Cohomology of small categories
In this interesting paper, a new cohomology theory for categories is defined, and its properties are explored. The coefficients are called natural systems. Explicitly, a natural system is a functor \(D: F\mathcal C\to\mathcal{A}b\) where \(F\mathcal C\) is the twisted morphism category of Dwyer and Kan (objects of \(F\mathcal C\) are the morphisms of \(\mathcal C\), and arrows of \(F\mathcal C\) from \(f\) to \(g\) are pairs \((\alpha,\beta)\) of morphisms of \(\mathcal C\) satisfying \(g=\alpha f(\beta)\). Derivations \(\mathcal C\to D\) and inner derivations are defined, and \(H^1(\mathcal C,D)\) is derivations modulo inner derivations. Similarly, \(H^2(\mathcal C,D)\) is interpreted in terms of equivalence classes of linear extensions of \(\mathcal C\) by \(D\); algebraic and topological examples are given in section 3. Computations are facilitated by using ``covers'' of \(\mathcal C\) by subcategories, as well as by an isomorphism \(H^n(\mathcal C,D)=\mathrm{Ext}^n_{F\mathcal C}(Z,D)\). For appropriate choices of \(\mathcal C\) and \(D\), one retrieves the earlier theories of Hochschild, Mitchell, Roos, Quillen, Grothendieck, and Watts, as well as ordinary group cohomology (whose techniques are used throughout the paper). Understanding of this theory should lead to a deeper understanding of various algebraic and topological problems.
scientific article; zbMATH DE number 3941767
scientific article