FOM: foundations of category theory

[Walter Felscher]

On two notes by Andrej Bauer from February 20th


1. Matters historical


In order to emphazise the foundational relevance of category theory,
Mr. Bauer writes :
                                                         Let me just
    mention adjoint functors here. That alone is enough to establish
    category theory as a foundational field, since it exposes a concept
    that is not treated by set theory and is without a doubt of
    a general foundational interest.

Before category theory arrived, situations of adjointness were
considered and, in particular, what categorists have chosen to call
"Freyd's adjoint functor theorem" was proved (inclusive the so-called
solution set condition) by

    Pierre Samuel: On universal mappings and free tological groups.
                   Bull.Amer.Math.Soc. 54 (1948) 591-598 .

Of course, Samuel did not use the words 'category' and 'functor'.
His approach was reproduced in

    N.Bourbaki: Th/eorie des ensembles.  Chapitre 4  Structures.
                Paris, Herman 1957

in paragraph 3 , Applications universelles, on pages 43-50 of that
book. Again, Bourbaki did not use the terminology of categories and
functors. Rephrased in that, the adjoint functor theorem then appeared
in the unpublished Princeton thesis of Freyd 1960 (which I have never
seen) and was reproduced in Freyd's "Abelian Categories" of 1964. For
a proof using the notions of category and functor  -  but not that of
adjointness  -  one may also compare the Proceedings of the Berkeley
symposium "The Theory of Models" of 1963 , pp.427-428 .


At another place, Mr. Bauer writes in a similar vein

    There are concepts that are fundamental but are not at all exposed,
    clarified, or easily studied within set theory. These are the basic
    category-theoretic concepts, such as "natural transformation",
    "adjoint functor", "limit", and "colimit".

But limits and colimits were, under the names of projective and
inverse (or injective) limits, studied decades before category theory
rose its head.

It seems, unfortunately, characteristic for some (though not all)
authors in category theory, and agressively influential ones at that,
that for them world began to exist only when it was formulated in
the language of categories. Presentations, one of them once wrote,
that do not fit into this worldview "must be rigidly suppressed".



2. Matters foundational


Explaining the objects of mathematics to consist of sets, by now has
become a rather tedious exercise in trivialities; nothing much new
happened since Wiener and Kuratowski gave the definition of ordered
pairs and since the authors of the 40ies learned to use set notions
when speaking about polynomials, eliminating the linguistic reference
to something 'written down'.

Places from which to take a comprehensive view of larger parts of
mathematics have been discovered from time to time. Einar Hille, in
the preface of his "Functional Analysis and Semi-Groups" of 1948,
wrote that he would see semigoups everywhere. More to the point, in
the 30ies, Garret Birkhoff and Oystein Ore developed lattice theory.
Lattices are everywhere. In algebra they permit a unified treatment of
topics such as the various Jordan-Hoelder-Schreier theorems and the
Remak-Krull-Schmid theorems. They appear as function lattices in
functional analysis and integration theory. They appear in logic in
the algebraic semantics of non-classical logics.  They appear in the
lattice of degrees of unsolvability.

But then, looking at the world through the categorical eye, became
most exiting experience; it taught to see an abundance of non-trivial
examples in a new and enlightening way. Chevalley's marvellous
"Fundamental Concepts of Algebra" of 1956 was a first presentation of
this view (although the notion of category was never explicitly
mentioned in it). In 1962 both Chevalley and Lawvere proposed the
functorial description of universal algebra (which had been nascent in
the homological papers by Eckmann and Hilton), which in the latter's
hands gave a much more efficient and managable treatment of Philip
Hall's 'clones of operations'.  Since then, it has been particularly
the insight and inventiveness of Lawvere which continuously opened up
new aspects, from topoi to differential geometry.  And as categorical
notions can be made to capture much more details than, say, lattice
theory, "The Category of Categories as a Foundation of Mathematics"
was proposed as well.

In the same way in which every mathematician knows to speak the
reductionist language of sets, he should be aware of the categorical
point of view wherever it matters (e.g. in the universal property of
products). If naive set formation is viewed as a "foundation" for
mathematics, then the notions about categories may be viewed as such
a "foundation" as well, and "in its own right" if eyes are kept
closely shut.

Because already on this level of naive practise of mathematics, set
language will fare better than category language. No description of
categories can avoid to speak about collections, e.g. the Hom(a,b)
formed by the morphisms from a to b  - in other words: about sets.
[Dana Scott, in his "Axiomatizing Set Theory", Proceedings of the UCLA
symposium 'Axiomatic Set Theory' 1967 , vol.2 , pp. 207-214, has shown
how such a necessity leads by itself to the axioms of set theory.]
Yes, set notions can be reconstructed in category language, but at the
price of introducing ever more complicated and arbitrary notions such
as subobject classifiers and the like. After thirty years now,
category theory has failed to provide a naive "foundation" for
mathematics of a simplicity comparable to that stated in set language.

But even on that level, Mr. Bauer's statement

    To a degree, what is and isn't a fundamental concept is a matter
    of opinion

(it sounds as if it were taken from Feyerabend) is an incorrect
relativization. What makes a concept fundamental is its wealth of
applications and the ease of its use. And again category language
fails when compared with that of sets.

Yet the categoricists want more: acknowledgement as serious workers in
the foundations of mathematics. No lattice theorist ever would claim
that his field is concerned with foundations.  Implication can be
viewed as a relative-pseudomplement, but is it of foundational impact ?
Talking about the same phenomenon, but expressing it with adjoint
functors, categoricists suddenly claim it to be so.

The gazing at mathematical phenomena in order to express them in
categorical language does not even touch the foundations of mathematics
in the standard sense. Reading Mr. Bauer depreciatingly classify
fields of mathematical logic as "popular matters"

    There is more to foundations of mathematics than just questions about
    consistency, proof strength, independence results, and other such
    popular matters

makes me wonder what kind of instruction he has been receiving at CMU .
And are we really to believe that knowledge about natural transformations
is only distantly as relevant for the foundations of mathematics than
is Frege's observation that it are the free variables which make
quantifier rules work at all ?  The notion of adjoint functor versus
that of cut-free proof, or of recursive (computable) function ?


W.F.