Comparing dimensions of a small category
type: Item
instance of: scholarly article
title: Comparing dimensions of a small category
zbMATH Open document ID: 0916.18009
author: A. A. Khusajnov
published in: Siberian Mathematical Journal
publication date: 1999-05-11T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18G20
Mathematics Subject Classification ID: 20L05
zbMATH DE Number: 1286464
zbMATH Keywords: cohomology of small categories
zbMATH Keywords: dimension of objects and categories
zbMATH Keywords: cohomological dimension
zbMATH Keywords: Hochschild-Mitchell dimension
zbMATH Keywords: Baues-Wirsching dimension
zbMATH Keywords: monoids with cancellation
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description: scientific article; zbMATH DE number 1286464