Feedback for linearly distributive categories: traces and fixpoints

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.

[1]  R. Ho.  Algebraic Topology , 2022 .

[2]  Michael Barr.  Duality of vector spaces , 1976 .

[3]  D. A. Higgs,et al.  Nuclearity in the category of complete semilattices , 1989 .

[4]  Jean-Yves Girard,et al.  Geometry of interaction 2: deadlock-free algorithms , 1990, Conference on Computer Logic.

[5]  Hongde Hu,et al.  Coherence completions of categories and their enriched softness , 1997, MFPS.

[6]  R. A. G. Seely,et al.  Weakly distributive categories , 1997 .

[7]  P. Panangaden,et al.  Nuclear and trace ideals in tensored-categories , 1998, math/9805102.

[8]  J. Robin B. Cockett,et al.  ! and ? – Storage as tensorial strength , 1996, Mathematical Structures in Computer Science.

[9]  Ross Street,et al.  Traced monoidal categories , 1996 .

[10]  J.R.B. Cockett,et al.  Applications of Categories in Computer Science: Weakly distributive categories , 1992 .

[11]  Masahito Hasegawa,et al.  Models of Sharing Graphs , 1999, Distinguished Dissertations.

[12]  Philip J. Scott,et al.  Linear Läuchli Semantics , 1996, Ann. Pure Appl. Log..

[13]  A. Joyal,et al.  The geometry of tensor calculus, I , 1991 .

[14]  R. A. G. Seely,et al.  Linearly distributive functors , 1999 .

[15]  R. Seely,et al.  Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories. , 1997 .

[16]  R. Blute,et al.  Natural deduction and coherence for weakly distributive categories , 1996 .

[17]  Alex K. Simpson.  A Characterisation of the Least-Fixed-Point Operator by Dinaturality , 1993, Theor. Comput. Sci..

[18]  Jean-Yves Girard,et al.  Geometry of Interaction 1: Interpretation of System F , 1989 .

[19]  Radha Jagadeesan,et al.  New Foundations for the Geometry of Interaction , 1994, Inf. Comput..

[20]  Samson Abramsky,et al.  Retracing some paths in Process Algebra , 1996, CONCUR.