Dagger category
Encyclopedia
In mathematics
, a dagger category (also called involutive category or category with involution ) is a category
equipped with a certain structure called dagger or involution. The name dagger category was coined by Selinger.
equipped with an involutive, identity-on-object functor
.
In detail, this means that it associates to every morphism
in 
its adjoint 
such that for all 
and 
,
Note that in the previous definition, the term adjoint is used in the linear-algebraic sense, not in the category theoretic sense.
Some reputable sources additionally require for a category with involution that its set of morphisms is partially ordered and that the order of morphisms is compatible with the composition of morphisms, that is a<b implies
for morphisms a, b, c whenever their sources and targets are compatible.
, a morphism
is called
The terms unitary and self-adjoint in the previous definition are taken from the category of Hilbert spaces
where the morphisms satisfying those properties are then unitary
and self-adjoint
in the usual sense.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a dagger category (also called involutive category or category with involution ) is a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
equipped with a certain structure called dagger or involution. The name dagger category was coined by Selinger.
Formal definition
A dagger category is a categoryCategory (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
equipped with an involutive, identity-on-object functorFunctor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
.In detail, this means that it associates to every morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
in its adjoint such that for all and ,
Note that in the previous definition, the term adjoint is used in the linear-algebraic sense, not in the category theoretic sense.
Some reputable sources additionally require for a category with involution that its set of morphisms is partially ordered and that the order of morphisms is compatible with the composition of morphisms, that is a<b implies
for morphisms a, b, c whenever their sources and targets are compatible.Examples
- The categoryCategory (mathematics)In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
Rel of sets and relationsCategory of relationsIn mathematics, the category Rel has the class of sets as objects and binary relations as morphisms.A morphism R : A → B in this category is a relation between the sets A and B, so ....
possesses a dagger structure i.e. for a given relationRelation (mathematics)In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
in Rel, the relation 
is the relational converseInverse relationIn mathematics, the inverse relation of a binary relation is the relation that occurs when you switch the order of the elements in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'...
of
.
- A self-adjoint morphism is a symmetric relationSymmetric relationIn mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
.
- A self-adjoint morphism is a symmetric relation
- The categoryCategory (mathematics)In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
FdHilb of finite dimensional Hilbert spacesCategory of finite dimensional Hilbert spacesIn mathematics, the category FdHilb has all finite dimensional Hilbert spaces for objects and linear transformations between them.-Properties:This category* is monoidal,* possesses finite biproducts, and* is dagger compact....
also possesses a dagger structure: Given a linear map
, the map 
is just its adjointHermitian adjointIn mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations...
in the usual sense.
Remarkable morphisms
In a dagger category, a morphismMorphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
is called
- unitary if
; - self-adjoint if
(this is only possible for an endomorphismEndomorphismIn mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
).
The terms unitary and self-adjoint in the previous definition are taken from the category of Hilbert spaces
Category of finite dimensional Hilbert spaces
In mathematics, the category FdHilb has all finite dimensional Hilbert spaces for objects and linear transformations between them.-Properties:This category* is monoidal,* possesses finite biproducts, and* is dagger compact....
where the morphisms satisfying those properties are then unitary
Unitary transformation
In mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation....
and self-adjoint
Self-adjoint
In mathematics, an element x of a star-algebra is self-adjoint if x^*=x.A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation...
in the usual sense.