Split Short Five Lemma for Clots and Subtractive Categories

Abstract

It is well-known that a pointed variety is classically ideal determined (or “BIT speciale”) if and only if it satisfies the split short five lemma (i.e. if and only if it is a protomodular category in the sense of D. Bourn). A much weaker property than being classically ideal determined is “subtractivity”, defined as follows: a variety with a constant 0 is said to be subtractive if its theory contains a binary term s satisfying s(x,x) = 0 and s(x,0) = x. In the case of a pointed variety (i.e. when 0 is the unique constant), this condition can be reformulated purely categorically (as many other similar term conditions), which gives rise to the notion of a subtractive category. In the present paper we show that in a certain general categorical context subtractivity is equivalent to a special restriction of the split short five lemma to the class of clots, i.e. monomorphisms that are pullbacks of reflexive relations R→Y×Y along product injections (1Y,0): Y→Y×Y.