FOM: Categorical foundations for linearity

[Colin Mclarty]

       	To show what is involved, and what it achieves, and maybe 
to give Sol ammunition to fire at me, I will describe categorical 
foundations for linear transformations, which indeed are defined 
before linear spaces on this approach. These grew from Grothendieck's 
early 1950s effort to organize homology theory in preparation for 
solving the Weil conjectures in number theory. It worked (see 
Grothendieck's 1958 Fields Medal and Deligne's in what, 1976?) and 
as expected had much wider spinoffs. Grothendieck, with help from 
Serre, saw the role of categories of linear spaces more widely 
than anyone and in more detail than anyone unless it is Serre. 
                
       	He arrived (by a path Mac Lane had earlier taken) at first 
order axioms for "Abelian categories". He also gave second order 
axioms for various kinds of Abelian categories, including a very 
useful second order property sufficient (but not necessary) for 
an Abelian category to have "enough injectives". Having enough 
injectives is itself a first order property, with some connection 
to the axiom of choice and I don't know how far that has been explored.
                
      	Abelian categories are in wide use today and not only formal
use. I have seen number theory graduate students who profess a
dislike for category theory, and who could not give the axioms 
themselves, who nonetheless take it for granted that to think
about homological algebra you should "Think of your favorite Abelian
category with enough injectives". They have a good informal grasp of
what you can do in such a category, and how.
                
       	The axioms begin with the category axioms, which I gave
in an earlier post as axioms C1-3. First I will extend this to a 
theory of linear transformations and then show how to specify 
linearity over a given ring or field. The linear spatial structure 
of any object in an Abelian category can then be defined in terms 
of the linear transformations around it.
                
DEFINITION: Arrows f and g are "parallel" iff there exist u and v
such that f:u-->v and g:u-->v.
                
       	We suppose operations + and - on arrows:
                
AXIOM L1: f+g is defined iff f and g are parallel, -f is always 
defined.
                
       	For any identities u and v there is an arrow 0 such that,
for any arrows f, g, h all u-->v we have
                
           f+g = g+f    f+0=f    f+(-f)=0   f+(g+h)=(f+g)+h
                
and for any arrows q and r
                
                    q(f+g) = qf+qg        (f+g)r = fr+gr
                
Informally: the arrows from any u to any v form an Abelian group and
composition prserves the group operations.
                
AXIOM L2: There is an identity v such that v=0 as an arrow v-->v.
       	For any identities u and v there is an identity vSu (if I
had the typeface I'd write a circled plus sign for S) and arrows
                
                            i                  j
                     ------------->    <---------------
                   u                uSv                  v
                     <-------------    --------------->
                            q                  r
                
such that: qj=0 and ri=0 and iq=u and jr=v and (qi)+(rj) = uSv
            
Informally, think of i and j as inserting two copies of the real line
R as axes in the coordinate plane R^2, and q and ar as projection the 
plane onto each axis.
            
AXIOM L3: For every arrow f there is an arrow k called a "kernel"
of f, such that fk=0, and for any arrow g with fg=0 there is a unique 
q such that qk=g. And dually for every arrow f there is an arrow c 
called a "cokernel" of f, such that cf=0 and for every g with 
gf=0 there is a unique q such that cq=g.
      	By definition, given c and k as described, and any cokernel c'
of k, there is some arrow r with f=rc'. To complete this axiom we
require that r is a kernel of c.
             
Informally, k inserts the largest subspace of the domain of f, such 
that f is 0 all over that subspace. And c is the coarsest projection 
of the codomain of f that kills the whole image of f.
            
      	I claim these axioms make a categorical foundation for the 
theory of linear transformations. They give a considerable algebra. 
Extending them by a very weak second order arithmetic gives
a tremendous amount of the very substantial appendix on homological 
algebra in Eisenbud's COMMUTATIVE ALGEBRA.
         
    	To further specify a category of transformations linear
over a ring add:
            
AXIOM L4: There is an identity r with these properties: Given any
parallel arrows f and g, say both going u-->v, if not(f=g) then there
is some k:r-->u with not-(fk=gk). Given any c:u-->v whioh is cokernel
to some arrow, and any f:r-->v, there is some g:r-->u with f=cg. And
for any r' with these properties, r is the kernel of some arrow
f:r'-->w. 
        
In the standard terminology, r is a "projective generator" and is a 
minimal one.
    
    	This makes our category a category of transformations over the
ring of arrows f:r-->r, where addition is addition of arrows, and
multiplication is composition. Call those arrows "scalars". For any
identity arrow u define an "element" of u (or a "vector" in u, if 
this ring happens to be a field) to be an arrow x:r-->u. Addition of 
elements is obvious, and to scalar multiply x by c form the composite 
xc:r-->r-->u.
    
    	Suppose you want linearity over a given ring R. If R has a
first order axiomatization (say, the ring of integers, or rationals,
or Gaussian integers) then relativize that axiomatization to arrows
f:r-->r, inerpeting addition by addition and multiplication by
composition. Adjoin that to the axioms above and you have first order
axioms for a category of transformations linear over R.

	Naturally, whatever means you need to define R you will also
need to adjoin to the axioms above to axiomatize a category of 
R-linear transformations.

	Further, suppose you want "the" category of "all" R-linear
transformations. Then you have to decide what means you will use to
define "all". You might decide you mean "all the ones that exist in ZF".
Then, obviously, you will have to put something about ZF into your
axioms: You will add an axiom saying every set of identity arrows
of the category (meaning, every set that exists in your model of ZF) has
a coproduct in the category; and you will add that the covariant
hom functor (taking values in your model of ZF) represented by r
preserves these colimits. 

	Maybe this is all that Sol meant by saying some set or class
theory always bleeds into supposedly categorical foundations: There
are after all classes and sets out there, and you can choose to relate
whatever subject you are founding to them--and when you do, you will
have to use sets and classes! That is undeniably true. And in practice
we often do want to so use some notion of collections. But I do not 
think there is any one notion of collection that is always used. These
notions themselves can be given categorical foundations (as in
the Elementary Theory of the Category of Sets). And none of this in
any way weakens the point that you can also found substantial parts
of various subjects in their own terms, using categorical methods.

	And none of it changes the point that you can describe
linear transformations before describing linear spaces, for some
purposes it is far the easiest way, and there are young number 
theorists out there today with no taste for foundations or category
theory who routinely do it.

Colin McLarty