(n, m)-category

May 2026 May 2026 In mathematics, specifically in category theory, an (n, m)-category is an n-category all of whose j-morphisms for are invertible. This notion has orthographic variation, such as (m, k)-category, (m, r)-category and etc. An (n, m)-category is considered a generalization of n-category and n-groupoid, and further (skeletal in poset case) (0, 1)-category can be defined from the notion of proset and poset. In an infinity category theory, assume that all j-morphisms of dimension greater than a certain dimension to be invertible, for example, in a quasi-category, all morphisms dimension are invertible, and this is called an (∞, 1)-category. We also have other (∞, 1)-category models, such as simplicial category, Segal category, and complete Segal space. More generally, an (∞, n)-category is a generalization of an (∞, 1)-category, where each k-morphism is invertible for . Definition of (n, m)-category Definition of (n, m)-category given by (weak) n-category An (n, m)-category is an (weak) n-category all of whose j-morphisms for are invertible. Definition of (n, m)-category given by ∞-category An (n, m)-category can be defined even if a instead of regarding an n-category as enriched over (n − 1)-categories, one return to regarding it as an ∞-category in which all cells of dimension are identities. The following definition give the characterization of (n, m)-categories, which includes the case of prosets. An (n, m)-category is an ∞-category such that:Baez Shulman 2010 Definition 8. All j-morphisms for exist and are unique wherever possible. In particular, this implies that all parallel (n + 1)-morphisms are equal. All j-morphisms for are invertible. Example A (0, 0)-category is up to equivalence the same as a set. A (1, 0)-category is a 1-groupoid. A groupoid is an ordinary category in which every morphism is invertible. A (2, 0)-category is a 2-groupoid. An ∞-groupoid is an (∞, 0)-category. An (n, 0)-category is a n-groupoid. An n-groupoid is an n-category where all morphisms are equivalences.Lurie 2008 Example 1.3.6. A (1, 1)-category is an ordinary category. A (2, 2)-category is a 2-category. A (2, 1)-category is a 2-category in which all 2-morphisms are invertible. For example, Lurie used this for the weakening of a 2-category and called it a bicategory, but in the terminology of standard 2-category theory, the 2-morphisms of a bicategory are not required to be invertible (He called this a strict bicategory).Definition 2.2.8.5. and Warning 2.2.8.6 in Kerodon An (n, n)-category is an n-category.Lurie A A (0, 1)-category (a.k.a thin category) can be seen as a proset (if it has a skeleton, then up to equivalence it is a posetrelation between preorders and (0,1)-categories in nLab). An (∞, 1)-category is a not-necessarily-quasi-category ∞-category in which all n-morphisms for are equivalences. An (∞, 2)-category has several models. For the equivalence of all models known in 2022 of the (∞, 2)-category, see Figure 1 by Gagna–Harpaz–Lanari.Gagna Harpaz Lanari 2022 Figure 1. See also Joyal's theta category Notes References Baez John C. Shulman Michael Lectures on n-Categories and Cohomology Towards Higher Categories The IMA Volumes in Mathematics and its Applications 2010 152 1–68 10.1007/978-1-4419-1524-5_1 math/0608420 978-1-4419-1523-8 Gagna Andrea Harpaz Yonatan Lanari Edoardo On the equivalence of all models for (∞,2)‐categories Journal of the London Mathematical Society October 2022 106 3 1920–1982 10.1112/jlms.12614 1911.01905 Lurie Jacob On the Classification of Topological Field Theories Current Developments in Mathematics 2008 129–280 10.4310/CDM.2008.v2008.n1.a3 0905.0465 Simpson Carlos Homotopy Theory of Higher Categories Fundamental elements of n -categories 2011 51–64 10.1017/CBO9780511978111.004 978-0-521-51695-2 Lurie Jacob Jacob Lurie On the Classification of Topological Field Theories Current Developments in Mathematics 2009 2008 129–280 0905.0465 10.4310/CDM.2008.v2008.n1.a3 Bergner Julia E. Rezk Charles Comparison of models for (∞ ,n )–categories, I Geometry & Topology 2013 17 4 2163–2202 10.2140/gt.2013.17.2163 1204.2013 External links Barwick Clark Clark Barwick Schommer-Pries Christopher On the Unicity of the Homotopy Theory of Higher Categories 2011 math.AT 1112.0040 2025 Haugseng Rune Yet another introduction to ∞-categories https://runegha.folk.ntnu.no/naivecat_web.pdf 2406.05425 Loubaton Félix Categorical Theory of (∞,ω)-Categories 2024 math.CT Remark 11.4.0.5. https://kerodon.net/tag/009R Kerodon 2.2.8 The Homotopy Category of a Bicategory (Definition 2.2.8.5. and Warning 2.2.8.6.) https://kerodon.net/tag/02GD Kerodon Definition 2.2.8.5. and Warning 2.2.8.6 in Kerodon (n,r)-category https://ncatlab.org/nlab/show/(n,r)-category nLab (n,1)-category https://ncatlab.org/nlab/show/%28n%2C1%29-category nLab (0,1)-category https://ncatlab.org/nlab/show/%280%2C1%29-category nLab (2,1)-category https://ncatlab.org/nlab/show/%282%2C1%29-category nLab poset – As a category with extra properties https://ncatlab.org/nlab/show/partial+order#AsACategoryWithExtraProperties nLab relation between preorders and (0,1)-categories https://ncatlab.org/nlab/show/relation+between+preorders+and+%280%2C1%29-categories nLab relation between preorders and (0,1)-categories in nLab Is there an accepted definition of (∞,∞) category? https://mathoverflow.net/questions/134057/is-there-an-accepted-definition-of-infty-infty-category MathOverflow en Bergner Julia E. A survey of models for (∞, n)-categories 2018 math.AT 1810.10052 (infinity,n)-category https://ncatlab.org/nlab/show/(infinity,n)-category ncatlab.org (∞,n)-category in Japanese Category:Categories in category theory Category:Higher category theory Category:Homotopy theory