Important Classes of Categories

ここでは, 圏に様々な形容詞を付けたものを集めた。 ある性質をみたす圏の class であることもあるし, monoidal category のように圏に構造を付加したものであることもある。 整理しないで, 思い付いたもの (目にしたもの) を並べてみた。

• small category
• Abelian category
• exact category
• triangulated category
• locally presentable and accessible category
• acyclic category
• monoidal category
• enriched category
• internal category
• closed category and closed monoidal category
• model category
• simplicial category
• dg category
• \(C^{*}\)-category
• von Neumann category or \(W^{*}\)-category
• skeletal category
• category of interest [Orz72]
• extensive category and lextensive category [CLW93]
• dagger category [Sel07b; Sel07a]
• gaunt category [BS21]
• restriction category [CL02]
• supercategory [KKO13; Mur18]
• locally generated category [DR22]
• moment category [Ber22]
• adhesive category [LS04]
• involutive category [Yau20]

圏の概念を一般化したものについては, 次にまとめた。

• 圏の概念の一般化

References

[Ber22]

Clemens Berger. “Moment categories and operads”. In: Theory Appl. Categ. 38 (2022), Paper No. 39, 1485–1537. arXiv: 2102.00634. url: https://doi.org/10.5486/pmd.1991.38.1-2.06.

[BS21]

Clark Barwick and Christopher Schommer-Pries. “On the unicity of the theory of higher categories”. In: J. Amer. Math. Soc. 34.4 (2021), pp. 1011–1058. arXiv: 1112 . 0040. url: https://doi.org/10.1090/jams/972.

[CL02]

J. R. B. Cockett and Stephen Lack. “Restriction categories. I. Categories of partial maps”. In: Theoret. Comput. Sci. 270.1-2 (2002), pp. 223–259. url: http://dx.doi.org/10.1016/S0304-3975(00)00382-0.

[CLW93]

Aurelio Carboni, Stephen Lack, and R. F. C. Walters. “Introduction to extensive and distributive categories”. In: J. Pure Appl. Algebra 84.2 (1993), pp. 145–158. url: http://dx.doi.org/10.1016/0022-4049(93)90035-R.

[DR22]

I. Di Liberti and J. Rosický. “Enriched locally generated categories”. In: Theory Appl. Categ. 38 (2022), Paper No. 17, 661–683. arXiv: 2009.10980.

[KKO13]

Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh. “Supercategorification of quantum Kac-Moody algebras”. In: Adv. Math. 242 (2013), pp. 116–162. arXiv: 1206.5933. url: https://doi.org/10.1016/j.aim.2013.04.008.

[LS04]

Stephen Lack and Paweł Sobociński. “Adhesive categories”. In: Foundations of software science and computation structures. Vol. 2987. Lecture Notes in Comput. Sci. Berlin: Springer, 2004, pp. 273–288. url: http://dx.doi.org/10.1007/978-3-540-24727-2_20.

[Mur18]

Daniel Murfet. “The cut operation on matrix factorisations”. In: J. Pure Appl. Algebra 222.7 (2018), pp. 1911–1955. arXiv: 1402.4541. url: https://doi.org/10.1016/j.jpaa.2017.08.014.

[Orz72]

G. Orzech. “Obstruction theory in algebraic categories. I, II”. In: J. Pure Appl. Algebra 2 (1972), 287–314, ibid. 2 (1972), 315–340. url: https://doi.org/10.1016/0022-4049(72)90008-4.

[Sel07a]

P. Selinger, ed. Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005). Vol. 170. Electronic Notes in Theoretical Computer Science. Held at DePaul University, Chicago, IL, June 30–July 1, 2005. Elsevier Science B.V., Amsterdam, 2007, front matter+199.

[Sel07b]

Peter Selinger. “Dagger compact closed categories and completely positive maps”. In: Electronic Notes in Theoretical computer science 170 (2007), pp. 139–163.

[Yau20]

Donald Yau. Involutive category theory. Vol. 2279. Lecture Notes in Mathematics. Springer, Cham, [2020] ©2020, pp. xii+243. isbn: 978-3-030-61203-0; 978-3-030-61202-3. url: https://doi.org/10.1007/978-3-030-61203-0.