Subject: Re: pratt cat

From: categories <[email-masked]>
Date: Fri, 31 Oct 1997 15:39:28 -0400 (AST)

Date: Fri, 31 Oct 1997 01:03:22 -0800
From: Vaughan R. Pratt <[email-masked]>

Peter's response to my question about categories having all the common
properties of toposes and abelian categories was a very pleasant
surprise.

First I was not at all expecting such a definite answer.

Second I did not expect a representation theorem, especially not such a
nice one as "every such category is a product of an abelian category
and a topos".  This is the converse of the automatic (by Horn) fact
that such a product is such a category, where (for this converse at
least) the language can be taken to be as large as the full Horn theory
common to abelian categories and toposes.

Third I did not expect at all that the class would be finitely
axiomatizable (but on the other hand neither did I have any reason to
suppose not).

Fourth and very nice, unless I'm overlooking something, Peter's
representation theorem implies that my original question (does the
common *elementary* theory allow anything other than abelian categories
and toposes?) can now be answered in the negative.  The elementary
sentence "either all the objects are A-type or all the objects are
T-type" is obviously true in both abelian categories and (pre)toposes,
and rules out hybrids like Set x Ab.  I'd been wondering what
elementary sentence might fail for that example, and Peter's A-type and
T-type constructs lead straight to it.

Minor remarks:

The trick of multiplying by 0 to tease out both the A- and T-components
at once is beautiful, not just intrinsically but also because it
reveals a certain duality between A and T that is not at all apparent
(to me anyway) from their separate definitions.  The actual
multiplication projects out the T part without disturbing the A part,
and then the pushout 

                    AX		(where AX = 0xX)
                   /  \
                  0    X
                   \  /
                    TX

neatly quotients out that surviving A part by forcing the left side
through the zero object while on the right AX->X communicates only the
A part which is therefore the only part that the pushout quotients out
from 0 + X (if that's not too clumsy a way of putting it).

The upshot is then that the AX's form (as an abelian category) a
coreflective subcategory of the whole while the TX's form (as a
pretopos) a reflective subcategory.  That's really beautiful!

The following is presumably the key to proving completeness of Peter's
axiomatization...

>VAE) For every  X  there's a map  TX -> X  such that 
>
>                    AX     TX
>                      \   /
>                        X

...by permitting any model to be constructed directly as a
product of an abelian category and a topos (not even "subcategory"
seems to be needed here, so this is an even stronger representation
theorem than the representation of a Boolean algebra as a subalgebra of
a power set).

>And note that the type-A objects can not be 
>reflective: if  1  has a map to any type-A object the entire category
>collapses.

Whoops, a bug here if I'm understanding things.  Only the T part
collapses, if the category were abelian to begin with then there'd be
no collapse.

>Define  TX -> X  as the image of rX.

This was lovely (getting rid of the E in VAE above).

Some questions.

1.  Can this representation theorem be extended to the full Horn
theory, i.e. any number of alternations of quantifiers?  (Note that my
disjunction---every object is A-type or every object is T-type---is not
a Horn sentence.)  (Now that I look again, I guess the answer must be
yes, since products preserve *all* Horn sentences.)

2.  In a text that treated both toposes and abelian categories, would
your axiomatization be a good starting point prior to doing either?
You could then quickly define an abelian category to be a category of
this general type with the additional property of being strongly
connected.  I don't see right off how to define a pretopos as slickly.

3.  What about bringing in the closed structure?  Might that permit
some simplification of your axiomatization?  For example it looks as
though you could define a T-type object to be one having a unique map
to the tensor unit.  Then you could answer the last sentence of the
previous paragraph simply with "the tensor unit is terminal."

Vaughan Pratt