Vaughan Pratt wrote:
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.
Dear Vaughan, There's a big reason (there are also some little reasons, but I'll mention them later) why this doesn't match some accepted categorical definitions, and it's to do with the elements of C(1, 1+1). The topological condition is often stated differently: that every map X -> 1+1 factors via 1. Thus C(X,1+1) <= C(1,1+1). I think in most contexts you would want to say that, if anything is connected, 1 is, but you can easily find C(1,1+1) > 2. A simple example is with C = Set^2, where C(1,1+1) = 4 (two coproduct injections, and two more mixed morphisms). Then with this C, the alternative definition gives a useful notion of "fibrewise connectedness" for spaces over 2 and it's really just connectedness in the internal mathematics of (the topos) Set^2. Your definition is external. I would say don't persevere with your definition unless you really don't mind if 1 is disconnected. The different definition of "every map to 1+1 factors via 1" has been quite successful. That was the big reason. The little reasons I alluded to are that it is often useful to require every map to 0 also to factor via 1. That excludes 0 itself from connectedness. This is similar to saying 1 is not prime. Once you have the 0 and 2 cases for X, then for every finite n (= 1 + ... + 1) you have all maps X -> n factor via 1 - at least, if coproduct is well enough behaved w.r.t. limits. In constructive locale theory the standard definition is stronger and requires that for every discrete I, every map X -> I must factor via 1. This allows "infinite n". (Classically this can be deduced from the 0 and 2 cases.) All the best, Steve.