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simplicial classifying space

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Homotopy theory

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see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Group Theory

Contents

Idea

For GG a simplicial group, there is a reduced simplicial set, traditionally denoted W¯G\overline W G and called the classifying space or classifying complex of GG, which is a model for the delooping of GG and such that the functor W¯()\overline{W}(-) is right adjoint to the standard simplicial loop space-construction GG (here denoted by LL to avoid a clash of notations).

This pair of adjoint functors

SimplicialGroupsW¯LSimplicialSets red 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.

Definition

In components

Definition

(standard universal principal simplicial complex)
For GG a simplicial group, one writes

WGSimplicialSets W G \;\in\; SimplicialSets

for the the simplicial set whose

  • underlying sets are

    (WG) nG n×G n1××G 0 (W G)_n \;\coloneqq\; G_{n} \times G_{n-1} \times \cdots \times G_0
  • face maps are given by

    (1) d i(g n,g n1,,g 0) {(d i(g n),d i1(g n1),,d 0(g ni)g ni1,g ni2,,g 0) if i<n (d n(g n),d n1(g n1),,d 1(g 1)) if i=n \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) s i(g n,g n1,,g 0) (s i(g n),s i1(g n1),,s 0(g ni),e,g ni1,,g 0), \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 ee denotes the respective neutral element.

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

(3)G×WG WG (h n,(g n,g n1,,g 0)) (h ng n,g n1,,g 0). \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 GG-action (3) that singles out the model WGW 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 GG a simplicial group, its standard simplicial classifying complex is the quotient of WGW G (Def. ) by its GG-action (3)

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

The corresponding quotient coprojection, whose fiber is, manifestly, GG

(5)G fib WG W¯G(WG)/G \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 GG-universal principal bundle (see below).

This means, under the isomorphism

(W¯G) n(WG) n/G nG n/G n×G n1××G 0G n1××G 0, \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 WGW G imply the following structure maps on the simplicial classifying complex:

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

    (W¯G) nG n1××G 1×G 0; (\overline{W}G)_n \;\coloneqq\; G_{n-1} \times \cdots \times G_1 \times G_0 \,;
  • face maps are given by:

    (6) d i(g n1,,g 0) {(g n2,,g 0) if i=0 (d i1(g n1),,d 0(g ni)g ni1,g ni2,,g 0) if 0<i<n (d n1(g n1),,d 1(g 1)) if i=n; \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) s i(g n1,,g 0) {(e,g n1,,g 0) if i=0 (s i1(g n1),,s 0(g ni),e,g ni1,,g 0) if 0<i. \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.)

Remark

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

(WG) n(W¯G) n+1 (W G)_{n} \;\simeq\; (\overline{W}G)_{n+1}
d i WG =d i+1 W¯G s i WG =s i+1 W¯G. \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 WGW G is the décalage of W¯G\overline{W}G, in the form

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

See the example there.

Example

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

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

Example

(simplicial classifying space of an ordinary group)
In the special case that

GGroupsconstSimplicialGroups 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 WGW G is that whose

  • underlying sets are

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

    d i(g n,g n1,,g 0){(g n,g n1,,g nig ni1,g ni2,,g 0) if i<n (g n,g n1,,g 1) if i=n 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 n1,,g 1)(g n,g n1,,g n1,e,g ni1,,g 0), 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

WG=N(G×GG) W G \;=\; N \big( G \times G \rightrightarrows G \big)

with the nerve of the action groupoid of GG acting on itself by right multiplication (isomorphic to the pair groupoid on the underlying set of GG):

Finally this means that the simplicial classifying complex (4) of an ordinary group is isomorphic to the nerve of its delooping groupoid:

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

Via total simplicial sets

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

W¯:[Δ op,Groups][Δ op,B][Δ op,Groupoids][Δ op,N][Δ op,SimplicialSets]σ *SimplicialSets. \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:

Properties

In all of the following, GG is any simplicial group.

Basic properties

Proposition

(simplicial classifying spaces are Kan complexes)
The underlying simplicial set of any simplicial classifying W¯G\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. W¯()\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 GG on a Kan complex XX, the canonical coprojection from the Borel construction to the simplicial classifying space is a Kan fibration:

(WG×X)/GFibW¯G. (W G \times X)/G \xrightarrow{ \;\in Fib \; } \overline{W}G \,.

In particular:

Proposition

The coprojection WGW¯GW 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=GX = 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.):

G WG Fib W¯GAAAAAAAAAA(WG×(G *))/G. \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 WGW 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 W¯G\overline{W} G are those of GG, shifted up in degree by one:

π n(G)π n+1(W¯(G)). \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.)

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

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

π n+1(WG)π n+1(W¯G)π n(G)π n(WG). \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 π n(WG)\pi_n(W G) is trivial for all nn, so that this collapses to short exact sequences:

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

which exhibit the claim to be proven.

Proposition

Let 𝒢 1ϕ𝒢 2\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 W¯𝒢 1W¯(ϕ)W¯𝒢 2\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2 is a Kan fibration if and only if ϕ\phi is a surjection on connected components: π 0(ϕ):π 0(𝒢 1)π 0(𝒢 1)\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 W¯G\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 W¯()\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 QuW¯Ω(sSet 0) inj. Groups(sSet)_{proj} \underoverset {\underset{\overline{W}}{\longrightarrow}} {\overset{ \Omega }{\longleftarrow}} {\;\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;\;} (sSet_{\geq 0})_{inj} \,.

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

(e.g. Goerss & Jardine 09, V Prop. 6.3)

Slice model structure

The slice model category of the classical model structure on simplicial sets over the simplicial classifying complex W¯G\overline{W}G is Quillen equivalent to the Borel model structure for GG-equivariant homotopy theory:

(SimplicialSets Qu) /W¯G QuGActions(SimplicialSets) proj. \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 W¯\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 W¯\overline{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 LL is also discussed by Kan (there denoted “GG”) 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 W¯G\overline{W}G with the Borel model structure:

Generalization to simplicial presheaves:

Last revised on July 5, 2021 at 08:48:12. See the history of this page for a list of all contributions to it.