FOM: Set theory vs category theory

Steve Awodey awodey at
Wed Jan 28 12:17:09 EST 1998

In reply to:
>Date: Tue, 27 Jan 1998 11:58:20 +0100 (MET)
>From: Soren Moller Riis <smriis at>
>Subject: FOM: Set theory vs category theory
>I am wondering: Why is there no representation theorem for category theory?

Where did you get this idea?  The well-known Cayley representation for
groups is a special case of a representation theorem for categories
(indeed, groups are special categories): 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, and send each
arrow f:c -> d in C to the function Cf : C_c -> C_d  given by C_f(g) = fg.)
I don't suppose your question has anything to do with the issues of size
that sometimes come up in category theory.

>are many different categories with many different properties. If the
>axioms of a
>category are well stated there ought to be a representation theorem.
>Or is this against the ideology?

Ignoring the non sequitur, why do you question whether the axioms for
categories are well-stated?  Of course they are.  And what is meant by
"ideology" here?  If there is there an "algebraic ideology" espoused by
such theories as that of semi-groups and lattices, then the theory of
categories adheres to it too, but beyond that I can make no sense of your

Steve Awodey

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