Dagger symmetric monoidal category
Encyclopedia
A dagger symmetric monoidal category is a monoidal category
which also possesses a dagger structure
; in other words, it means that this category comes equipped not only with a tensor
in the category theoretic
sense but also with dagger structure
which is used to describe unitary morphism
and self-adjoint morphisms
in
that is, a form of abstract analogues of those found in FdHilb, the category of finite dimensional Hilbert spaces
. This type of category
was introduced by Selinger as an intermediate structure between dagger categories
and the dagger compact categories
that are used in categorical quantum mechanics
, an area which now also considers dagger symmetric monoidal categories when dealing with infinite dimensional quantum mechanical concepts.
which also has a dagger structure
such that for all
, 
and all 
and 
in 
,
Here,
and 
are the natural isomorphisms from the symmetric monoidal structure.
are examples of dagger symmetric monoidal categories:
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
which also possesses a dagger structureDagger category
In mathematics, a dagger category is a category equipped with a certain structure called dagger or involution...
; in other words, it means that this category comes equipped not only with a tensor
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
in the category theoretic
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
sense but also with dagger structure
Dagger category
In mathematics, a dagger category is a category equipped with a certain structure called dagger or involution...
which is used to describe unitary morphism
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
and self-adjoint morphisms
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
that is, a form of abstract analogues of those found in FdHilb, the category of finite dimensional Hilbert spacesCategory 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....
. This type of 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...
was introduced by Selinger as an intermediate structure between dagger categories
Dagger category
In mathematics, a dagger category is a category equipped with a certain structure called dagger or involution...
and the dagger compact categories
Dagger compact category
In mathematics, dagger compact categories first appeared in 1989 in the work of Doplicher and Roberts on the reconstruction of compact topological group from their category of finite-dimensional continuous unitary representations...
that are used in categorical quantum mechanics
Categorical quantum mechanics
Categorical quantum mechanics is the study of quantum physics using methods from mathematics and computer science, notably category theory, that emphasize compositionality...
, an area which now also considers dagger symmetric monoidal categories when dealing with infinite dimensional quantum mechanical concepts.
Formal definition
A dagger symmetric monoidal category is a symmetric monoidal category which also has a dagger structureDagger category
In mathematics, a dagger category is a category equipped with a certain structure called dagger or involution...
such that for all
, and all and in ,
;
;
;
and
.
Here,
and are the natural isomorphisms from the symmetric monoidal structure.Examples
The following categoriesCategory (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...
are examples of dagger symmetric monoidal categories:
- 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 ....
where the tensor is given by the productProduct (category theory)In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
and where the dagger of a relation is given by its relational converse. - 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....
is a dagger symmetric monoidal category where the tensor is the usual tensor productTensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of Hilbert spaces and where the dagger of a linear map is given by its hermitian 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...
.