Involutive Categories and Monoids, with a GNS-Correspondence

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. A part of the so-called Gelfand–Naimark–Segal (GNS) construction is identified as an isomorphism of categories, relating states on involutive monoids and inner products. This correspondence exists in arbritrary involutive symmetric monoidal categories.

[1]  Bart Jacobs,et al.  Categorical Logic and Type Theory , 2001, Studies in logic and the foundations of mathematics.

[2]  G. M. Kelly,et al.  Two-dimensional monad theory , 1989 .

[3]  Ana Sokolova,et al.  The Microcosm Principle and Concurrency in Coalgebra , 2008, FoSSaCS.

[4]  Anders Kock,et al.  Closed categories generated by commutative monads , 1971, Journal of the Australian Mathematical Society.

[5]  William Arveson,et al.  A Short Course on Spectral Theory , 2001 .

[6]  Samson Abramsky,et al.  Categorical quantum mechanics , 2008, 0808.1023.

[7]  E. Beggs,et al.  Bar Categories and Star Operations , 2006, math/0701008.

[8]  J. M. EGGER,et al.  On involutive monoidal categories , 2011 .

[9]  Bart Jacobs.  Coalgebraic Walks, in Quantum and Turing Computation , 2011, FoSSaCS.

[10]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[11]  W. Arveson.  An Invitation To C*-Algebras , 1976 .

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

[13]  M. Barr,et al.  Toposes, Triples and Theories , 1984 .

[14]  Ichiro Hasuo,et al.  Tracing Anonymity with Coalgebras , 2008 .

[15]  Sylvia Pulmannová,et al.  New trends in quantum structures , 2000 .

[16]  Ana Sokolova,et al.  Coalgebraic Components in a Many-Sorted Microcosm , 2009, CALCO.

[17]  Adam Grabowski,et al.  Orthomodular Lattices , 2008, Formaliz. Math..

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

[19]  Alexander Katovsky,et al.  Category Theory , 2010, Arch. Formal Proofs.

[20]  Robin Milner,et al.  Theories for the Global Ubiquitous Computer , 2004, FoSSaCS.

[21]  James Dolan,et al.  Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes , 1997 .

[22]  Sally Popkorn,et al.  A Handbook of Categorical Algebra , 2009 .

[23]  Prakash Panangaden,et al.  Dagger Categories and Formal Distributions , 2010 .

[24]  Dov M. Gabbay,et al.  Handbook of Quantum logic and Quantum Structures , 2007 .

[25]  S. Lack,et al.  The formal theory of monads II , 2002 .

[26]  A. Kock.  Bilinearity and Cartesian Closed Monads. , 1971 .