In [this paper](http://arxiv.org/abs/1403.4325) 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. 



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> 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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Mayer-Vietoris sequence for twisted R-homology
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. 


