FOM: Representations of categories: some references

[Jean-Pierre Marquis]

Dear Soren Moller Riis,

this is just a note to complete Vaughan Pratt's answer to your query
concerning representation theorems in category theory.

Representation theorems played and still play an important part in category
theory.  Already in the sixties, the Lubkin-Heron-Freyd-Mitchell
representation theorem for *abelian* categories set the stage and is an
important result.  (See, for instance, Freyd's book "Abelian Categories" or
Mitchell's "Theory of Categories".  Notice that the theorems one is looking
for are theorems for abstract structured categories, e.g. abelian
categories, regular categories, pretoposes, toposes, etc. usually in
functor categories over Set, the category of sets.

Here are some papers on the subject with reference to the literature.

- Barr, M., 1986, Representation of Categories, JPAA, 41, 113-137.

- Barr, M., & Makkai, M., 1987, On representations of Grothendieck toposes,
Canadian J. Math., 39, 168-221.

- Joyal, A. & Tierney, M., 1984, An extension of the Galois Theory of
Grothendieck, Memoirs of the AMS, no. 304, Providence.

Since you mention representations of C*-algebras and boolean algebras, I
should point out that, in the first case, these results can be presented in
a categorical setting in a profitable manner, e.g.

Johnstone, P., 1982, Stone Spaces, Cambridge University Press.

and that some people are presently lifting these results to categories,
i.e. they are "categorifying" them, e.g.

Baez, J., 1997, Higher-dimensional algebra II: 2-Hilbert spaces, Advances
in Mathematics, 127, 125-189;

and, in the second case, that in a categorical framework, completeness
theorems are representation theorems (this was suggested by A. Joyal in the
early 70's), e.g. see for instance:

Makkai, M. & Reyes, G., 1995, Completeness theorems for intuitionistic and
modal logic in a categorical setting, Annals of Pure and Applied Logic, 72,
25-101.

As to whether all this is related to the foundations of mathematics... I
will let you decide.

Sincerely,

Jean-Pierre Marquis
Departement de Philosophie
Universite de Montreal
Interests: philosophy of mathematics, foundations of mathematics,
epistemology, metaphysics.