Relations in operational categories
type: Item
instance of: scholarly article
title: Relations in operational categories
zbMATH Open document ID: 0870.18004
DOI: 10.1016/S0022-4049(96)00166-1
published in: Journal of Pure and Applied Algebra
publication date: 1997-07-27T00:00:00Z
review text: Ever since functions were declared to be universally defined and single valued, other binary relations have been pushed into the background of mathematics. A notable exception is the construction of the so-called connecting homomorphism in homological algebra. There are two well-known methods for constructing the connecting homomorphism in homological algebra. Either one constructs it as a binary relation which is shown to be universally defined and single valued, or one makes use of the so-called two-square lemma, which provides an isomorphism between two invariants associated with adjacent commutative squares. Both of these two constructions can be generalized to arbitrary `Goursat categories', namely operational categories satisfying the condition that the relative product of any relation with its converse is transitive. By an `operational category' we mean a category \({\mathcal C}\) accomplished by a category of set-valued functors from \({\mathcal C}\). In order to state the results, one has to define `relations' in operational categories and one has to generalize the notion of `exactness' from short sequences to forks and the notion of `commutativity' to squares in which two arrows are doubled. The proof of the general two-square lemma involves a construction which closely resembles that of `partial equivalence relation' in theoretical computer science and contains the latter as a special case.
Mathematics Subject Classification ID: 18B10
zbMATH DE Number: 1004401
zbMATH Keywords: connecting homomorphism
zbMATH Keywords: binary relation
zbMATH Keywords: operational category
zbMATH Keywords: two-square lemma
zbMATH Keywords: partial equivalence relation
reviewed by: Hua-Ping Yu
MaRDI profile type: Publication
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author: J. Lambek
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description: scientific article; zbMATH DE number 1004401