Re: What is the right abstract definition of "connected"?

[Vaughan Pratt <pratt@cs.stanford.edu>]

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).