Finite Dimensional Vector Spaces Are Complete for Traced Symmetric Monoidal Categories

We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two different arrows in the free traced category there always exists a strong traced functor into FinVectk which distinguishes them. Therefore two arrows in the free traced category are the same if and only if they agree for all interpretations in FinVectk.

[1]  Cristian S. Calude,et al.  Discrete Mathematics and Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

[2]  Zoltán Ésik,et al.  Some varieties of iteration theories , 1984, Bull. EATCS.

[3]  Aimo Tietäväinen.  Proof of a conjecture of S. Chowla , 1975 .

[4]  T. Kerler.  On braided tensor categories , 1994, hep-th/9402018.

[5]  Boris A. Trakhtenbrot,et al.  On the Power of Compositional Proofs for Nets: Relationships Between Completeness and Modularity , 1996, Fundam. Informaticae.

[6]  Andrzej Blikle.  Review: B. A. Trahtenbrot, On Operators Realizable in Logical Nets , 1962 .

[7]  Cristian S. Calude,et al.  Discrete Mathematics and Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

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

[9]  Alexander Moshe Rabinovich,et al.  Synchronous Circuits over Continuous Time: Feedback Reliability and mpleteness , 2004, Fundam. Informaticae.

[10]  Alexander Moshe Rabinovich,et al.  Nets and data flow interpreters , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[11]  Masahito Hasegawa,et al.  Models of sharing graphs : a categorical semantics of let and letrec , 1999 .

[12]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[13]  Alex K. Simpson.  Categorical completeness results for the simply-typed lambda-calculus , 1995, TLCA.

[14]  Dominic R. Verity,et al.  Traced monoidal categories , 1996, Mathematical Proceedings of the Cambridge Philosophical Society.

[15]  Jirí Adámek,et al.  What Are Iteration Theories? , 2007, MFCS.

[16]  Masahito Hasegawa,et al.  Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi , 1997, TLCA.

[17]  Sam Lindley,et al.  Extensional Rewriting with Sums , 2007, TLCA.

[18]  Sergei Soloviev,et al.  Proof of a Conjecture of S. Mac Lane , 1996, Ann. Pure Appl. Log..

[19]  Richard Statman,et al.  Completeness, invariance and λ-definability , 1982, Journal of Symbolic Logic.

[20]  Gordon D. Plotkin,et al.  Complete axioms for categorical fixed-point operators , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[21]  John Power,et al.  Symmetric Monoidal Sketches and Categories of Wirings , 2004, Electron. Notes Theor. Comput. Sci..

[22]  Alexander Moshe Rabinovich,et al.  Nets of processes and data flow , 1988, REX Workshop.