Dear Vaughan ============================================================== An object of a category is *connected* when its every morphism to a nonempty coproduct factors through an inclusion thereof. ============================================================== Your proposed definition above is precisely the notion of *abstractly unary* from my J.Algebra '69 paper. It was so termed (instead of *connected*) since it does not need a terminal object to state it (precisely your motivation) and since one does not want to restrict to binary coproducts. When there is a terminal object, and when the coproducts considered are just the binary ones, it is enough to consider morphisms into the coproducts 1+1 (as I show in my thesis) and, in that case, it should be simply called *connected*. In another guise, this is the definition of *connected* given in Cats and Alligators, and it is the one directly inspired by topology. I see no reason to change the terminology. In short, your connected objects I have called abstractly unary. They came about in connection with atoms. An object A in a cocomplete (concrete) category E is an *atom* if HOM(A,-):E--->Set preserves colimits. More objectively, if E has exponentiation, Lawvere uses the notion of an *A.T.O.* instead, meaning that the functor (-)^A : E---> E has a right adjoint (the "amazing right adjoint"). I hope this helps, Cordially, Marta