(n, m)-category

In mathematics, specifically in category theory, an (n, m)-category is an n-category all of whose j-morphisms for $j>m$ are invertible. This concept 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.

This notion is also used in infinity categories; for example, in a quasi-category, all morphisms dimension $\geq 2$ 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 $k>n$.

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 $j>m$ 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 $>n$ are identities. This definition allows us to define the characterization of (n, m)-categories, which includes the case of prosets.

An (n, m)-category is an ∞-category such that:

• All j-morphisms for $j>n$ exist and are unique wherever possible. In particular, this implies that all parallel (n + 1)-morphisms are equal.
• All j-morphisms for $j>m$ 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.

• 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. Some authors use this to define a weak 2-category, but in standard terminology, 2-category refers to a strict bicategory, where 2-morphisms are not required to be invertible.

• An (n, n)-category is an n-category.

• 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 poset).

• An (∞, 1)-category is a not-necessarily-quasi-category ∞-category in which all n-morphisms for $n>1$ 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.

See also

• Joyal's theta category

References

• Baez, John C.; Shulman, Michael (2010). "Towards Higher Categories". Lectures on n-Categories and Cohomology. The IMA Volumes in Mathematics and its Applications. Vol. 152. pp. 1–68. arXiv:math/0608420. doi:10.1007/978-1-4419-1524-5_1. ISBN 978-1-4419-1523-8.
• Gagna, Andrea; Harpaz, Yonatan; Lanari, Edoardo (October 2022). "On the equivalence of all models for (∞,2)‐categories". Journal of the London Mathematical Society. 106 (3): 1920–1982. arXiv:1911.01905. doi:10.1112/jlms.12614.
• Lurie, Jacob (2008). "On the Classification of Topological Field Theories". Current Developments in Mathematics: 129–280. arXiv:0905.0465. doi:10.4310/CDM.2008.v2008.n1.a3.
• Simpson, Carlos (2011). "Fundamental elements of n -categories". Homotopy Theory of Higher Categories. pp. 51–64. doi:10.1017/CBO9780511978111.004. ISBN 978-0-521-51695-2.
• Lurie, Jacob (2009). "On the Classification of Topological Field Theories". Current Developments in Mathematics. 2008: 129–280. arXiv:0905.0465. doi:10.4310/CDM.2008.v2008.n1.a3.
• Bergner, Julia E.; Rezk, Charles (2013). "Comparison of models for (∞ ,n )–categories, I". Geometry & Topology. 17 (4): 2163–2202. arXiv:1204.2013. doi:10.2140/gt.2013.17.2163.

External links

• Barwick, Clark; Schommer-Pries, Christopher (2011). "On the Unicity of the Homotopy Theory of Higher Categories". arXiv:1112.0040 [math.AT].
• Haugseng, Rune (2025). "Yet another introduction to ∞-categories" (PDF).
• Loubaton, Félix (2024). "Categorical Theory of (∞,ω)-Categories". arXiv:2406.05425 [math.CT].
• "Remark 11.4.0.5". Kerodon.
• "2.2.8 The Homotopy Category of a Bicategory (Definition 2.2.8.5. and Warning 2.2.8.6.)". Kerodon.
• "(n,r)-category". nLab.
• "(n,1)-category". nLab.
• "(0,1)-category". nLab.
• "(2,1)-category". nLab.
• "poset – As a category with extra properties". nLab.
• "relation between preorders and (0,1)-categories". nLab.
• "Is there an accepted definition of (∞,∞) category?". MathOverflow.
• Bergner, Julia E. (2018). "A survey of models for (∞, n)-categories". arXiv:1810.10052 [math.AT].
• "(infinity,n)-category". ncatlab.org.
• (∞,n)-category in Japanese