FOM: For categorical foundations
Torkel Franzen
torkel at sm.luth.se
Wed Nov 19 06:21:33 EST 1997
Colin Mclarty says:
>Similarly, categorical foundations do not begin with a class of
>objects and one of arrows. They begin by positing arrows and a composition
>relation, and stating assumptions about them.
You speak of "categorical foundations", whereas Feferman's comments
refer to "foundations of category theory". This perhaps reflects a
difference in what the two of you have in mind.
I don't think anybody will dispute that category theory can be
formulated by positing arrows and a composition relation, and stating
assumptions about them. (There seems to be a typo in your
presentation, since your axiom C1 is a logical truth.) In putting
forward a formal theory of categories, we don't need to use any
formal theory of sets or functions.
But now suppose I ask about the meaning and justification of the
basic theory of categories. The answer I'm used to is (essentially)
that these axioms sum up, in a way that turns out be very useful,
basic aspects and properties of structures in different parts of
mathematics. By looking at a number of examples, I get to understand
how this might be so. Mappings of various kinds, I find, seem to be
the basic source of inspiration for and instantiation of the
categorical axioms.
Thus the "foundations of category theory", in the sense of the
explanations (of the concepts involved) and justifications (for using
these axioms) which we invoke when putting forward the theory (unless,
of course, we try to be thoroughly formalistic about the whole thing)
seem to involve other basic mathematical concepts in an essential way.
Therefore, even if theories of those other basic concepts can be
cast in categorical terms, categories seem to be secondary in terms
of our understanding of mathematics.
Of course similar remarks can be made about set theory. In
explaining and justifying the axioms of set theory we also invoke
mathematical examples, and the definitions in set theory of
e.g. ordered pairs, functions and natural numbers are motivated by our
prior understanding of these notions. Still, in presenting the
foundations of set theory, we invoke a concept of "a bunch of things"
which is not abstracted from various mathematical instantiations of
that concept, but is understood and pictured as a primitive notion.
We can use this primitive notion to define concepts that serve us
well, in most contexts, as set-theoretical interpretations of the
notions of ordered pairs, functions, natural numbers, and so on.
In category theory, as I understand it, there is no similar primitive
notion to be had.
This of course is not to deny that others may, right this moment
or in the future, have such a basic or primitive understanding of
category theory, corresponding to my basic or primitive understanding
of set theory.
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