nLab
simplicial principal bundle

Idea

Simplicial groups model all ∞-groups in ∞Grpd. Accordingly principal ∞-bundles in ∞Grpd (a discrete $\mathrm{\infty }$-bundle) should be modeled by Kan complexes $E\to X$ 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 $G$ be a simplicial group. For $P$ a Kan complex, an action of $G$ on $E$

$$\rho :E\times G\to E$$\rho : E \times G \to E

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

Example

The canonical action

$$G\times G\to G$$G \times G \to G

of any simplicial group on itself is principal.

Definition

(simplicial principal bundle)

For $G$ a simplicial group, a morphism $P\to X$ of Kan complexes equipped with a $G$-action on $P$ is called a $G$-simplicial principal bundle if

• the action is principal;

• the base is isomorphic to the quotient $E/G:={\lim }_{\to }(E\times G\stackrel{\stackrel{\rho }{\to }}{\underset{p_1}{\to }E})$ by the action:

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

Properties

General

Proposition

A simplicial $G$-principal bundle $P\to X$ is necessarly a Kan fibration.

Proof

This appears as (May, Lemma 18.2).

Twisted cartesian products

Proposition

Let $E\to B$ be a twisted cartesian product of the simplicial set $B$ with a simplicial group $G$. Then with respect to the canonical $G$-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 $\overline{W}G$. This is discussed below.

The universal simplicial $G$-principal bundle

Recall from generalized universal bundle that a universal $G$-principal simplicial bundle should be a principal bundle $EG\to BG$ such that every other $G$-principal simplicial bundle $P\to X$ arises up to equivalence as the pullback of $EG$ along a morphism $X\to BG$.

A standard model for the delooping Kan complex $BG$ for $G$ a simplicial group goes by the name

$$\overline{W}G.$$\bar W G \,.

This is described at simplicial group - delooping. The following establishes a model for the universal simplicial bundle over this model of $BG$.

Definition

Definition

For $G$ a simplicial group, define the simplicial set $WG$ to be the decalage of $\overline{W}G$

$$WG:=\mathrm{Dec}\overline{W}G.$$W G := Dec \overline{W}G \,.

By the discussion at homotopy pullback this means that for $X_{•}$ any Kan complex, an ordinary pullback diagram

$$\begin{array}{ccc}{P}_{•}& \to & WG\\ \downarrow & & \downarrow \\ X_{•}& \stackrel{g}{\to }& \overline{W}G\end{array}$$\array{ P_\bullet &\to& W G \\ \downarrow && \downarrow \\ X_\bullet &\stackrel{g}{\to}& \overline{W}G }

in sSet exhibits $P_{•}$ as the homotopy pullback in ${\mathrm{sSet}}_{\mathrm{Quillen}}$ / (∞,1)-pullback in ∞Grpd

$$\begin{array}{ccc}{P}_{•}& \to & *\\ \downarrow & ⇙& \downarrow \\ X_{•}& \stackrel{g}{\to }& \overline{W}G\end{array},$$\array{ P_\bullet &\to& * \\ \downarrow &\swArrow& \downarrow \\ X_\bullet &\stackrel{g}{\to}& \overline{W}G } \,,

i.e. as the homotopy fiber of the cocycle $g$.

Definition

We call $P_{•}:=X_{•}{\times }^{g}WG$ the simplicial $G$-principal bundle corresponding to $g$.

Properties

Proposition

Let $\{\varphi :X_n\to G_{(n-1)}\}$ be the twisting function corresponding to $g:X_{•}\to \overline{W}G$ by the above discussion.

Then the simplicial set $P_{•}:=X_{•}{\times }_{g}WG$ is explicitly given by the formula called the twisted Cartesian product $X_{•}{\times }^{\varphi }{G}_{•}$:

its cells are

$$P_n=X_n\times G_n$$P_n = X_n \times G_n

with face and degeneracy maps

• $d_i(x,g)=(d_{i}x,d_{i}g)$ if $i>0$

• $d_0(x,g)=(d_{0}x,\varphi (x)d_{0}g)$

• $s_i(x,g)=(s_{i}x,s_{i}g)$.

References

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

One useful reference is

• Peter May, Simplicial Objects in Algebraic Topology (djvu).

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

The fact that every PTCP $X{\times }_{\varphi }G\to X$ defined by a twisting function $\varphi$ arises as the pullback of $WG\to \overline{W}G$ along a morphism of simplicial sets $X\to \overline{W}G$ can be found there by combining

1. the last sentence on p. 81 which asserts that pullbacks of PTCPs $X{\times }_{\varphi }G\to X$ along morphisms of simplicial sets $f:Y\to X$ yield PTCPs corresponding to the composite of $f$ with $\varphi$;

2. section 21 which establishes that $WG\to \overline{W}G$ is the PTCP for some universal twisting function $r(G)$.

3. lemma 21.9 states in the language of composites of twisting functions that every PTCP comes from composing a cocycle $Y\to \overline{W}G$ with the universal twisting function $r(G)$. 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

• Tim Porter, Crossed Menagerie

On page 239 there it is mentioned that

$$G\to WG\to \overline{W}G$$G \to W G \to \overline{W}G

is a model for the loop space object fiber sequence

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

One place in the literature where the observation that $WG$ is the decalage of $\overline{W}G$ 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

• David Roberts, Danny Stevenson, Simplicial principal bundle in parameterized spaces (arXiv:1203.2460)

• Danny Stevenson, Classifying theory for simplicial parametrized groups (arXiv:1203.2461)