeneracy maps at higher levels are given as follows:

The face and degeneracy maps are given by

• $d_0(h_{n-1},\dots ,h_0)=(h_{n-2},\dots ,h_0)$;

• for $0<i<n$, $d_i(h_{n-1},\dots ,h_0)=(d_{i-1}{h}_{n-1},d_{i-2}{h}_{n-2},\dots ,d_0{h}_{n-i}{h}_{n-i-1},h_{n-i-2},\dots ,h_0)$;

and

• $d_n(h_{n-1},\dots ,h_0)=(d_{n-1}{h}_{n-1},d_{n-2}{h}_{n-2},\dots ,d_1{h}_1)$;

whilst

• $s_0(h_{n-1},\dots ,h_0)=({\mathrm{id}}_{\mathrm{dom}(h_{n-1})},h_{n-1},\dots ,h_0)$;

and,

• for $0<i\le n$, $s_i(h_{n-1},\dots ,h_0)=(s_{i-1}{h}_{n-1},\dots ,s_0{h}_{n-i},{\mathrm{id}}_{\mathrm{cod}(h_{n-i})},h_{n-i-1},\dots ,h_0)$.

References

A standard reference for the case of abelian simplicial groups is chapter 5 of

• Gorss-Jardine, Simplicial homotopy theory (web)

Section 1.3.3 of

• Tim Porter, The Crossed Menagerie

discusses simplicial groups in the context of nonabelian algebraic topology.

The algorithm for finding the horn fillers in a simplicial group is given in the proof of

theorem 17.1, page 67 of

• P. May, Simplicial Objects in Algebraic Topology .

The theorem that every simplicial group is a Kan complex is originally due to

• J. C. Moore, Algebraic homotopy theory, lecture notes, Princeton University, 1955–1956