Contemporary Mathematics

Volume 318, 2003

Diagrammatic Morphisms

D. N. Yetter

ABSTRACT. A

historical perspective on the subject matter of the Special Ses-

sion is offered.

There is, of course, the question of why there should have been an AMS Special

Session with the title "Diagrammatic Morphisms in Algebra, Category Theory, and

Topology", much less one with a proceedings volume. Morphisms at least have a

clear place, being the very stuff of category theory, and quite prominenent in algebra

in the homo- (iso- , endo-, and auto-) varieties, and in topology in the homeo- and

diffeo- types. But why diagrammatic morphisms?

It is the purpose of this introduction to take the prospective reader of this

proceedings volume on a brief historical tour of potential answers to this question.

It is hoped that the reader will find this useful in putting the various papers in this

volume, and those they cite, in a more intellectually satisfying context. As this is

our purpose, we at most indicate vaguely the relevant definitions, expecting that

the reader will be diligent enough to pursue the citations.

Prehistory

That algebra and geometry (of which topology is simply the "flabbiest" sort) are

intimately related has been a central theme of mathematics at least since Descartes

provided the world with the coordinate plane. At a surface level, equations ( alge-

braic gadgets) can be used to cut out loci (geometric gadgets). At a deeper level

there is a duality between geometric objects and algebraic objects: smooth man-

ifolds and rings of 0

00

-functions; affine schemes and commutative rings; compact

Hausdorff spaces and commutative C* -algebras;. . .

Diagrammatic morphisms in all their guises are the central features of a different

connection between algebra and geometry: a connection in which geometric objects

of interest (or relative versions thereof) become the elements of algebraic structures.

In most cases the geometric objects become the morphisms of categories. These

categories, then, are not foundational tools, but algebraic objects of interest in their

own right.

The importance of this direct connection from geometry (particularly low-

dimensional topology) to algebra has come to light mostly through developments in

1991

Mathematics Subject Classification.

Primary 01A65.

@

2003 American Mathematical Society

http://dx.doi.org/10.1090/conm/318/05540