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In this paper Ando, Blumberg, Gepner, Hopkins and Rezk define the twisted $R$-Homology of a ring spectrum $R$ together with a map $f \colon X \to R$-$Line$ to be $$ R^f_n(X) = \pi_0(map_R(\Sigma^nR, Mf)) \cong \pi_n(Mf) $$ where $Mf$ is the Thom spectrum associated to the above map. The latter is defined as the colimit of $X \to R$-$Line \to R$-$Mod$ in an $\infty$-categorical sense.

If this definition deserves to be called twisted $R$-homology, it should satisfy the corresponding version of the Mayer-Vietoris sequence (using $M(\left.f\right|_A)$, $M(\left.f\right|_B)$ for a decomposition $X = A \cup B$). Why is this true?

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Here is a sketch. First, here is the argument for untwisted homology that I want to base the argument for twisted homology off of. Every categorical thing I say below is $\infty$-categorical by default, e.g. every colimit is an $\infty$-colimit and so forth.

The untwisted $R$-homology spectrum of a space $X$ with coefficients in a ring spectrum $R$ is the colimit of the constant diagram $X \to \text{Mod}(R)$ with constant value $R$. Taking $R$-homology spectra defines a functor from spaces to $R$-module spectra which is itself cocontinuous (because colimits commute with colimits), and in particular which sends pushout squares to pushout squares. But $\text{Mod}(R)$, being stable, has the property that pushout squares are also pullback squares. A pullback square of $R$-module spectra can be converted into a fiber sequence of $R$-module spectra, and then we can apply the long exact sequence in homotopy.

The argument works essentially without modification for twisted $R$-homology, except that the domain category is no longer spaces but, say, pairs of a space $X$ and a local system of $R$-module spectra on $X$ (there is no particular reason to restrict our attention to local systems of $R$-lines). Again taking $R$-homology is a cocontinuous functor and hence again sends pushout squares to pushout squares, which again are also pullback squares.

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  • $\begingroup$ Thanks! Is there a way to see directly that the definition of the Thom spectrum in May and Sigurdsson (which is proven to be equivalent to the above one in the paper mentioned in the question) is compatible with homotopy colimits? $\endgroup$
    – Ulrich Pennig
    Commented Jan 5, 2015 at 12:27
  • $\begingroup$ Unfortunately I'm not at all familiar with May-Sigurdsson. Maybe ask that as a separate question? $\endgroup$
    – Qiaochu Yuan
    Commented Jan 6, 2015 at 21:47
  • $\begingroup$ For the record: The "commutation of colimits"-theorem that is needed in this situation is phrased as Corollary 4.2.3.10 in Higher Topos Theory. In fact, the Mayer-Vietoris setup is used as a motivating example in the beginning of the chapter (which reminds me that I need to take a closer look at that book! :-) $\endgroup$
    – Ulrich Pennig
    Commented Jan 18, 2015 at 20:38

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