Subject: Re: is union associative? Ilya Beylin asked for which categories is union associative, where union is defined by: Consider the following diagram A |a V b W <----- B and define union of two morphisms a U b := Pushout(Pullback(X)) (I'm a little troubled with a non-idempotent operation being called a union.) It's true in any pre-topos. For a proof I would specialize to the case that W is the terminator (move to the slice category if it isn't). Let me use @ for the binary operation on isomorphism types characterized by the pushout diagram A x B / \ A B (add downward arrows) \ / A @ B (where it's understood that the top half is a product diagram}. Given a subobject X' in X recall that X/X' is characterized by the pushout X' / \ X Spt(X') \ / X/X' where Spt(X'), the support of X', is the image of X; -> 1. Then A @ B = (A + B)/(AxSpt(B) + Spt(A)xB)). Beyond the pre-topos case, I have no idea of just what's needed. For a counterexampe take the category of Z-sets (an object is a set with an automorphism) in which all orbits have three or fewer elements, take A to be a two-element orbit, B to be a three-element orbit. Then A @ (B @ B) is the terminator and (A @ B) @ B is isomoprhic to A + 1. (B @ B is the terminator and in any category 1 @ X = 1. A @ B is A + B.)