nLab twisting function

Idea

The notion of twisting functions is an explicit simplicial formula for a cocycle with values in a simplicial group. The twisted product induced from the twisting function is an explicit simplicial formula for a simplicial principal bundle, a model for a discrete principal ∞-bundle classified by this cocycle.

In a fibre bundle or more generally in a fibration, the fibre ‘twists’ as one goes around a loop in the base space, (the standard simple example is the Möbius band). Such fibre bundles are usually restricted to being ‘locally trivial’, that is locally a product of an open set in the base with the fibre.

In the setting of simplicial homotopy theory, one can attempt to construct analogues of fibre bundles by starting with a base simplicial set $X_\bullet$ and a fibre $F_\bullet$ and trying to ‘deform’ the simplicial product $X_\bullet \times F_\bullet$ to get some non-trivial fibred object. In the Cartan seminars of the period 1956–57 (numdam), (about pages 1–10), a neat solution was described: by leaving all but one of the face maps of the product alone, and deforming the last. The result is a “twisted cartesian product” (see below). The deformation required a simplicial automorphism of the fibre of course, and the resulting twisting function went from the base $X_\bullet$ to the automorphisms of $F_\bullet$, completely mirroring the topological example. The simplicial identities force the twisting function to obey certain equations.

Definition

Twisting function

Let $X_\bullet$ be a simplicial set and $G_\bullet$ a simplicial group. Then a twisting function $\phi :X_\bullet\to G_\bullet$ is a family of maps $\phi=\{\phi_n : X_n\to G_{n-1}\}_{n\gt 1}$ such that

$$\array{ d_0 \phi(x) = (\phi(d_0 x))^{-1} \phi(d_1 x)\\ d_i \phi(x) = \phi(d_{i+1}x), i\gt 0,\\ s_i\phi(x) = \phi(s_{i+1}x), i\geq 0,\\ \phi(s_0 x) = 1_G. }$$

Twisted Cartesian products

Given a simplicial set $F_\bullet$ with left $G_\bullet$-action, one then defines a twisted Cartesian product, (TCP), $X_\bullet \times_\phi F_\bullet$ with $(X_\bullet \times_\phi F_\bullet)_n = X_n\times F_n$ and

$$\array{ d_i(x,f) = (d_i x, d_i f), i\gt 0\\ d_0 (x,f) = (d_0 x, \phi(x)(d_0 f)),\\ s_i(x,y) = (s_i x,s_i y). }$$

Thus the only difference from the usual Cartesian product of simplicial sets is in $d_0$.

Properties

• A twisting function $\phi :X_\bullet\to G_\bullet$ corresponds exactly to a simplicial map from $X$ to $\overline{W}(G_\bullet)$ delooping of the simplicial group. There is a universal twisting function $\overline{W}(G_\bullet)_\bullet\to G_\bullet$. See simplicial principal bundle for more.

• By the adjunction between $W$-bar and the Dwyer-Kan loop groupoid functor, a twisting function also corresponds to a morphism of simplicial groupoids $G(X_\bullet)\to G_\bullet$.

• A twisting function is an analogue of a twisting cochain in the context of simplicial sets although historically the development went the other way, it seems as the basic theory of twisting cochains was introduced by Ed Brown in his paper

Literature

• Henri Cartan, Sur la théorie de Kan, Séminaire Henri Cartan, 9 (1956-1957), Exposé No. 1, 19 p. numdam

• Edgar H. Brown Jr. Twisted tensor products. I. Ann. of Math. (2) 69 (1959) 223–246.

The link between Kan fibrations and simplicial fibre bundles, and thus TCPs is neatly summarised in

• E. Curtis, Simplicial Homotopy Theory, Advances in Math. 6 (1971) 107 – 209 MR279808 doi

For one of the foundational papers:

• M. G. Barratt, V. K. A. M. Gugenheim, J. C. Moore, On semisimplicial fibre-bundles, Amer. J. Math. 81 1959, 639–657. MR0111028 (22 #1895)

Detailed exposition is also in the book

• P. May, Simplicial objects in algebraic topology, University of Chicago Press, 1967 djvu