Relations in operational categories (Q678847)

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.