Involutive categories and monoids, with a GNS-correspondence. (English) Zbl 1259.81032
Summary: 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 arbitrary involutive symmetric monoidal categories.
MSC:
| 81R15 | Operator algebra methods applied to problems in quantum theory |
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 06F05 | Ordered semigroups and monoids |
| 46K05 | General theory of topological algebras with involution |
| 55T20 | Eilenberg-Moore spectral sequences |
| 46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
References:
| [1] | Abramsky, S., Blute, R., Panangaden, P.: Nuclear and trace ideals in tensored categories. J. Pure Appl. Algebra 143, 3–47 (2000) · Zbl 0946.18004 · doi:10.1016/S0022-4049(98)00106-6 |
| [2] | Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Engesser, K., Gabbai, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures, pp. 261–323. North Holland/Elsevier/Computer Science, Amsterdam (2009) · Zbl 1273.81014 |
| [3] | Arveson, W.: A Short Course on Spectral Theory. Springer, Berlin (2002) · Zbl 0997.47001 |
| [4] | Awodey, S.: Category Theory. Oxford Logic Guides. Oxford Univ. Press, London (2006) · Zbl 1100.18001 |
| [5] | Baez, J.C., Dolan, J.: Higher dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math. 135, 145–206 (1998) · Zbl 0909.18006 · doi:10.1006/aima.1997.1695 |
| [6] | Barr, M., Wells, Ch.: Toposes, Triples and Theories. Springer, Berlin (1985). Revised and corrected version available from URL: www.cwru.edu/artsci/math/wells/pub/ttt.html |
| [7] | Beggs, E.J., Majid, S.: Bar categories and star operations. Algebr. Represent. Theory 12, 103–152 (2009) · Zbl 1180.18002 · doi:10.1007/s10468-009-9141-x |
| [8] | Blackwell, R., Kelly, G.M., Power, A.J.: Two-dimensional monad theory. J. Pure Appl. Algebra 59, 1–41 (1989) · Zbl 0675.18006 · doi:10.1016/0022-4049(89)90160-6 |
| [9] | Blute, R., Panangaden, P.: Dagger categories and formal distributions. In: Coecke, B. (ed.) New Structures in Physics. Lect. Notes Physics, vol. 813, pp. 421–436. Springer, Berlin (2010) · Zbl 1253.81006 |
| [10] | Borceux, F.: Handbook of Categorical Algebra. Encyclopedia of Mathematics, vols. 50–52. Cambridge Univ. Press, Cambridge (1994) |
| [11] | Coumans, D., Jacobs, B.: Scalars, monads and categories (2010). arXiv:1003.0585 |
| [12] | Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht (2000) · Zbl 0987.81005 |
| [13] | Egger, J.M.: Involutive monoidal categories (2010). Available via http://homepages.inf.ed.ac.uk/jegger |
| [14] | Hasuo, I.: Tracing anonymity with coalgebras. PhD thesis, Radboud Univ. Nijmegen (2010) |
| [15] | Hasuo, I., Heunen, C., Jacobs, B., Sokolova, A.: Coalgebraic components in a many-sorted microcosm. In: Kurz, A., Tarlecki, A. (eds.) Conference on Algebra and Coalgebra in Computer Science (CALCO 2009). Lect. Notes Comp. Sci., vol. 5728, pp. 64–80. Springer, Berlin (2009) · Zbl 1239.68044 |
| [16] | Hasuo, I., Jacobs, B., Sokolova, A.: The microcosm principle and concurrency in coalgebra. In: Amadio, R. (ed.) Foundations of Software Science and Computation Structures. LNCS, vol. 4962, pp. 246–260. Springer, Berlin (2008) · Zbl 1139.68042 |
| [17] | Jacobs, B.: Categorical Logic and Type Theory. North Holland, Amsterdam (1999) · Zbl 0911.03001 |
| [18] | Jacobs, B.: Coalgebraic walks in quantum and Turing computation. In: Hofmann, M. (ed.) Foundations of Software Science and Computation Structures. Lect. Notes Comp. Sci., vol. 6604, pp. 12–26. Springer, Berlin (2011) · Zbl 1326.68192 |
| [19] | Jacobs, B.: Dagger categories of tame relations (2011). arXiv:1101.1077 · Zbl 1294.68107 |
| [20] | Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983) · Zbl 0512.06011 |
| [21] | Kock, A.: Bilinearity and cartesian closed monads. Math. Scand. 29, 161–174 (1971) · Zbl 0253.18006 |
| [22] | Kock, A.: Closed categories generated by commutative monads. J. Aust. Math. Soc. XII, 405–424 (1971) · Zbl 0244.18007 · doi:10.1017/S1446788700010272 |
| [23] | Manes, E.G.: Algebraic Theories. Springer, Berlin (1974) · Zbl 0489.18003 |
| [24] | Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin (1971) · Zbl 0232.18001 |
| [25] | Street, R.: The formal theory of monads. J. Pure Appl. Algebra 2, 149–169 (1972) · Zbl 0241.18003 · doi:10.1016/0022-4049(72)90019-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
