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
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