TITLE & AUTHOR
==============

  For want of awareness of the Yoneda Lemma, ...

                F.E.J. Linton

ABSTRACT
========

Half a century ago -- or more -- while the Yoneda Lemma 
was still -- or not yet even -- in its infancy, results 
were being proved from scratch, or by brute force, that, 
in retrospect, were just instances of the Yoneda lemma 
at work.

We reminisce over some forgotten or overlooked examples
of such results.

CONTENTS
========

0) Personal timeline -- Yoneda

1) Cohomology Operations

2) Homotopy Operations

3) Caratheodory's Arithmetic of Functionoids

4) (Elementary) Row (or Column) Operations on Matrices

------------------------------------------------------

0) Personal timeline -- Yoneda
==

Circa anno 1960/61, Albrecht Dold's Ritt Instructorship
Algebraic Topology lectures at Columbia University.

Standard (combinatorial) simplices for simplicial sets
as contravariant functors on Δ -- needs Yoneda
again and again, both "straight YL" and "Y density L"
for gaining geometric realization as colimit of cells.

Among other things: what Dold called "super-naturality":

whatever follows not so much merely from n.t.(A~, F) = FA
but from naturality of that in F and A, and from the fact
that when Y: R --> P is full and faithful and F: X --> P
is given along with objects f_o(x) in R and isomorphisms
f_o(x) = F(x), there's exactly one way to fatten f_o up
to a functor f: X --> R with those isomorphisms becoming
a natural isomorphism. ("Any way of representing all the
values of a functor X --> P is automatically natural --
'supernaturally' natural.")

1)
==

First example: Steenrod squares. H^n(X; G) --> H^n(X; G). 
Once H^n(X; G) = π(X, K(G, n)), such operations boil 
down to cohomology classes of K(G, n)'s, members of 
H^n(K(G, n); G). Natural progression, then, to ... :

2)
==

M. Kervaire (lectures at NYU, circa 1960 +/-), for pointed 
spaces, say.

Homotopy operation of type (n_1, ..., n_k; n) was a scheme 
ψ that, w/ each space *X* = (X, x), linked an operation 
ψ_*X*: π_n_1(X, x) x ... x π_n_k(X, x) --> π_n(X, x), 
naturally in (X, x).

Such things, Kervaire took two full lectures to show, 
boiled down to elements of π_n(S^n_1 V ... V S^n_k).  

In essence, those two full lectures beat around these two 
bushes:

1) π_n1(X, x) x ... x π_nk(X, x) = 
 π((S^n1 V ... V S^nk), (X, x)) 
(first functor is representable); and

2) Yoneda: those n.t.'s boil down to the value of that 
second functor π_n at that wedge (or bouquet) of spheres.

Involved chasing the sequence ([id], ..., [id]) into the 
domain of ψ_wedge-of-spheres:

π_n_1(S^n_1) x ... x π_n_k(S^n_k)
                                 -->
π_n1((S^n1 V ... V S^nk)) x ... x π_nk((S^n1 V ... VS^n_k))

and then applying ψ -- and verifying that association was 
bijective without explicitly relying on 1).

Contortionism :-) .

3) Caratheodory "functionoids" and their kin (spectral 
resolutions, spectral measures)
==

Monotone families E_λ of poset elements (indexed by 
real, or even just rational, λ), mod equivalence 
given by E </= F iff λ < μ => F_λ &sub; E_μ . 
(f </= g iff {x/g(x) < λ} &sub; {x/f(x) < μ}.)

Hundred-page exposition of how to add/subtract/multiply 
these creatures, using all sorts of "astuces" like the 
functions α and β (that model inf_X (f(x)) as α(X), 
sup_X (f(x)) as β(X)) but derived from the family {E_λ} 
(as if that were f^-1(-oo, λ}).

All sorts of technical hoops to jump through, instead of 
recognizing (with A. Goetz, 20 years later): it's just 
φ: IB_IR --> A, σ-homomomorphism, and the m'ble functions 
on IR act (Lawvere-theory-wise) hereon, whence:

add/subtract/multiply/sup/inf all available trivially, as 
also numerous infinitary operations such as:

conditional (bounded) countable sups 

  (b, x_1, ..., x_n, ...) |-> sup (|b| ^ x_i),
                                 i

summable countable linear combinations of bounded sequences

(for each abs. summable seq. {a_i} and each sequence {x_i} 
w/ bound b > 0 as above,

Σ_i [a_i (|b| ^ x_i v (-|b|))] ),

countable disjoint sups (x_1, ..., x_n, ...)
                               |-> 
                x_1 + (x_2)|(comp(supp(x_1))) + ... 

(from real function  'first non-zero x_i'), ... .

All told, action of the clone of meas'ble real functions of 
at most countably many real v'bles, in the actual form of a 
clone of σ-homomorphisms amongst the boolean σ-algebras 
IB_IR^n = n●(IB_IR) (0 </= n </= &alef:_0).

References:
----------
http://planetmath.org/?op=getobj&from=objects&id=10406 -- 
Projections as noncommutative characteristic functions

http://planetmath.org/encyclopedia/LatticeOfProjections.html 
-- The Lattice of Projections

http://planetmath.org/?op=getobj&from=objects&id=11646 -- 
Proof of Borel Functional Calculus

http://planetmath.org/encyclopedia/BorelFunctionsOfANormalOperator.html 
-- Borel Functional Calculus (Borel Functions of a Normal 
Operator)

http://planetmath.org/encyclopedia/SpectralMeasure.html -- 
Spectral Measure

---

Spectral measures: http://planetmath.org/encyclopedia/SpectralMeasure.html

Borel Fnct'l Calc: http://planetmath.org/encyclopedia/BorelFunctionsOfANormalOperator.html
and: 
http://planetmath.org/?op=getobj&from=objects&id=11646

http://planetmath.org/encyclopedia/LatticeOfProjections.html

Projections as noncommutative characteristic functions: 
http://planetmath.org/?op=getobj&from=objects&id=10406

Kakutani, S.: Abstract [ L-spaces | M-spaces ]

Andrzej Goetz: Caratheodory functions ... Rozprawy Mat.

C. Caratheodory, Mass und Integral und Ihre 
Algebraisierung, Birkhauser, Basel, 1956; Engl. Tr. 
Chelsea, NY, 1963.

Problems: need good explanation why Borel functions on the 
spectrum of A "act" on A qua sa op in vN alg, whereas 
continuous functions on spec(A) act on A qua sa op in the 
C*-alg it generates.

need good description for spectral measure of A + B 
(two non-c sa op's -- or even just two non-c sa proj's).

4)
==

Think of k x n matrices (k rows, each n entries long) as 
"maps" from integer n to integer k
(Think of n x 1 matrices (single n-length columns) as 
"elements" of n or as "maps" from integer 1 to integer n.)

Traditionally, (invertible) "elementary row operations" on 
k-rowed matrices were of 3 sorts:

●  interchanging two distinct rows;
●  adding a fixed multiple of one row to *another* row;
●  replacing a given row by a fixed, non-zero -- i.e., 
     invertible -- (scalar) multiple of itself.

And, for each one, R, one showed that there's an invertible 
k x k square matrix E for which: for all k x n A: 

   R(A) = E●A .

Ad hoc: given R, produce E deus-ex-machina fashion from 
secret knowledge, and show it works;

Somewhat motivated: suggest checking whether E = R(I_kxk) 
fills the bill;

Yoneda in reverse: confirm, for each R, that 

   R(B●A) = R(B)●A 

and then *know* that R(I_kxk) fills the bill.

In fact, Yoneda puts n.t.(Mat(-, k), Mat(-, m)) in bij've 
corr. w/ {Mat(-, m)}(k) = Mat(k, m), and, in particular, 
n.t.(Mat(-, k), Mat(-, k)) with Mat(k, k) in natural, hence 
invertibility/inverse-preserving, way.

In fact, for fixed j, 1 x k [δ_i,j]_i serving to 
extract j^th row alone, ... .

( Row_j(B●A) = Row_j(B)●A )