In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
in mathematics, a 2-category is a
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
with "
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
between
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s.
The concept of a strict 2-category was first introduced by
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory.
He was an early member of the Bourbaki group, and is known for his work on the differentia ...
in his work on
enriched categories
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the ...
in 1965. The more general concept of bicategory (or weak 2-category), where composition of morphisms is
associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
only up to a 2-isomorphism, was introduced in 1967 by
Jean Bénabou
Jean Bénabou (1932 – 11 February 2022) was a French mathematician, known for his contributions to category theory. He directed the Research Seminar in Category Theory at the Institut Henri Poincaré
The Henri Poincaré Institute (or IHP fo ...
.
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:
* a
class
Class, Classes, or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used d ...
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, the 1-cells and the ordinary compositions form a category,
* for each
,
together with the vertical compositions is a category,
* the 2-cells together with the horizontal compositions form a category; namely, an object is a 0-cell and the hom-set from
to
is the set of all 2-cells of the form
with 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 sources (see also
Higher category theory
In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...
). Vertical compositions and horizontal compositions are also written as
.
The interchange law can be drawn as a
pasting diagram as follows:
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
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an Object (cate ...
''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
In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".
The adjectives such as "pseudo-" and "lax-" ...
. The coherent condition here is similar to those needed for
monoidal categories
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left an ...
; thus, for example, a monoidal category is the same as a weak 2-category with one 0-cell.
In higher category theory, if ''C'' is an
∞-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a Category (ma ...
(a
weak Kan complex
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. Th ...
) 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
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-c ...
, with natural transformations serving as 2-morphisms.
The objects (''0-cells'') are all small categories, and for objects and the hom-set
acquires a structure of a category as a
functor category
In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...
. A vertical composition is 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
In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a Category (mathematics), category to one another ...
is a 2-category with natural transformations that map to the identity.
Grpd
Like Cat,
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
* '' Group'' with a partial fu ...
s (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 set
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name is meant to suggest that preorders are ''almost'' partial orders, but not quite, as they are not necessar ...
s 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 object
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* H ...
s in some category is a 2-category.
Boolean monoidal category
Consider a simple
monoidal category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an Object (cate ...
, such as the monoidal preorder Bool based on the
monoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being .
Monoids are semigroups with identity ...
M = (,
∧
Wedge (∧) is a symbol that looks similar to an in-line caret (^). It is used to represent various operations. In Unicode, the symbol is encoded and by \wedge and \land in TeX. The opposite symbol (∨) is called a vel, or sometimes a (des ...
, T). As a category this is presented with two objects 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 become morphisms, and the morphism ''g'' becomes a natural transformation (forming a
functor category
In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...
for the single hom-category B(''x'', ''x'')).
Coherence theorem
*Every bicategory is "biequivalent" to a 2-category. This is an instance of
strictification In mathematics, specifically in category theory, a strictification refers to statements of the form “every weak structure of some sort is equivalent to a stricter one.” Such a result was first proven for monoidal categories by Mac Lane, and it ...
(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. Then the following are equivalent:
*
is a (2, 1)-category; i.e., each 2-morphism is invertible.
*
is a weak Kan complex.
The Duskin nerve is an instance of the
homotopy coherent nerve
In category theory, a discipline within mathematics, the nerve ''N''(''C'') of a small category ''C'' is a simplicial set constructed from the objects and morphisms of ''C''. The geometric realization of this simplicial set is a topological space, ...
.
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. 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". 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''.
A lax functor in which all of the structure 2-morphisms, i.e. the
and
above, are invertible is called a
pseudofunctor
In mathematics, a pseudofunctor ''F'' is a mapping from a category to the category Cat of (small) categories that is just like a functor except that F(f \circ g) = F(f) \circ F(g) and F(1) = 1 do not hold as exact equalities but only up to '' coh ...
.
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 or
)
Related notion: double category
While a strict 2-category is a category enriched over Cat, a category
internal
Internal may refer to:
*Internality as a concept in behavioural economics
*Neijia, internal styles of Chinese martial arts
*Neigong or "internal skills", a type of exercise in meditation associated with Daoism
* ''Internal'' (album) by Safia, 2016 ...
to Cat is called a
double category.
See also
*
''n''-category
*
Doctrine (mathematics) In mathematics, specifically category theory, a doctrine is roughly a system of theories ("categorical analogues of fragments of logical theories which have sufficient category-theoretic structure for their models to be described as functors"). Fo ...
*
Pseudofunctor
In mathematics, a pseudofunctor ''F'' is a mapping from a category to the category Cat of (small) categories that is just like a functor except that F(f \circ g) = F(f) \circ F(g) and F(1) = 1 do not hold as exact equalities but only up to '' coh ...
*
String diagram
*
2-Yoneda lemma
Footnotes
References
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Further reading
*
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External links
*
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{{Category theory
Higher category theory