Third Mac Lane Cohomology via Categorical Rings

Abstract

In the fifties, Saunders Mac Lane invented a cohomology theory of rings using the cubical construction introduced earlier by Eilenberg and himself to calculate stable homology of Eilenberg-Mac Lane spaces. As shown in [9], this theory coincides with the topological Hochschild cohomology for Eilenberg-Mac Lane ring spectra. In particular, the third dimensional cohomology group is expected to provide classification of 2-types of ring spectra. Some algebraic models for such 2-types have been constructed in [1]. In this paper we consider one such algebraic model of different kind which in our opinion is especially straightforwardly related to 3-cocycles in Mac Lane cohomology. This is the notion of categorical ring—a category carrying the structure of a ring up to some natural isomorphisms satisfying certain coherence conditions. Our axioms for the categorical ring present a slightly modified version of the notion of Ann-category due to Quang [10]. Axioms we use reflect defining relations of Mac Lane 3-cocycles. Our main result is Theorem 4.4 which asserts that for any ring R and any R-bimodule B there is a bijection H(R;B) ≈ Crext(R;B)

Cite this paper

@inproceedings{Jibladze2006ThirdML, title={Third Mac Lane Cohomology via Categorical Rings}, author={Mamuka Jibladze and George Janelidze}, year={2006} }