Dear Vaughan, Lawvere and Janelidze have each argued for many years (in somewhat different contexts) that notions of connectedness and cohesion should be understood as relative. This impacts on both your questions: how should connectedness be defined, and what sort of answers should be allowed to the question ``how many connected components does X have?'' --- the second question becomes ``what is the codomain of the pi_0 functor?'' Steve Vickers mentioned the example Set^2. He said that the terminal object (1,1) is obviously connected. But it is equally obviously not connected: (1,1)=(1,0)+(0,1). The latter point of view comes from thinking of Set^2 as a Set-topos, where the connected components functor becomes the functor Set^2-->Set given by homming out of (1,1). The former point of view comes from thinking of Set^2 as defined over itself; then, as Steve says, (1,1) becomes almost tautologically connected, since pi_0 is just the identity functor Set^2-->Set^2. If crng is the category of finitely presentable commutative rings with no non-trivial nilpotents, then there is a lovely pi_0:crng^op-->set_f. For in this case every ring R splits as R_1 x R_2 x ... x R_n, where the R_i have no non-trivial idempotents. It is these R_i which are your connected components. For a larger category of commutative rings, you have to expand your notion of connected component to something like Stone spaces. For a locally connected topos E, defined over S, the inverse image functor e^*:S-->E has not just a right adjoint e_* but also a left adjoint e_!, which serves as pi_0. But one can describe just in terms of e_! -| e^* (i.e. without mention of e_*, and without all of the topos structure) the sorts of abstract properties needed for a good pi_0. This is the starting point for Janelidze's Galois theory. If E is infinitarily extensive (small coproducts, which are stable under pullback and disjoint), then a good notion of connectedness of an object X is that the hom-functor E(X,-):E-->Set preserves coproducts. This includes the locally connected topos case, which in turn includes your case of directed graphs. The case of crng is a finitary version. Regards, Steve Lack.