FOM: Categorical foundations, reply to Feferman
cxm7 at po.cwru.edu
Thu Nov 20 14:49:39 EST 1997
Reply to message from sf at Csli.Stanford.EDU of Thu, 20 Nov
>I think the argument which I made in "Categorical foundations and
>foundations of category theory" [CFFCT], and which McLarty
>strenuously objects to, has not been properly understood, either by him or
I'm sure this is true, at least of me. This post helps me
understand, though I continue to disagree.
> To elaborate a bit, let me quote from CFFCT, p.150: "...when
>explaining the general notion of structure and of particular kinds of
>structures such as groups, rings, categories, etc. we implicitly *presume
>as understood* the ideas of *operation* and *collection*; e.g. we say that
>a group consists of a collection of objects together with a binary
>operation satisfying such and such conditions.
When I describe groups or group theory I do begin this way. I
don't know a categorical foundation for group theory and the topic has
not attracted much effort.
When I describe categories and functors I MAY begin this way.
I certainly do when I want specifically the small categories in the
universe of sets or some other topos (construing *collection* as
set, or object of the topos). I often begin more or less this way
in informal descriptions for people used to set theoretic foundations.
I do not begin this way in giving categorical foundations, nor in
my book (most of which is not foundational but does reflect my
view of the subject). There I start with assumptions on arrows and
objects, formalizable as first order axioms.
>Then to follow category
>theory beyond the basic definitions, we must deal with questions of
>completeness, which are formulated in terms of collections of morphisms.
Many categories, including all toposes, have an internal
sense of completeness: If a collection of objects exists in the topos,
then it has a Cartesian product within the topos. This is the
relevant notion for many purposes, and the only notion that arises for
a category construed as a foundation.
>Further to verify completeness in concrete categories, we must be able to
>form the operation of Cartesian product over collections of its
>structures. Thus at each step we must make use of the unstructured
>notions of operation and collection to explain the structural notions to
A "concrete category" by the standard definition is a category
together with a faithful functor to the category of sets. So of course
to study concrete categories you must posit sets. Feferman may intend a
broader sense of "a category completely specified in terms of some
larger foundation" (such as his theory of operations and collections).
And certainly then you will need to use that larger foundation to study
the category: for example, does every collection of objects of the
category, which collection exists according to the larger foundation,
have a Cartesian product in the topos?
But none of this bears on the question of whether you must use
such a larger foundation to describe a category. I still claim that
Lawvere first and then I and others have published first order
categorical axioms suitable as foundations for set theory, category
theory, various aspects of differential geometry, and other subjects.
> The *logical* and *psychological priority* if not primacy of
>the notions of operation and collection is thus evident."
How does *priority* differ from *primacy* here? This may be an
important gap in my understanding of your view.
>These questions of priority take place in ordinary mathematical parlance
>and understanding. What do you have to understand first before you
>explain the next thing? You can't explain what a linear transformation is
>before you've explained what a linear space is.
This is the example that most struck me, because linear spaces
and transformations are precisely the part of mathematical practice
where categorical foundations first arose and still have the most impact.
I will give the categorical foundations in another post this afternoon.
>McLarty wants to say in place of "what is a category", "what are the
>first order axioms of categories?"
No, no more than you want to say in place of "what is an
operation", "what are the first order axioms of operations". This is
the same point that you explain to Vaughan--the first order axioms
formalize informal insights. The fact of being first order
expressible does show that no appeal to classes is being snuck in.
>and likewise for sets, etc., thus
>seemingly putting everything on a par: no priorities at all, and no need
>to posit collections and operations. Similarly, supposedly, for groups,
>rings, topological spaces, etc. But if we want to be able to say: the
>set of all permutations of a set of n elements forms a group under
>composition, where are we? This is a specific structure. What are its
>elements? What is the operation?
I can't see why you'd say I want "no priorities at all". As to
collections and operations, I "need to posit" them very often--when they
are what I want to use.
In fact it seems to me that category theory IS the general
theory of operations. When I begin a categorical foundation for a
subject by saying "There are arrows, and they compose, and..." I am
making that subject into one theory of operations--but not a single,
putatively all encompassing theory such as you propose. Nor do I
object to such broad theories as yours, or ZF, as theories in their
I only say I need not posit such a theory of operations and
collections to found the theory of sets, or categories, or linear spaces,
or smooth spaces, and other things.
If we want to say "the set of all permutations of an n element
set forms a group under composition" then evidently we are in some
context where we can already talk about sets--it might be an informal
context, or we might have formalized it using ZF, or the Elementary
Theory of the Category of Sets. In any case the answers to your
questions are the same: The group is a set of functions, it is the
subset of the set of all endofunctions of our set containg just the
invertible ones. The operation is composition. What kind of answer did
you expect? Do you think ETCS has some problem giving these answers?
>be regarded as innocuous, even banal, if it didn't mention categories at all.
>It's just because of the program of re-expressing all mathematical notions
>in terms of categorical notions that led to saying: "Well, sets can be
>explained in terms of the category SET, so you see, categories are more
>basic than sets" that has led to the objections. Much as those who pursue
>this program (the categorical ideologues) would like
>to think that they have kicked away the traces, they only have to face up
>to the fact in example after example that there is an ineliminable residue
>of the notions of collection and operation.
You keep saying this, but you do not give actual examples. You
refer to some example in very general terms, and then do not find any
specific residue but claim there must be one. And so far as I can see,
your argument for this claim is just to say that indeed there must be.
Maybe my coming post on linear spaces could be your example.
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