From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: Re: pratt cat
Date: Fri, 31 Oct 1997 15:39:28 -0400 (AST) [thread overview]
Message-ID: <Pine.OSF.3.90.971031153919.12899A-100000@mailserv.mta.ca> (raw)
Date: Fri, 31 Oct 1997 01:03:22 -0800
From: Vaughan R. Pratt <pratt@cs.Stanford.EDU>
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
next reply other threads:[~1997-10-31 19:39 UTC|newest]
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1997-10-31 19:39 categories [this message]
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1997-10-31 2:32 categories
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