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 $G(X_\bullet)\to G_\bullet$.

Remark

Twisting functions are the analogue of twisting cochains in the context of simplicial sets; but twisting cochains were introduced by Brown 1959, whilst twisting functions were discussed already in Cartan 1956.

Literature

• Henri Cartan, Sur la théorie de Kan, Séminaire Henri Cartan 9 (1956-1957) talk no. 1 [numdam:SHC_1956-1957__9__A1_0]

• Edgar H. Brown Jr. Twisted tensor products. I. Ann. of Math. 69 2 (1959) 223-246 [doi:10.2307/1970101, jstor:1970101]

• Michael G. Barratt, Victor K.A.M. Gugenheim, John C. Moore, On semisimplicial fibre-bundles, Amer. J. Math. 81 (1959) 639-657 [doi:10.2307/2372920, jstor:2372920, MR0111028]

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

• Edward Curtis, Simplicial Homotopy Theory, Advances in Math. 6 (1971) 107-209 [MR279808 doi]

Review:

• Peter May, §IV in: Simplicial objects in algebraic topology, University of Chicago Press (1967) [ISBN:9780226511818, djvu, pdf]