FOM: For categorical foundations

[Colin Mclarty]

    Reply to message from sf at csli.Stanford.EDU of Mon, 17 Nov
>
>The argument was made in the paper "Categorical foundations and
>foundations of category theory" in the volume _Logic, Foundations of
>Mathematics and Computability Theory_ (Proc. LMPS V conference),
>Vol. 1, Reidel, 1977, pp.149-169.
    
    	Yes, thank you. I think this paper is still the best articulated
argument against categorical foundations and I guess (the list will
correct me if I am wrong) it is entirely persuasive to people who
were inclined to agree with the conclusion to begin with. I admit it
is entirely unpersuasive to me and others who were inclined against the
conclusion. I hope that discussing it here can break this unhelpful
"entirety" to a more usable debate. 
     

>Basically, the argument is that the
>notion of category is defined in terms of the notions of collection and
>operation, viz. the collection of all the "objects" of the category, the
>collection of all its "morphisms", and the operations of domain,
>co-domain, and composition applied to morphisms and of 1_a applied to 
>objects a.  (Alternatively, one can deal with just the collection of
>morphisms alone.).  Thus the foundations of category theory must be given
>in terms of some sort of theory of operations (or functions) and
>collections (or classes), though there are alternatives to standard
>set-theoretical formalisms for that purpose (one such is given op.cit.).
    
    	This seems to me quite wrong. It seems parallel to another
mistake, which Feferman would not make, whereby one would object to set
theory or any theory of operations and collections by saying:
    
    	The theory of operations and collections is circular
    	since it begins by saying "consider the collection of all
    	collections, and the collection of all operations". 
    
    	Of course set theoretic foundations do not begin by saying 
"there is a class of all sets" nor will Feferman begin by saying there
is a collection of all collections. Set theoretic foundations begin
by positing sets and a membership relation, and stating assumptions about
them. 
    
    	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. (As Feferman notes there
is a trivial choice of using objects and arrows, or just arrows. For most
purposes I use objects. But here I want to emphasize that the information
is all about arrows so I will use only them.)
    
    	I will give the assumptions explicitly. Since composition is not 
defined for all pairs of arrows you might like to use a relation symbol 
Cfgh to say "h is the composite of f and g", but I will use operator 
notation gf=h for visual clarity. 
    
DEFINITION: An arrow u is an "identity" iff, for every arrow f with
the composite fu defined, fu=f.
    
AXIOM C1: If u is an identity, then for any arrow g with the composite
gu defined, gu=g.
    
AXIOM C2: For every arrow f there is a unique identity u with the
composite fu defined, and a unique identity v with vf defined. We
write f:u-->v to say u and v are identities and the composites are
defined.
    
AXIOM C3: For any arrows f, g, h, if gf and hg are defined then the
associativity equation holds and both sides are defined:
    
                       h(gf) = (hg)f
    
    	These are first order axioms for a theory of arrows (i.e. for
a category).
    
    	One well known first order extension of this theory specializes
it to categorical set theory, another to a categorical theory of
categories and functors. I can give references if people need them
(or just get my book ELEMENTARY CATEGORIES, ELEMENTARY TOPOSES now
in paperback from Oxford and if you don't want to read the whole thing, 
look in the bibliography). I plan to write up categorical foundations 
for linear spaces and transformations for the list later, but I want 
to stop this post now.

	I hope we can untangle the many issues of whether categorical
foundations are logically impossible, or too arcane, or psychologically 
unsatisfying, or too derivative or whatever. For the moment I just
want to argue that categorical axiomatizations of major subjects
(e.g.g set theory, or category theory) are logically possible with
no dependence on any prior theory (e.g. sets, or operations 
and collections) in the same way that set theory can be axiomatized
without assuming any prior theory. I would like to ask how 
controversial my claims before this paragraph seem--and whether 
people think I've connected with Feferman's quoted criticism.
    
    Colin McLarty