Who needs category theory?

07/30/2018 ∙ by Andreas Blass, et al. ∙ University of Michigan 0

In computer science, category theory remains a contentious issue, with enthusiastic fans and a skeptical majority. Categories were introduced by Samuel Eilenberg and Saunders Mac Lane as an auxiliary notion in their general theory of natural equivalences. Here we argue that something like categories is needed on a more basic level. As you work with operations on structures, it may be necessary to coherently manipulate isomorphism (or more generally homomorphism) witnesses for various properties of these operations, e.g.associativity, commutativity and distributivity. A working mathematician, to use Mac Lane's term, is well advised to be aware of the coherent witness-manipulation problem and to know that category theory is an appropriate framework to address the problem. Of course, the working mathematician in question may be a computer scientist or physicist.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

Code Repositories

Lecturas_GLC

Lecturas del Grupo de Lógica Computacional


view repo
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

References

  • [1] John Baez, in Opinions of Category Theory, https://www.arsmathematica.net/2006/06/24/opinions-of-category-theory/, June 29, 2006.
  • [2] Andreas Blass and Yuri Gurevich, ‘‘On quantum computation, anyons, and categories,’’ in Martin Davis on Computability, Computational Logic, and Mathematical Foundations, 209--241, Springer 2016.
  • [3] Andreas Blass and Yuri Gurevich, ‘‘Coherence for braided distributivity,’’ arXiv:1807.11403.
  • [4] Andreas Blass and Yuri Gurevich, ‘‘Witness algebra and anyon braiding,’’ arXiv:1807.10414.
  • [5] Samuel Eilenberg and Saunders Maclane, ‘‘General theory of natural equivalences,’’ Trans. Amer. Math. Society 58:2 (1945), 231--294.
  • [6] Peter J. Freyd, ‘‘Abelian categories,’’ in Reprints in Theory and Applications of Categories, No. 3 (2003), http://www.tac.mta.ca/tac/reprints/, originally published by Harper & Row in 1964.
  • [7] Miguel Laplaza, ‘‘Coherence for categories with associativity, commutativity and distributivity,’’ Bull. Amer. Math. Soc. 78 (1972), 220--222.
  • [8] Miguel Laplaza, ‘‘Coherence for distributivity,’’ in Coherence in Categories, Springer Lecture Notes in Mathematics 281 (1972), 29--65.
  • [9] Saunders Mac Lane, ‘‘Coherence and canonical maps,’’ Symposia Mathematica, IV (1970), 231--241.
  • [10] Saunders Mac Lane, ‘‘Categories for the working mathematician,’’ Springer 1971.
  • [11] Prakash Panangaden and Éric O. Paquette, ‘‘A categorical presentation of quantum computation with anyons,’’ Chapter 15 in New Structures for Physics, ed. Bob Coecke, Springer Lecture Notes in Physics 813 (2011), 983--1025.
  • [12] Zhenghan Wang, Topological Quantum Computation, CBMS Regional Conference Series in Mathematics, vol. 112, American Mathematical Society (2010).