Cohomology of small categories

In this paper we introduce and study the cohomology of a small category with coefficients in a natural system. This generalizes the known concepts of Watts [23] (resp. of Mitchell [17]) which use modules (resp. bimodules) as coefficients. We were led to consider natural systems since they arise in numerous examples of linear extensions of categories; in Section 3 four examples are discussed explicitly which indicate deep connection with algebraic and topological problems: (1) The category of Z//p2-modules, p prime. (2) The homotopy category of Moore spaces in degree n, n22. (3) The category of group rings of cyclic groups. (4) The homotopy category of Eilenberg-MacLane fibrations. We prove the following results on the cohomology with coefficients in a natural system: (5) An equivalence of small categories induces an isomorphism in cohomology. (6) Linear extensions of categories are classified by the second cohomology group HZ. (7) The group H’ can be described in terms of derivations. (8) Free categories have cohomological dimension 5 1, and category of fractions preserve dimension one. (9) A double cochain complex associated to a cover yields a method of computation for the cohomology; two examples are given. The results (7) and (8) correspond to known properties of the Hochschild-Mitchell cohomology, see [7] and [ 171. In the final section we discuss the various notions of cohomology of small categories, and we show that all these can be described in terms of Ext functors studied in the classical paper [l l] of Grothendieck.