FOM: Trad.vers.Funct.Algebra #1

[Walter Felscher]

Traditional versus Functorial Algebra  # 1


-1. Background


The content of this and the following articles is not
claimed to concern the foundations of mathematics; at best,
it may concern the foundations of universal algebra.

Yet here on fom, there have been several contributions which
discussed complaints about a loose way in which to talk
about the sameness of Boolean algebras and Boolean rings;
they arose in connection with the general contention that
our categorical brethren do have neither the means nor the
methods to contribute to FOM .  It appears astonishing to me
that Mr Simpson, who formulated those complaints, chose the
examples he did, because if anything, it is the situation of
Boolean algebras (and propositional connectives) in which
the advantages of the functorial approach to algebra may
become convincing.  Yet it appears even more astonishing
that the proponents of categorical techniques on fom seem to
be unable to take this matter up; hence in the following I
shall take it upon myself to point out the connections
between the traditional and the functorial approach to
algebra.



0.  Introduction

Everybody understands me when I speak of a 'Boolean algebra' -
without that I list its basic operations  [cup, cap,
minus; cup, minus; cap, minus; the previous and the constant
Zero; the previous and the constant One ... ] or its
defining equations. Also, everybody knows that I can
describe them just as well as 'Boolean rings'.

So we speak of Boolean algebras, knowing that we can refer
to the fundamental facts about them - e.g. the
representation theorems - without making precise which
particular kind of basic operations and equations we would
choose to prove them: what matters is that the various
definitions are intertranslatable AND that the homomorphisms
are always the same. The situation then is widely analogous
to that of geometry where we have ellipses and hyperbolas,
metric and affine transformations: geometric objects which
we describe coordinate free and without references to
particular shapes of matrices or normal forms of quadratic
equations. Boolean algebras, therefore, appear as analogous
algebraic objects, and the fundamental facts about them are
independent of a particular 'coordinatization' through basic
operations and defining equations.

The first to attempt a coordinate-free approach to algebras
was Philipp Hall from Cambridge, of fame in group theory,
apparently during the 1940ies. He never seems to have
published about it; a few references to Hall's basic notion
of 'clones' appeared during the 1960ies in publications by his
former students P.Higgins and P.M.Cohn. It is obvious from
this public non-reaction that Hall's ideas did not find
resonance; there is, to my knowledge, only one publication, by
Walter D.Neumann, in which algebras under a clone were
studied seriously.

In the meantime, geometers had invented the concept of
category and put it to use; the first, long paper I remember
was by Charles Ehresmann 1956 who applied them to describe
the complicated situations occurring for differentiable
manifolds; not the concept, but the use of categorical
techniques appears in Chapter 4 of N.Bourbaki's Th‚orie des
Ensembles of 1957 which, in its para 3 , contains what
categorists later chose to name "Freyd's adjoint functor
theorem".

In a series of papers on homotopy, beginning in 1960,
B.Eckmann and P.Hilton developed as one of their tools a
coordinate-free, functorial description of groups. This same
idea for the case of general, equationally defined algebras
was developed independently

  (a) by Claude Chevalley (presented in November 1962 in a
  joint Berkeley-Stanford colloquium talk 'Definition of
  Algebraic Structures' at Stanford, the precise date of
  which Mr Pratt should have no difficulties to unearth from
  Stanford documents; a detailed handwritten manuscript of
  Chevalley's apparently was lost at sea in summer 1963) and

  (b) by F.W.Lawvere (dissertation 1963, Columbia U., and
  Proc.Nat.Acad. 50, 869-872, cf. also Lawvere's
  presentation in July 1963 at the Model Theory conference
  at Berkeley).

In what follows, I shall use, essentially, the terminology
of Lawvere's. As the functorial description is indeed very
simple, I can state it already here, assuming only that the
reader knows

  what is a category, an object and a morphism, composition
  of morphisms f and g being denoted as g # f ,

  what is a functor between categories,

  when an object b_m is called the m-fold product (power) of
  an object b_1 in category: when there are morphims p_(m,i)
  from b_m to b_1 , i<m , such that for any object a and for
  any family of morphisms g_i from a to b_1 , i<m , there is
  a unique morphism g from a to b_m such that  g_i = p_(m,i) # g
  for all i<m .

A diagram B shall be a category whose objects are b_0, b_1, ... ,
such that b_m is the m-th power of b_1 with projections
p_(m,i).

A B-algebra shall be a product-preserving functor F from B
into the category of sets; the image of b_1 under F is
called the set underlying F . A homomorphism between
B-algebras is nothing but a natural transformation of
functors.

That's all.

   For instance, assume that B contains a morphism m from
   b_2 to b_1 . Assume that in B the morphisms from b_3 to b_1

     m # [ p_(3,0) , m # [p_(3,1), p_(3,2)] ]     and
     m # [ m # [p_(3,0), p_(3,1)] , p_(3,2) ]

   are the same. Then for a B-algebra F the map F(m) is a
   2-ary associative operation on the underlying set of F .

More examples are mentioned in Mr Pratt's contribution from
March 19th .

What remains to be discussed now are two theorems:

   Given a diagram B , then there is a suitable signature
   and a set Q of equations such that the B-algebras
   'correspond' to the (traditional) algebras defined by the
   the equations Q  [ i.e. the category of B-algebras is
   isomorphic to that of the equational class defined by Q ,
   and this isomorphism commutes with the underlying-set
   functors ].

   Given an equational class _A of algebras then there is a
   diagram B such that the algebras of _A  'correspond' to
   the B-algebras.