FOM: Books on categorical foundations: a baker's dozen

Vaughan Pratt pratt at cs.Stanford.EDU
Fri Feb 27 14:15:58 EST 1998

From: Harvey Friedman <friedman at>
HF>Neither Steve nor I has any commitment to "every object is a set," although
HF>it is often very convenient to do so. The problem is that at the moment
HF>there is no comprehensive categorical foundations, or any comprehensive non
HF>set theoretic foundations.

This claim, frequently made on FOM, reflects not so much on the coherence
of categorical foundations as on the technical narrowness of those making
the claim.  The claim is more than adequately refuted by the substantial
body of literature on categorical foundations.  Besides the many papers
on the subject, books on categorical foundations appear these days at
the rate of one every eighteen months or so.

I attach below a list of a baker's dozen of books on categorical

What is your criterion for "comprehensive"?  If a comprehensive
bibliography whose books tell essentially the same story doesn't meet
it then I would suggest that either your criterion is set unreasonably
high or you are grossly underestimating the coherence, technical depth,
and overall mathematical quality of the picture jointly painted by this
literature.  This is serious work, which you do a serious injustice in
dismissing so casually.

Vaughan Pratt



Johnstone, "Topos Theory", Academic Press, 1977.

Goldblatt, "Topoi: the categorical analysis of logic", North-Holland,

Makkai and Reyes, "First Order Categorical Logic", Springer-Verlag, 1977.

Barr and Wells, "Toposes, Triples, and Theories," Springer-Verlag, 1985.

Lambek and Scott, "Introduction to Higher-Order Categorical Logic,"
Cambridge, 1986.

Bell, "Toposes and Local Set Theories," Oxford, 1988.

Makkai and Pare, "The Foundations of Categorical Model Theory", AMS
CM104, 1989.

Freyd and Scedrov, "Categories, Allegories", North-Holland, 1990.

Moerdijk and Reyes, "Models for Smooth Infinitesimal Analysis",
Springer-Verlag, 1991.

McLarty, "Elementary Categories, Elementary Toposes," Oxford, 1992.

Mac Lane and Moerdijk, "Sheaves in Geometry and Logic", Springer-Verlag,

Borceux, "Handbook of Categorical Algebra" (three volumes), Cambridge,

Lawvere and Schanuel, "Conceptual Mathematics", 1997.

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