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Similar questions have already been asked here and here but not exactly in the direction I need.

I have a (small) index category $\mathcal{I}$ which is not cofiltered, and I need to consider categories of projective systems, indexed over $\mathcal{I}$, with values in some $R-\mathbf{Mod}$ (for several rings $R$, all commutative). I won't need to change $\mathcal{I}$, so I don't need a whole theory about all possible weird projective systems, but I am trying to find references for things like

  • The categories of $R-\textbf{Mod}$-valued $\mathcal{I}$-projective systems is abelian;
  • The componentwise tensor-product is a direct limit;
  • The componentwise tensor-product satisfies the natural adjunction with respect to an internal $\operatorname{Hom}$;
  • A description of injective/projective objects (in particular, of flat objects to study commutativity of taking componentwise tensor-product and inverse limit over $\mathcal{I}$).

All references I know, SGA 4 in primis, require that $\mathcal{I}$ is filtered, but this is certainly not my case. I would like to avoid reinventing the wheel (expecially because I can end up with a square one), and would be very grateful if someone knows references for the above facts or knows if they are true/false beyond the filtered case.

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  • $\begingroup$ Are you referring to something like "freely adjoining colimits to a category"? In Lurie's Higher Topos Theory, section 5.3.6, he discusses a generalization to $\infty$-categories. However, I don't know results about the monoidal structures (tensor product in your case). Maybe this is related. $\endgroup$
    – user20948
    Commented Oct 18, 2019 at 8:21
  • $\begingroup$ Well, I am most probably after something much more down-to-earth. First of all, I would like to work in usual abelian categories. Secondly, I don't need to add all colimits, I have an explicit object (componentwise tensor product of two projective systems) and I would like to know if it is a colimit, i.e. if it verifies the usual universal property. $\endgroup$
    – Filippo Alberto Edoardo
    Commented Oct 18, 2019 at 16:47
  • $\begingroup$ A sidenote: If you adjoin all colimits, then you end up with the presheaf category (by Yoneda lemma). The section in Lurie's book is about adjoining a certain family of colimits. $\endgroup$
    – user20948
    Commented Oct 18, 2019 at 18:07
  • $\begingroup$ Good point. I will have a look at Lurie's book. Thanks. $\endgroup$
    – Filippo Alberto Edoardo
    Commented Oct 18, 2019 at 18:16

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The fact that categories of projective (or inductive) systems over arbitrary categories in abelian categories are again abelian is explained in Grothendieck's Tohoku paper (Sections 1.6 and 1.7).

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  • $\begingroup$ Excellent, thank you very much. This is very helpful. Do you have an idea on where I can find, for instance, a statement discussing (termwise) tensor products of projective/inductive systems in such generality? $\endgroup$
    – Filippo Alberto Edoardo
    Commented Oct 18, 2019 at 7:01
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    $\begingroup$ Dear @Filippo, unfortunately I don't. $\endgroup$
    – Fred Rohrer
    Commented Oct 19, 2019 at 9:03

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