nLab
simplicial principal bundle

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Simplicial groups model all ∞-groups in ∞Grpd. Accordingly principal ∞-bundles in ∞Grpd (a discrete \infty-bundle) should be modeled by Kan complexes EXE \to X equipped with a principal action by a simplicial group. It is suficient to assume the action to be strict. This yields the notion of simplicial principal bundles .

Definition

Definition

(principal action)

Let GG be a simplicial group. For PP a Kan complex, an action of GG on EE

ρ:E×GE \rho : E \times G \to E

is called principal if it is degreewise principal, i.e. if for all nn \in \mathbb{N} the only elements gG ng \in G_n that have any fixed point eE ne \in E_n in that ρ(e,g)=e\rho(e,g) = e are the neutral elements.

Example

The canonical action

G×GG G \times G \to G

of any simplicial group on itself is principal.

Definition

(simplicial principal bundle)

For GG a simplicial group, a morphism PXP \to X of Kan complexes equipped with a GG-action on PP is called a GG-simplicial principal bundle if

  • the action is principal;

  • the base is isomorphic to the quotient E/G:=lim (E×Gp 1Eρ)E/G := \lim_{\to}(E \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to} E}) by the action:

    E/GX. E/G \simeq X \,.

Properties

General

Proposition

A simplicial GG-principal bundle PXP \to X is necessarly a Kan fibration.

Proof

This appears as (May, Lemma 18.2).

Twisted cartesian products

Proposition

Let EBE \to B be a twisted cartesian product of the simplicial set BB with a simplicial group GG. Then with respect to the canonical GG-action this is a simplicial principal bundle.

This is (May, prop. 18.4).

Remark

It is simplicial principal bundles of this form that one is mainly interested in. These are the objects that are classified by the evident classifying space W¯G\bar W G. This is discussed below.

The universal simplicial GG-principal bundle

Recall from generalized universal bundle that a universal GG-principal simplicial bundle should be a principal bundle EGBG\mathbf{E}G \to \mathbf{B}G such that every other GG-principal simplicial bundle PXP \to X arises up to equivalence as the pullback of EG\mathbf{E}G along a morphism XBGX \to \mathbf{B}G.

A standard model for the delooping Kan complex BG\mathbf{B}G for GG a simplicial group goes by the name

W¯G. \bar W G \,.

This is described at simplicial group - delooping. The following establishes a model for the universal simplicial bundle over this model of BG\mathbf{B}G.

Definition

Definition

For GG a simplicial group, define the simplicial set WGW G to be the decalage of W¯G\overline{W}G

WG:=DecW¯G. W G := Dec \overline{W}G \,.

By the discussion at homotopy pullback this means that for X X_\bullet any Kan complex, an ordinary pullback diagram

P WG X g W¯G \array{ P_\bullet &\to& W G \\ \downarrow && \downarrow \\ X_\bullet &\stackrel{g}{\to}& \overline{W}G }

in sSet exhibits P P_\bullet as the homotopy pullback in sSet QuillensSet_{Quillen} / (∞,1)-pullback in ∞Grpd

P * X g W¯G, \array{ P_\bullet &\to& * \\ \downarrow &\swArrow& \downarrow \\ X_\bullet &\stackrel{g}{\to}& \overline{W}G } \,,

i.e. as the homotopy fiber of the cocycle gg.

Definition

We call P :=X × gWGP_\bullet := X_\bullet \times^g W G the simplicial GG-principal bundle corresponding to gg.

Properties

Proposition

Let {ϕ:X nG (n1)}\{\phi : X_n \to G_{(n-1)}\} be the twisting function corresponding to g:X W¯Gg : X_\bullet \to \overline{W}G by the above discussion.

Then the simplicial set P :=X × gWGP_\bullet := X_\bullet \times_{g} W G is explicitly given by the formula called the twisted Cartesian product X × ϕG X_\bullet \times^\phi G_\bullet:

its cells are

P n=X n×G n P_n = X_n \times G_n

with face and degeneracy maps

  • d i(x,g)=(d ix,d ig)d_i (x,g) = (d_i x , d_i g) if i>0i \gt 0

  • d 0(x,g)=(d 0x,ϕ(x)d 0g)d_0 (x,g) = (d_0 x, \phi(x) d_0 g)

  • s i(x,g)=(s ix,s ig)s_i (x,g) = (s_i x, s_i g).

References

Here are some pointers on where precisely in the literature the above statements can be found.

One useful reference is

There the abbreviation PCTP ( principal twisted cartesian product ) is used for what above we called just twisted Cartesian products.

The fact that every PTCP X× ϕGXX \times_\phi G \to X defined by a twisting function ϕ\phi arises as the pullback of WGW¯GW G \to \overline{W}G along a morphism of simplicial sets XW¯GX \to \overline{W}G can be found there by combining

  1. the last sentence on p. 81 which asserts that pullbacks of PTCPs X× ϕGXX \times_\phi G \to X along morphisms of simplicial sets f:YXf : Y \to X yield PTCPs corresponding to the composite of ff with ϕ\phi;

  2. section 21 which establishes that WGW¯GW G \to \bar W G is the PTCP for some universal twisting function r(G)r(G).

  3. lemma 21.9 states in the language of composites of twisting functions that every PTCP comes from composing a cocycle YW¯GY \to \bar W G with the universal twisting function r(G)r(G). In view of the relation to pullbacks in item 1, this yields the statement in the form we stated it above.

An explicit version of the statement that twisted Cartesian products are nothing but pullbacks of a generalized universal bundle is on page 148 of

On page 239 there it is mentioned that

GWGW¯G G \to W G \to \overline{W}G

is a model for the loop space object fiber sequence

G*BG. G \to * \to \mathbf{B}G \,.

One place in the literature where the observation that WGW G is the decalage of W¯G\overline{W}G is mentioned fairly explicitly is page 85 of

  • John Duskin, Simplicial methods and the interpretation of “triple” cohomology, number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc. (1975)

Discussion of topological simplicial principal bundles is in

Revised on December 28, 2012 11:25:38 by Tim Porter (95.147.236.158)