Comparing dimensions of a small category
For a small category \({\mathcal C}\), the following three notions of dimension are known: the cohomological dimension \(\text{c.d.} {\mathcal C}\), the Hochschild-Mitchell dimension \(\text{dim} {\mathcal C}\), and the Baues-Wirsching dimension \(\text{Dim} {\mathcal C}\). The cohomological dimension is the best known among them and it can be characterized by the following condition: \(\text{c.d.} {\mathcal C}\leq n\) whenever \(\lim^k F=0\) for every \(k>n\) and every functor \(F\: {\mathcal C} \to \text{Ab}\). The Hochschild-Mitchell dimension equals to the minimal length of the projective resolvent of the bimodul \(L{\mathcal C}\) in the abelian category of bimoduls on \({\mathcal C}\). The category of factorizations \({\mathcal C}'\) is also connected with \({\mathcal C}\) whose objects are the morphisms of \({\mathcal C}\). The Baues-Wirsching dimension of the category \({\mathcal C}\) equals the cohomological dimension of the category \({\mathcal C}'\). The inequalities \(\text{c.d.} {\mathcal C}\leq\text{dim} {\mathcal C}\leq\text{Dim} {\mathcal C}\) are known. In the article under review, it is proved that \(\text{dim} {\mathcal C}=\text{Dim} {\mathcal C}\) for an arbitrary small category with cancellation. \textit{B. Mitchell} raised the hypothesis that monoids with cancellation of cohomological dimension 1 are partially free. \textit{B. V. Novikov} constructed a counterexample to this hypothesis. The author proves that the above Mitchell hypothesis remains true if we substitute the cohomological dimension by the Hochschild-Mitchell dimension.
scientific article; zbMATH DE number 1286464
scientific article