Bill on John (and his adequate subcats)

Bill on John (and his adequate subcats)

[Peter Freyd]

                 John Isbell's Adequate Subcategories
                            F. W. Lawvere
                   Submitted to the Topology Atlas
               (in response to Mel Henriksen's request)

For mathematicians of my age, the theory of rings of continuous
functions was one of the first exciting research topics we
encountered. Many results of that theory have appeared, but its
ramifications for category theory are still not fully worked out. A
crucial link was provided by John Isbell's contributions around 1960
on the theme of adequate subcategories.

Briefly, a subcategory  A  of a larger category is adequate if every
object  X  of the larger category is canonically the colimit of the
category  A/X  of objects of  A  equipped with structural maps to  X;
John's equivalent definition was that the truncated Yoneda embedding
of the whole category into the category of set-valued contravariant
functors on  A  is actually a full embedding. The following language
is suggestive:

  (a) The objects of  A  are figure-types,
  (b) the objects of  A/X  are particular figures in  X, and
  (c) the morphisms of  A/X  (commutative triangles in the big
      category) are incidence relations between figures.

Thus adequacy means that the large category in question consists of
objects entirely determined by their  A-figures and incidence
relations, and that

  (d)  the morphisms in the whole category are nothing but the
       "geometrically continuous" ones in the sense that they map
       figures to figures without tearing the incidence relations.

For example, if  A  is the category of countable compact spaces then
A  is adequate in many large categories constructed in attempts to
capture the notion of topological space; in this case a morphism can
be identified as a mapping that preserves sequential limits. That
example is one of many illustrating that typical large categories of
mathematics often have quite small adequate subcategories; it had been
studied by Fox in 1945 at the instigation of Hurewicz, who sought a
rational notion of function space for use in algebraic topology and
functional analysis. In fact, for any  A, each space of A-continuous
morphisms has its own cohesion, described again by  A-figures.

The dual notion of a co-adequate subcategory  C  leads to a
contravariant representation of the larger category that can be
described in terms of

  (a)  quantity-types,
  (b)  functions, and
  (c)  algebraic operations on functions.

The dual of the notion of geometric continuity (that is, a name for
naturality of maps of covariant functors instead of contravariant
ones) is

  (d)  "algebraic homomorphism"

These ideas of John Isbell became fused with the conceptions of
Kan, Grothendieck, and Yoneda (emerging in the same period 1958-1960),
to form a basic method of analyzing and constructing mathematical
categories. That method was used in the early 60's by Freyd, Gabriel,
Lawvere, Mitchell, and by Isbell himself, and became as natural as
breathing to many algebraists and topologists during the following
decades.

What does adequacy have to do with rings of continuous functions? The
theory of rings of continuous functions springs from a basic
philosophical hope to the effect that there should be a near-perfect
duality between space and quantity. Such duality questions can be
investigated for a great many different categories, but categorical
considerations suggest that they need to be brought down to earth in
certain respects.

John Isbell was also one of the main developers of the theory of
locales. This theory revealed that the traditional notion of
topological space is algebraic rather than geometric (in the sense of
the above analysis) with the infinitary algebras (frames) of open
sets playing the dominant role; this merely means that the Sierpinski
space, together with "all" its powers, constitutes a coadequate
subcategory of the category of sober spaces. John's insistent quest
for smallness (as a further requirement on co-adequate subcategories
C) brought this analysis qualitatively nearer to real mathematics.

If a small co-adequate subcategory is available, it can often be
reduced to a single object (for example by taking the product of its
objects; or more concretely, the Euclidean plane as a topological
object will often serve the purpose of co-adequacy). The endomorphisms
of that object then parameterize the unary operations whose
preservation by  C-homomorphisms serves to exclude ghosts from among
detected points and figures. Even among those operations a few may be
co-adequate, as the Stone-Weierstrass theorem had shown: addition,
multiplication, and conjugation can replace the monoid of all
continuous operations for that particular task, and there are many
variations on that theme. But apart from such details of presentation,
the implicit insight of Czech and Stone, of Hewitt and Nachbin,
apparently includes smallness of the algebraic theory  C  in terms of
which spaces are to be (co) analyzed.

There was a seeming barrier to the realization of that concrete
insight: set theory, in its striving for larger and larger cardinals,
had neglected to emphasize that all of the cardinals arising in
geometry and analysis in fact satisfy a useful smallness condition:
the category of countable sets is co-adequate in the category of all
small sets. That is essentially John Isbell's formulation; he proved
that it is equivalent to the condition that no small set has the kind
of ghost elements called Ulam measures. John knew full well that his
formulation for abstract sets would imply that many categories having
small adequate subcategories also have small co-adequate
subcategories, thus making possible the desired sort of dualities
between space and quantity.

The ideas of John Isbell contributed to the enlightened understanding
of mathematics by lifting some dark clouds of confusion, and they
continue to be actively developed and diffused.