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Is there a model structure on the category of topological spaces equipped with a local system (i.e. a functor from the fundamental groupoid to the category of abelian groups), such that the weak equivalences are the isomorphisms in homology with local coefficients?

If so, is there a reference in which the model structure is established?

Edit: A morphism $(X,F\colon\Pi(X)\rightarrow AB)\rightarrow(X,F\colon\Pi(X)\rightarrow AB)$ is a continuous map $\phi\colon X\rightarrow Y$ together with a natural transformation $F\implies \phi^*G$.

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  • $\begingroup$ Can you clarify what are the maps in this category? Precisely, if $f:(X,A)→(Y,B)$ is a map, are you requiring that $A→f^*B$ is an isomorphism of local systems? $\endgroup$
    – Denis Nardin
    Commented Oct 7, 2016 at 16:47
  • $\begingroup$ No. Just a natural transformation, not necessarily an isomorphism. I edited the question accordingly. $\endgroup$
    – local
    Commented Oct 7, 2016 at 18:54
  • $\begingroup$ Is this category cocomplete? It's fairly easy to construct filtered colimits, but I don't see why pushouts exist. $\endgroup$
    – Karol Szumiło
    Commented Oct 7, 2016 at 20:02
  • $\begingroup$ @KarolSzumiło It is the total category of the Grothendieck fibration classified by the functor sending $X$ to the category of local systems on $X$, so it is cocomplete by general principles (in particular the pushouts are pushouts in spaces together with the pushout of the pushforwards of the local systems). $\endgroup$
    – Denis Nardin
    Commented Oct 7, 2016 at 20:44
  • $\begingroup$ @DenisNardin You are right. I got a little confused by these pushforwards at first, but they are just left Kan extensions along the induced functors between fundamental groupoids. In that case the construction of the model structure seems rather easy. I will add an answer. $\endgroup$
    – Karol Szumiło
    Commented Oct 7, 2016 at 22:08

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