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

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

Steve Vickers wrote:
> 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.

Thanks, Steve, this is great.  I didn't want to go out on a limb with
C(X,1+1) <= 2 (or = 2) if it was buggy, good to know about the C(1,1+1)
 > 2 problem.

This also takes care of my concern about situations where 1+1 = 1, since
your definition as stated makes Boolean algebras etc. connected.

Presumably my taking the anarchist side (no unity) in the definition of
locally cartesian closed obligates me to ask for the right formulation
of "connected" in the absence of 1.  How about the following?

=================================================================
An object of a category is *connected* when its every morphism to a
nonempty coproduct factors through an inclusion thereof.
=================================================================

This eliminates all assumptions about the category -- if there are no
nontrivial coproducts every object is connected by default (any morphism
to a trivial coproduct factors through its one inclusion), reasonable
when there is no recognizable (by the coproduct test) example of
disconnectedness in the category to compare with.

It also accomodates:

> 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".

with the same benefits - constructive I suppose (how is that judged
exactly?), and allows infinite comparisons.

If necessary one could qualify "coproduct" with "small" but
methodologically it would seem preferable to let such size limits be set
by a larger context.

The effect of

> 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.

can be had by omitting "nonempty" from the definition.  While this might
seem a very natural omission, my concern with it is not so much 0 itself
as the objects with morphisms to 0, e.g. all Boolean algebras except 1,
which this definition would therefore make not connected.  Stone spaces
being totally disconnected, it just seems plain wrong to have their
duals not connected either when they are so obviously connected, like
totally (except 1, which is, like, connected but not totally, being dual
to the empty Stone space, which is, like, disconnected but not totally).

In the geometric duality of points and lines in the plane, two points
are disconnected unless they coincide, while two lines are connected
unless they are parallel.  And an undirected graph and its complement
either both contain an N or neither do, and in the latter case you can
ask Google the following.

   Is an N-free graph connected if and only if its complement is
disconnected?

Google will confirm that it is, no need to click on any of the links it
returns.  (You may have to read several of Google's "answers" though
since Google isn't yet smart enough to just say yes, or even to give the
most direct "answer" first.)  Graphs with an N are the undirected graph
counterpart of the empty Stone space and the one-element Boolean
algebra, being neither totally connected nor totally disconnected.

Incidentally it's amazing just how many questions Google "knows" the
answer to.  Like all oracles though it tends to be a little erratic on
questions involving future events.  Google's staggering R&D budget
notwithstanding, asking it whether Hillary will win the election is
about as useful as asking the 8-ball: you're way better off asking the
people who place sub-Google-sized ($100) bets on such questions.  And
asking NSF for funding for your research into questions you propose to
answer by asking Google has even lower odds than asking Google.

Vaughan