Autonomous category
Encyclopedia
In mathematics
, an autonomous category is a monoidal category
where dual object
s exist.
where every object has a left (resp. right) dual
. An autonomous category is a monoidal category where every object has both a left and a right dual
. Rigid category
is a synonym for autonomous category.
In a symmetric monoidal category, the existence of left duals is equivalent to the existence of right duals, categories of this kind are called compact closed categories
.
The concepts of *-autonomous category
and autonomous category are directly related, specifically, every autonomous category is *-autonomous. A *-autonomous category may be described as a linearly distributive category with (left and right) negations; such categories have two monoidal products
linked with a sort of distributive law. In the case where the two monoidal products coincide and the distributivities are taken from the associativity isomorphism of the single monoidal structure, one obtains autonomous categories.
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...
, an autonomous category is a monoidal category
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...
where dual object
Dual object
In category theory, a branch of mathematics, it is possible to define a concept of dual object generalizing the concept of dual space in linear algebra.A category in which each object has a dual is called autonomous or rigid.-Definition:...
s exist.
Definition
A left (resp. right) autonomous category is a monoidal categoryMonoidal 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...
where every object has a left (resp. right) dual
Dual object
In category theory, a branch of mathematics, it is possible to define a concept of dual object generalizing the concept of dual space in linear algebra.A category in which each object has a dual is called autonomous or rigid.-Definition:...
. An autonomous category is a monoidal category where every object has both a left and a right dual
Dual object
In category theory, a branch of mathematics, it is possible to define a concept of dual object generalizing the concept of dual space in linear algebra.A category in which each object has a dual is called autonomous or rigid.-Definition:...
. Rigid category
Rigid category
In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual X* and a morphism 1 → X ⊗ X* satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or...
is a synonym for autonomous category.
In a symmetric monoidal category, the existence of left duals is equivalent to the existence of right duals, categories of this kind are called compact closed categories
Compact closed category
In category theory, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space...
.
The concepts of *-autonomous category
*-autonomous category
In mathematics, a *-autonomous category C is a symmetric monoidal closed category equipped with a dualizing object \bot.-Definition:Let C be a symmetric monoidal closed category...
and autonomous category are directly related, specifically, every autonomous category is *-autonomous. A *-autonomous category may be described as a linearly distributive category with (left and right) negations; such categories have two monoidal products
linked with a sort of distributive law. In the case where the two monoidal products coincide and the distributivities are taken from the associativity isomorphism of the single monoidal structure, one obtains autonomous categories.