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simplicial classifying space

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Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

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 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

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

    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 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 n1,,g 0)(s i(g n),s i1(g n1),,s 0(g n1),e,g ni1,,g 0), 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 ee denotes the respective neutral element.

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

(1)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{\,.} }

The quotient of WGW G by this GG-action (1) is denoted

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

and the quotient coprojection

WGW¯G W G \longrightarrow \overline{W} G

is known as the standard model for the simplicial GG-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

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 (2) 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

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

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

Proposition

The coprojection WGW¯GW 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 WGW G is contractible.

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

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).

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¯\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 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 June 6, 2021 at 04:24:55. See the history of this page for a list of all contributions to it.