2-category

Generalization of category In category theory in mathematics, a 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transformation between functors. The concept of a strict 2-category was first introduced by Charles EhresmannEhresmann 2004 in his work on enriched categories in 1965.Ehresmann 1965 The more general concept of bicategory (or weak 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1967 by Jean Bénabou.Bénabou 1967 A (2, 1)-category is a 2-category where each 2-morphism is invertible. Definitions A strict 2-category By definition, a strict 2-category C consists of the data:Kelly Street 1974 § 1.2. a class of 0-cells, for each pairs of 0-cells , a set called the set of 1-cells from to , for each pairs of 1-cells in the same hom-set, a set called the set of 2-cells from to , ordinary compositions: maps , vertical compositions: maps , where are in the same hom-set, horizontal compositions: maps for and that are subject to the following conditions the 0-cells with 1-cells between them form a category under ordinary composition, for each 0-cells and , the 1-cells from to with 2-cells between them form a category under vertical composition, the 0-cells with 2-cells between 1-cells between them form a category under horizontal composition; namely, an object is a 0-cell and the hom-set from to is the set of all 2-cells of the form for some , the interchange law: , when defined, is the same as . The 0-cells, 1-cells, and 2-cells terminology is replaced by 0-morphisms, 1-morphisms, and 2-morphisms in some sources2-category in nLab (see also Higher category theory). Vertical compositions and horizontal compositions are also written as . The interchange law can be drawn as a pasting diagram as follows: 240px  =  240px  =  132px 132px 240px Here the left-hand diagram denotes the vertical composition of horizontal composites, the right-hand diagram denotes the horizontal composition of vertical composites, and the diagram in the centre is the customary representation of both. The 2-cell are drawn with double arrows ⇒, the 1-cell with single arrows →, and the 0-cell with points. Since the definition, as can be seen, is not short, in practice, it is more common to use some generalization of category theory such as higher category theory (see below) or enriched category theory to define a strict 2-category. The notion of strict 2-category differs from the more general notion of a weak 2-category defined below in that composition of 1-cells (horizontal composition) is required to be strictly associative, whereas in the weak version, it needs only be associative up to a coherent 2-isomorphism. As a category enriched over Cat Given a monoidal category V, a category C enriched over V is an abstract version of a category; namely, it consists of the data a class of objects, for each pair of objects , a hom-object in , compositions: morphisms in , identities: morphisms in that are subject to the associativity and the unit axioms. In particular, if is the category of sets with cartesian product, then a category enriched over it is an ordinary category. If , the category of small categories with product of categories, then a category enriched over it is exactly a strict 2-category. Indeed, has a structure of a category; so it gives the 2-cells and vertical compositions. Also, each composition is a functor; in particular, it sends 2-cells to 2-cells and that gives the horizontal compositions. The interchange law is a consequence of the functoriality of the compositions. A similar process for 3-categories leads to tricategories, and more generally to weak n-categories for n-categories, although such an inductive approach is not necessarily common today. A weak 2-category A weak 2-category or a bicategory can be defined exactly the same way a strict 2-category is defined except that the horizontal composition is required to be associative up to a coherent isomorphism. The coherent condition here is similar to those needed for monoidal categories; thus, for example, a monoidal category is the same as a weak 2-category with one 0-cell.Jurčo Sämann Wolf 2015 Definition 2.6. In higher category theory, if C is an ∞-category (a weak Kan complex) whose structure is determined only by 0-simplexes, 1-simplexes and 2-simplexes, then it is a weak (2, 1)-category; i.e., a weak 2-category in which every 2-morphism is invertible. So, a weak 2-category is an (∞, 2)-category whose structure is determined only by 0, 1, 2-simplexes. Examples Category of small categories The archetypal 2-category is the category of small categories, with natural transformations serving as 2-morphisms.Kelly Street 1974 § 1.3. The objects (0-cells) are all small categories, and for objects a and b the hom-set acquires a structure of a category as a functor category. A vertical composition isVertical composition in nLab the composition of natural transformations. Similarly, given a monoidal category V, the category of (small) categories enriched over V is a 2-category. Also, if is a category, then the comma category is a 2-category with natural transformations that map to the identity. Grpd Like Cat, groupoids (categories where morphisms are invertible) form a 2-category, where a 2-morphism is a natural transformation. Often, one also considers Grpd where all 2-morphisms are invertible transformations. In the latter case, it is a (2, 1)-category. Ord The category Ord of preordered sets is a 2-category since each hom-set has a natural preordered structure; thus a category structure by for each element x. More generally, the category of ordered objects in some category is a 2-category. Boolean monoidal category Consider a simple monoidal category, such as the monoidal preorder BoolFong Spivak 2018 based on the monoid M = ({T, F}, ∧, T). As a category this is presented with two objects {T, F} and single morphism g: F → T. We can reinterpret this monoid as a bicategory with a single object x (one 0-cell); this construction is analogous to construction of a small category from a monoid. The objects {T, F} become morphisms, and the morphism g becomes a natural transformation (forming a functor category for the single hom-category B(x, x)). Coherence theorem Every bicategory is "biequivalent"Leinster 1998 2.2 Biequivalence to a 2-category.Leinster 1998 2.3 The Coherence TheoremGurski 2013 2.2 Coherence for bicategoriesJohnson Yau 2021 Theorem 8.4.1 This is an instance of strictification (a process of replacing coherent isomorphisms with equalities.) Duskin nerve The Duskin nerve of a 2-category C is a simplicial set where each n-simplex is determined by the following data: n objects , morphisms and 2-morphisms that are subject to the (obvious) compatibility conditions.Khan 2023 Construction 1.5.1. Then the following are equivalent: Khan 2023 Theorem 1.5.2. is a (2, 1)-category; i.e., each 2-morphism is invertible. is a weak Kan complex (i.e., an ∞-category). The Duskin nerve is an instance of the homotopy coherent nerve. Functors and natural transformations By definition, a functor is simply a structure-preserving map; i.e., objects map to objects, morphisms to morphisms, etc. So, a 2-functor between 2-categories can be defined exactly the same way.Kelly Street 1974 § 1.4.2-functor in nLab In practice though, this notion of a 2-functor is not used much. It is far more common to use their lax analogs (just as a weak 2-category is used more). Let C,D be bicategories. We denote composition in "diagrammatic order".Composition in nLab A lax functor P from C to D, denoted , consists of the following data: for each object x in C, an object ; for each pair of objects x,y ∈ C a functor on morphism-categories, ; for each object x∈C, a 2-morphism in D; for each triple of objects, x,y,z ∈C, a 2-morphism in D that is natural in f: x→y and g: y→z. These must satisfy three commutative diagrams, which record the interaction between left unity, right unity, and associativity between C and D.Pseudofunctor in nLab A lax functor in which all of the structure 2-morphisms, i.e. the and above, are invertible is called a pseudofunctor. There is also a lax version of a natural transformation. Let C and D be 2-categories, and let be 2-functors. A lax natural transformation between them consists of a morphism in D for every object and a 2-morphism for every morphism in C satisfying some equations (see Lax natural transformation in nLab or Gray 1974). Related notion: double category While a strict 2-category is a category enriched over Cat, a category internal to Cat is called a double category. See also n-category Doctrine (mathematics) Pseudofunctor String diagram 2-Yoneda lemma Pasting theorem Footnotes References Jean Bénabou https://link.springer.com/chapter/10.1007/BFb0074299 10.1007/BFb0074299 Introduction to bicategories Reports of the Midwest Category Seminar Lecture Notes in Mathematics 1967 Bénabou Jean 47 1–77 978-3-540-03918-1 https://books.google.com/books?id=WKTk1n6ggKsC Generalised Algebraic Models 978-2-930344-78-2 Centazzo Claudia 2004 Presses univ. de Louvain https://books.google.com/books?id=Q2WnAAAAIAAJ Catégories et structures Ehresmann Charles 1965 Dunod, Paris 0213410 1199888 Charles Ehresmann Fong Brendan Spivak David I. 2018-10-12 Seven Sketches in Compositionality: An Invitation to Applied Category Theory math.CT 1803.05316 https://link.springer.com/book/10.1007/BFb0061280 10.1007/BFb0061280 Formal Category Theory: Adjointness for 2-Categories Lecture Notes in Mathematics 1974 391 978-3-540-06830-3 John W. Gray 10.1017/CBO9781139542333.003 Coherence for bicategories Coherence in Three-Dimensional Category Theory 2013 21–34 978-1-139-54233-3 Gurski Nick 2310.12456 Khan Adeel A. Lectures on algebraic stacks 2023 math.AG Max Kelly Ross Street Kelly G. M. Street Ross Kelly Gregory M. Review of the elements of 2-categories 10.1007/BFb0063101 978-3-540-06966-9 357542 75–103 Springer Lecture Notes in Mathematics Category Seminar: Proceedings of the Sydney Category Theory Seminar, 1972/1973 420 1974 Max Kelly Kelly Gregory Maxwell 0-521-28702-2 651714 Cambridge University Press, Cambridge-New York London Mathematical Society Lecture Note Series Basic concepts of enriched category theory 64 1982 http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf Laumon Gérard Gérard Laumon Moret-Bailly Laurent Champs algébriques Springer-Verlag Berlin, New York Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 978-3-540-65761-3 1771927 2000 39 10.1007/978-3-540-24899-6 Tom Leinster math/9810017 Leinster Tom Basic Bicategories 1998 2002.06055 10.1093/oso/9780198871378.001.0001 2-Dimensional Categories 2021 Johnson Niles Yau Donald 978-0-19-887137-8 Oxford University Press http://hdl.handle.net/1773/20977 EPrint Collection, University of Washington 1773/20977 Fibrations and Sheaves 13 December 2012 Warner Garth Jurčo Branislav Sämann Christian Wolf Martin Semistrict higher gauge theory Journal of High Energy Physics April 2015 2015 4 10.1007/JHEP04(2015)087 1403.7185 Further reading https://link.springer.com/chapter/10.1007/978-1-4419-1524-5_4 10.1007/978-1-4419-1524-5_4 A 2-Categories Companion Towards Higher Categories The IMA Volumes in Mathematics and its Applications 2010 Lack Stephen 152 105–191 978-1-4419-1523-8 math/0702535 2007math......2535L 10.1016/0022-4049(85)90087-8 Coherence for bicategories and indexed categories 1985 MacLane Saunders Saunders Mac Lane Paré Robert Journal of Pure and Applied Algebra 37 59–80 2.2 The Theory of 2-Categories https://kerodon.net/tag/007K Kerodon https://golem.ph.utexas.edu/category/2006/12/bicat_is_not_triequivalent_to.html External links Vertical composition in nLab https://ncatlab.org/nlab/show/vertical+composition Vertical composition in nLab ncatlab.org Leinster Tom January 9, 2007 A survey of the theory of bicategories https://webhomes.maths.ed.ac.uk/~tl/toronto/bicats.pdf Higher categories and their applications Fields Institute, Toronto 2-category in nLab https://ncatlab.org/nlab/show/2-category ncatlab.org 2-category in nLab 2-functor in nLab https://ncatlab.org/nlab/show/2-functor ncatlab.org 2-functor in nLab Bicategory in nLab https://ncatlab.org/nlab/show/bicategory ncatlab.org Bicategory in nLab Composition in nLab https://ncatlab.org/nlab/show/composition ncatlab.org Composition in nLab Lax natural transformation in nLab https://ncatlab.org/nlab/show/lax+natural+transformation ncatlab.org Lax natural transformation in nLab Pseudofunctor in nLab https://ncatlab.org/nlab/show/pseudofunctor ncatlab.org Pseudofunctor in nLab Andree Ehresmann Andrée Ehresmann Re: Who invented n-categories? categories.org.au 2004 https://categories.org.au/archives/list/categories-list@categories.org.au/message/IKN3VCPLQZF4MV56MIAJ4LKFERYJW3HI/ Ehresmann2004 Category:Higher category theory