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

["Stephen Lack" <S.Lack@uws.edu.au>]

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.