Chu spaces, defined by M. Barr at McGill and studied by his student P.
Chu in the 1970's, are rectangular matrices over a given set that
transform by duplicating and deleting columns, identifying equal rows,
and adjoining new rows. With these transformations Chu spaces form a
richly endowed category, being self-dual, bicomplete, and symmetric
monoidal closed, and hence a model of Girard's linear logic.
Our motivating interest in Chu spaces has been as a universal model of
computation. More recently and to our considerable surprise, we have
found that every algebraic and/or topological object of mathematics is
representable as a Chu space, in such a way that the above
transformations are in one-to-one correspondence with those functions
that are homomorphisms and/or continuous when definable traditionally.
Restating this in categorical language, the many diverse categories of
mathematics all embed fully and concretely in different regions of the
one category of Chu spaces, whose objects may then be understood as
spanning the gamut from sets as the most discrete objects of
mathematics to complete atomic Boolean algebras, aka antisets, as the
most coherent.
There being no explicit notion of either signature or theory here, this
implies that every Chu space must come with both signature and theory
built in implicitly, along with the ability to communicate
appropriately with objects having both similar and dissimilar
signatures. The talk will focus on explaining how such an
unsophisticated mechanism can exhibit such sophisticated behavior.
About the Speaker:
Vaughan Pratt is Professor of Computer Science at Stanford
University. From 1972 to 1982 he was on the Electrical Engineering
and Computer Science faculty at MIT. He helped found Sun Microsystems
in 1982 and is the designer of Sun's logo and the Pixrect graphics
system. He worked in natural language, analysis of algorithms, and
logics of programs in the 1970's, and concurrency modeling, computer
graphics, and digital typography in the 1980's. His current interest
is Chu spaces and their many applications. His bachelor's and master's
degrees are from Sydney University, and his Ph.D. (1972) is from
Stanford University, advisor Donald Knuth.
This is the final speaker in the Distinguished Lecturer Series of the
DIMACS Special Year on Logic and Algorithms.
For additional information about the DIMACS special year, please see the
following web site:
