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

["Jonathon Funk" <jfunk@uwichill.edu.bb>]

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).

If say b=1=terminal of B (and F(1)=1), then it is as if to say that if X is
connected (relative to F), then elements of any b'
are in bijection with (or at least onto) maps X --> F(b'): every such map is
thus `constant'.

For example, in this sense we may speak of a connected object X in a topos
E-->S relative to Delta: S--->E.
Jonathon

----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: "categories list" <categories@mta.ca>
Sent: Monday, October 08, 2007 4:18 PM
Subject: categories: What is the right abstract definition of "connected"?


> 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
>
>
>