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
• supercategory or involutive category
• skeletal category
• category of interest [Orz72]
• extensive category and lextensive category [CLW93]
• dagger category
• gaunt category [BS21]
• restriction category [CL02]
• locally generated category [DR22]
• moment category [Ber22]
• adhesive category [LS04]

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.

[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.

[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.