Twisted cohomology as sections of bundle of Eilenberg-Maclane spaces

Question

Let $X$ a space, and $E$ an multiplicative cohomology theory represented by a ring spectrum $K$, i.e. $E^\bullet(X)=[X,K]$. Also let $A$ be a local system of abelian groups. Cohomology with local coefficients or twisted cohomology $E^\bullet(X;A)$ can apparently be described as the set of sections of a bundle over $X$ with fiber $K$. Can you elucidate this construction for me? For example, let $A$ be the orientation sheaf of a manifold $M$, and consider ordinary cohomology with these coefficients. What is my bundle of $K(G,n)$'s? Also can you explain how we understand the twisting as a map to the Picard group of $K$? References welcome

Answer 1

I realize this is an old question but I thought I'd still chime in with an answer, partially because when I first encountered this idea I was also very confused by how the general construction works.

The example via the orientation bundle is good, but because the twist is low dimensional you don't really see the complexity that arises when you consider higher degree twists.

The $\infty$-Grothendieck construction is generally the right way to go, but right off the bat one has to address the question of where these bundles of spectra are supposed to live in the first place. The "correct" habitat for these generalized objects is in the tangent $\infty$-topos to spaces which I will denote by $$p:T(\infty Gpd)\to \infty Gpd \;.$$ More explicitly, the objects in $T(\infty Gpd)$ are so called spectrum objects which essentially are constructed by using a suspension operation which is "twisted" by a base space $B$ (you can read more about that here https://ncatlab.org/nlab/show/tangent+%28infinity%2C1%29-category). When $B=\ast$ you recover usual spectra. The map $p$ takes a spectrum object living over a base space $B$ and maps it to its underlying base space. So in a sense that can be made precise, there are fibers $T(\infty Gpd)_B$ of the bundle $p$ (taken at $B$) which encode the information of spectra parametrized over the space $B$.

This is all very abstract at first, but the machinery is actually very easy to utilize in practice. Most of the techniques one uses for bundles (i.e. local triviality, sheaf of sections, etc.) have analogues in this more general setting.

I'll illustrate by resonding to the question about Picard groupoid (or more generally $\infty$-groupoid). Let $R$ be a ring spectrum (at least $A_{\infty}$). Let's recall that the $\infty$-category of spectra is symmetric monoidal (as an infinity category) with respect to the smash product operation. Using that operation, we can consider modules in the same way that one considers modules via the tensor product and we can consider invertible module spectra in the same way we can consider invertible modules (i.e. there is a smash product inverse). Here though, everything is only taken up to higher homotopy coherence, so we don't have strict inverses, but only inverses up to higher homotopy coherence. So instead of organizing these invertible module spectra into a category, we are more naturally led to an $\infty$-groupoid ${\rm Pic}(R)$.

The space ${\rm Pic}(R)$ has a canonical bundle of spectra in $T(\infty Gpd)_{{\rm Pic}(R)}$ which lives over it. Via the $\infty$-Grothendiek construction (which works for spectra) this bundle is associated to the tautological $\infty$-functor $$F:{\rm Pic}(R)\to Sp\;,$$ which associates to an invertible module spectrum $L$ over $R$ to the spectrum $L$ and to morphisms morphisms of such spectra etc. Call this bundle of spectra $\lambda\to {\rm Pic}(R)$. It lives in the tangent $\infty$-topos $T(\infty Grp)$. This is a sort of universal bundle which classifies bundles of spectra with fiber $R$ in the following way. For any other space $B$, a map $\tau:B\to {\rm Pic}(R)$ specifies a twist for $R$ (i.e. an invertible $R$-module spectrum $L$). Then pullback of the universal bundle $\lambda\to {\rm Pic}(R)$ gives a bundle of spectra $E\to B$, now living over $B$.

The mapping space in $T(\infty Grp)_B$ actually forms an infinite loop space and so we have an enrichment over spectra. The spectrum of maps $Map_B(B,E)$ is then precisely the spectrum of sections of $E\to B$ and the homotopy groups of this spectrum are the $\tau$-twisted $R$-cohomology groups of $B$.

Finally, to really drive home the point as to why these objects really do behave like bundles, we need to consider the axiom of descent for an $\infty$-topos. In the present context, the key point is that if $B$ is a paracompact space and I take a good open cover $\{U_{\alpha}\}$ of $B$, then the homotopy colimit over the resulting Cech nerve is equivalent to B (by the Borsuk nerve theorem). If you take your bundle of spectra $E\to B$ and pullback by the induced maps from the nerve you simply get the trivial bundle $R\times U\to U$ (because the cover is good each of the $U_{\alpha}$'s and their intersections are contractible). Descent says that since $B$ is the colimit over the $U_{\alpha}$'s, then the bundle $E\to B$ must be the colimit of the locally trivial bundles $R\times U_{\alpha}\to U_{\alpha}$. So this bundle $E\to B$ can be glued from local data (as a bundle should). Of course if the twist has degree higher than 1, then this bundle is strictly general in the sense that one needs to take into account higher-fold intersections of the cover $\{U_{\alpha}\}$