# FOM: Set theory vs category theory

Vaughan Pratt pratt at cs.Stanford.EDU
Wed Jan 28 15:04:50 EST 1998

```From: awodey at cmu.edu (Steve Awodey)
>Every category is isomorphic to a category of sets and functions.
>(For the proof, associate to each object c in a category C  the set C_c
>of all arrows with codomain c, ...

I.e. that every category is concrete in the sense of my message yesterday.
This proof only goes through for small categories.  It does not
demonstrate for example that Grp is a concrete category, even though it
obviously is.

From: Jean-Pierre Marquis <Jean-Pierre.Marquis at uni-konstanz.de>
>Already in the sixties, the Lubkin-Heron-Freyd-Mitchell representation
>theorem for *abelian* categories set the stage and is an important result

Yes, it occurred to me afterwards that I should also have said something
about representation theorems for more than just general categories.
In particular the representability of any abelian category as a category
of R-modules is at the heart of the FOM debate of a couple of months ago
about whether one could define "linear transformation" without first
defining "vector space".  A vector space is just an R-module over a
ring R that has the additional property of being a field, a condition
that is easily added as a further restriction on abelian categories.
An abelian category so restricted is then representable as a category
of vector spaces.  This effectively defines the notion of vector space
in terms of that of linear transformation, the morphisms being all that
the axioms for an abelian category talk about.

Just how close vector spaces come to sets can be measured by the startling
new fact found two months ago by Peter Freyd that the only difference
between an abelian category and a topos is in the behavior of 0 (the
initial object).  Among AT categories, those satisfying the universal
Horn sentences of category theory common to the theories of abelian
categories and toposes, the abelian categories are those for which 0 is
isomorphic to 1 (the final object---0 and 1 are both the trivial group
in the category of abelian groups, the canonical abelian category),
whereas the toposes are those for which 0 is an annihilator for product
(Ax0 ~ 0 for all objects A, the familiar behavior of the empty set under
Cartesian product in the category of sets, the canonical topos).

I mentioned this fact before, but it is startling enough to bear repeating
even to this categorically skeptical audience.

Vaughan Pratt
CS Prof, Stanford
Interests: Foundations of computation and mathematics
Web site: http://boole.stanford.edu

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