Re: connectedness

[<wlawvere@buffalo.edu>]

Paul's remarks are quite cogent. Indeed, when
Steve Schanuel and I introduced the notion
of Extensive category, one of the main motivations
was the recognition that a rational theory of
connectedness requires a condition on the category
of not-necessarily connected things, and moreover
that a category like (K-rigs)^op satisfies this 
condition even though it is not exact. (Also there 
was the realization of a need for an algebraic 
geometry for some cases where K is not a ring,
indeed where it may satisfy 1+1=1).

When C^op is an algebraic theory, i.e., C has 
finite coproducts, then the algebras form a topos iff 
those coproducts satisfy extensivity (because 
then the attempt to consider "finite disjoint covers" 
actually succeeds to satisfy Grothendieck's condition
 for a"topology").

But a non-trivial dual question is "almost" stated by Paul:

 For which algebraic categories is the opposite extensive ?

Obvious extensions of K-rigs are M-K-rigs, where the 
given monoid M acts by K-rig homomorphism, and an
infinitesimal version of that where M is  a Lie algebra acting
by derivations.

A special case of the question is, given an algebraic category
that is coextensive, which varieties in it are also ? (Here I take
"variety" in the  original Birkhoff spirit, i.e., a full subcategory 
that is also algebraic for the special reason that it is defined
 by a quotient theory and is thus closed wrt subalgebras, which
a general full reflective algebraic subcategory would not be).
A sufficient condition is that the inclusion functor is also
COREFLECTIVE. Call these "core varieties".

Proposition : A core variety in a coextensive algebraic category
is also coextensive in  its own right.

Hence any core variety is a candidate to serve as the algebras
for an algebraic geometry. (Extensivity was the only distinctive
feature of rings mentioned in Gaeta's notes on Grothendieck's
Buffalo Lectures 1973, and indeed you can verify that the basic 
construction of a corresponding topos of spaces works, in 
particular that the algebras become algebras of functions on these).

For single-sorted theories, a core variety is defined by the 
imposition of further identities in one variable having the rare
property that the elements satisfying them form a subalgebra.
For example ( )^p=id in algebras of characteristic p.

Already for K=2, there are nontrivial core varieties in K-rigs.
The best known is the category of distributive lattices, the
corresponding topos of spaces being generated by the category of
finite posets. The core of any 2-rig is the DL defined by two equations, 
one of which is idempotence of the multiplication. But the other
 equation, taken alone, defines a larger core variety whose
spaces look like intervals, cubes,etc ; it is intimately related
to a less systematic subject burdened with the odd name "tropical".

Bill



On Fri Oct 12  9:53 , Paul Taylor  sent:

>Vaughan Pratt's original enquiry was actually in the context of
>graph theory (as I suspected at the time, and he subsequently
>confirmed), but I would like to add something from the point of
>view of constructive real analysis.
>
>First, though, I would like to underline something that Steve Lack
>(almost) said, namely that the category in which you index your
>components, and therefore also the one in which you define
>connectedness, need to be EXTENSIVE, ie their coproducts should
>be disjoint, and stable under pullback, and the initial object strict.
>
>Maybe we've over-done philology recently, but "component" means
>"putting together", where we expect the parts to cover the whole
>(coproduct), without overlapping (disjoint), to be distinguishable
>(like disjoint union, but unlike addition and disjunction).
>The modern notion of extensivity, in which Steve had a part,
>captures this idea very neatly.   The equivalence between definitions
>of connectedness based on 1+1 and on X+Y surely depends on stability
>under pullback, and the requirement that the choice between left
>and right be unique surely requires disjointness.   Maybe a close
>study of Marta Bunga's work on abstract connectedness would clarify
>this.
>
>Vaughan originally asked about various categories of algebras,
>and Steve mentioned commutative rings, but quietly turned their
>arrows around.   Stone duality would suggest to me that one should
>look for connectedness of algebras in their OPPOSITE category of
>"spaces", which I understand in a generic sense that includes
>sets, graphs, predomains, locales and affive varieties.
>

...