Dagger compact category

In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations (that is, Tannakian categories).[1] They also appeared in the work of John Baez and James Dolan as an instance of semistrict k-tuply monoidal n-categories, which describe general topological quantum field theories,[2] for n = 1 and k = 3. They are a fundamental structure in Samson Abramsky and Bob Coecke's categorical quantum mechanics.[3][4][5]

  1. ^ Doplicher, S.; Roberts, J. (1989). "A new duality theory for compact groups". Invent. Math. 98: 157–218. Bibcode:1989InMat..98..157D. doi:10.1007/BF01388849. S2CID 120280418.
  2. ^ Baez, J.C.; Dolan, J. (1995). "Higher-dimensional Algebra and Topological Quantum Field Theory". J. Math. Phys. 36 (11): 6073–6105. arXiv:q-alg/9503002. Bibcode:1995JMP....36.6073B. CiteSeerX 10.1.1.269.4681. doi:10.1063/1.531236. S2CID 14908618.
  3. ^ Abramsky, S.; Coecke, B. (2004). "A categorical semantics of quantum protocols". Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE. pp. 415–425. arXiv:quant-ph/0402130. CiteSeerX 10.1.1.330.7289. doi:10.1109/LICS.2004.1319636. ISBN 0-7695-2192-4. S2CID 1980118.
  4. ^ Abramsky, S.; Coecke, B. (2009). "Categorical quantum mechanics". In Engesser, K.; Gabbay, D.M.; Lehmann, D. (eds.). Handbook of Quantum Logic and Quantum Structures. Elsevier. pp. 261–323. arXiv:0808.1023. ISBN 978-0-08-093166-1.
  5. ^ Abramsky and Coecke used the term strongly compact closed categories, since a dagger compact category is a compact closed category augmented with a covariant involutive monoidal endofunctor.