horizontal composition in nLab
Article Images
Context
2-Category theory
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Higher category theory
Basic concepts
Basic theorems
- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle
Applications
Models
- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category
- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory
Morphisms
Functors
Universal constructions
Extra properties and structure
1-categorical presentations
Contents
Definition
In a 2-category, the composition of 2-morphisms along objects is called horizontal composition .
category: svg
This is in contrast to the vertical composition of 2-morphisms, which is their composition along 1-morphisms.
Properties
Horizontal and vertical composition are subject to the compatibility condition called the interchange law, which in this case and the higher categorical ones means that if we fix three objects , , and then the composition of 1-cells between them defines a functor.
Examples
- In Cat, horizontal composition is the Godement product of natural transformations.
References
Horizontal composition of natural transformations is first describes in
- Roger Godement, Appendix (pp. 269) of: Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]
for textbook accounts see most of those listed at category, such as
- Saunders MacLane, §II.5, p. 43 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
Last revised on August 18, 2023 at 15:20:12. See the history of this page for a list of all contributions to it.