I'd like to say that "connected" is defined on objects of any category C having an object 1+1 (coproduct of two final objects). X is connected just when C(X,1+1) <= 2. If this definition appears in print somewhere I can just cite it. If not is there a better or more standard generally applicable definition I can use? If C(X,1+1) = 2 is citable but not <= 2, have the proponents of =2 taken into account that no Boolean algebra is connected according to the =2 definition? This is because 1+1 ~ 1 in Bool, CABA, DLat, StoneDLat, etc. (dual to 0x0 ~ 0 in Set, Pos, etc.), forcing C(X,1+1) = 1. Boolean algebras and distributive lattices fail the =2 test not because they are disconnected in any natural sense but rather because they are hyperconnected. It seems unreasonable to say that hyperconnected objects are not connected. There is also the question of the object of connected components of an object. In Set and Grph, if X has k connected components then C(X,1+1) = 2^k for all X, a set (C being ordinary, i.e. enriched in Set). In Stone (Stone spaces) however this only holds for finite X, with k = X. For infinite X Stone(X,1+1) is the set of clopen sets of X, which can be countably infinite and hence not 2^k for any k. If we read 2^k as Stone(k,2), taking k = X and 2 the Sierpinski space this doesn't help. However Stone(k,1+1) is ideal: instead of treating the object of connected components of a Stone space k = X as a set we can treat them as a Boolean algebra, namely that of the clopen sets of X. These examples are worth bearing in mind when considering the appropriate general definition of number of connected components of an object, and whether even to treat it as a number (cardinal) or a more general object. Connectedness seems somehow more basic than finiteness because we can easily draw examples of connected and disconnected objects, whereas it requires a vivid imagination to see the boundary between finite and infinite objects one might try to draw on paper. This motivates making connectedness prior to finiteness. Another familiar and easily visualized notion with small examples is that of path. Define a *path* to be a connected directed graph having one vertex each of degree (0,1) and (1,0), and all others (1,1). (The degree (m,n) specifies the in-degree as m and the out-degree as n.) We can then define a finite set to be one in bijection with the set of vertices of some path. This seems more natural than defining it to be one such that every injection on itself is a surjection, because there are a lot of injections to worry about and how do you convince yourself that surjective injections don't kick in until omega? Those who are already wedded to some other definition of finite will want to check that this path-based definition draws the boundary in the same place as theirs. For what definitions of "finite" can this not be shown? And are any of them more palatable than the path-based definition? Vaughan