Abstract :
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 constructions generalize 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 here mean a category ß accompanied by a category of setvalued functors from ß. 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 PER in theoretical computer science and contains the latter as a special case.
