Dear Marta and Jonathan, As it turns out I really only needed the definition for categories of directed graphs, where "An object of a category is *connected* when its every morphism to a nonempty coproduct factors through an inclusion thereof" does exactly what I wanted there (if I haven't messed up my generalization of Steve Vickers' definition). This raises the interesting question however of whether the definitions you both mentioned differ from the above in the categories to which they apply, and if so which notion is preferable in those categories and why? What about Cat&Al's Sh(Y) for example? You both may have such examples; if not then I would argue that my definition has the advantages of generality and simplicity. Best, Vaughan Jonathan Funk wrote:
One suggestion is to say that an object X in a category C (with products) is connected relative to a functor F:B-->C if passing from maps m:b-->b' in B to maps XxF(b)-->F(b') (by composing the projection XxF(b)-->F(b) with F(m) ) is a bijection for every b,b' (or possibly just onto, not bijection, could be stipulated, but I don't know how inappropriate that would be).
If pullbacks exist X*: C-->C/X, then this is equivalent to X*F full and faithful (or just full).