Ragular epis that do not compose: In Cat, which is complete and cocomplete and is even tripleable over a category (graphs) tripleable over sets, regular epis do not compose. Let 2 be the category with two objects and one non-identity arrow from one to the other. Let N be the category with one object and a natural numbers of maps. Let N_2 be the category with one object and 2 arrows, one an identity and the other idempotent. The obvious map from N to N_2 is obviously a regular epi and so is the map from 2 to N that identifies the objects, but the composite is not. Michael ++++++++++++++++++ From: pavlovic@triples.Math.McGill.CA (Dusko Pavlovic)

DOES ANYONE KNOW of a category in which REGULAR EPIS DO NOT COMPOSE? (Maybe the opposite of the category of commutative rings?)

I don't suppose that this is what you mean, but isn't already something like --0--> --a--> --f--> --1--> --b--> a0=a1, fa=fb providing an answer to the question as it stands? - Dusko ++++++++++++++++++++++++++++++++++++++++++++++++