The following paper is available on RAG Seely's WWW home page at <http://www.math.mcgill.ca/~rags> or directly by ftp at <ftp://triples.math.mcgill.ca/pub/rags/linear/trace.ps.gz> or <ftp://triples.math.mcgill.ca/pub/rags/linear/trace.dvi.gz> Comments are most welcome; please send them to any of the authors. Any problems in obtaining the paper should be sent to rags@math.mcgill.ca. Feedback for linearly distributive categories: traces and fixpoints by R.F. Blute J.R.B. Cockett R.A.G. Seely ABSTRACT In the present paper, we develop the notion of a trace operator on a linearly distributive category, which amounts to essentially working within a subcategory (the "core") which has the same sort of "type degeneracy" as a compact closed category. We also explore the possibility that an object may have several trace structures, introducing a notion of compatibility in this case. We show that if we restrict to compatible classes of trace operators, an object may have at most one trace structure (for a given tensor structure). We give a linearly distributive version of the "geometry of interaction" construction, and verify that we obtain a linearly distributive category in which traces become canonical. We explore the relationship between our notions of trace and fixpoint operators, and show that an object admits a fixpoint combinator precisely when it admits a trace and is a cocommutative comonoid. This generalises an observation of Hyland and Hasegawa. This paper is presented to Bill Lawvere on the occasion of his 60th birthday. =================================== RAG Seely <rags@math.mcgill.ca> <http://www.math.mcgill.ca> [ NB - please use the "generic" email address above and not machine specific e-addresses like "rags@triples.math.mcgill.ca" ] ===================================