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We define local system cohomology of $(X,M_\rho)$, $\rho:\pi_1(X)\to Aut(M)$, as the cohomology associated to the complex $$Hom_{\mathbb{Z}[\pi_1]}(C_0\tilde X,M_\rho)\to Hom_{\mathbb{Z}[\pi_1]}(C_1\tilde X,M_\rho)\to \dots$$ This requires the existence of a universal cover for $X$. If we then want to write down a Mayer-Vietoris sequence, we run into the problem that disconnected spaces do not have universal covers, and hence not all terms in the sequence make sense in general.

How can we, while sticking to this definition of local system cohomology, make sense of the Mayer-Vietoris sequence?

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    $\begingroup$ You can't and shouldn't: choose a better definition of local system of coefficients and its cohomology. $\endgroup$
    – Oscar Randal-Williams
    Commented Jul 28, 2017 at 17:15
  • $\begingroup$ @OscarRandal-Williams I see. I'm not convinced that we can't though, since this definition supposedly coincides with the sheaf cohomology of the sheaf defined by the local system. Using this sheaf definition we do have a Mayer Vietoris sequence, and hence this should translate back to something phrased in terms of the definition I gave, right? $\endgroup$
    – user2520938
    Commented Jul 28, 2017 at 17:23
  • $\begingroup$ See Brown's "Cohomology of groups" for an algebraic treatment. Or use any book which covers sheaf cohomology, e.g. Bredon's "Sheaf theory". In terms of sheaves, lack of connectivity is a non-issue. $\endgroup$
    – Misha
    Commented Jul 28, 2017 at 20:18
  • $\begingroup$ @Misha Thanks for the recommendations. The fact that in one framework the connectivity of the space is significant, while in the other it isn't, is exactly what confuses me, since I've repeately been told that these frameworks are equivalent... Can you say anything about that? $\endgroup$
    – user2520938
    Commented Jul 28, 2017 at 20:39
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    $\begingroup$ You need fundamental groupoids in general. $\endgroup$
    – Fernando Muro
    Commented Jul 29, 2017 at 4:37

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