In mathematics, an autonomous category is a

monoidal category 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 ...

where

dual object In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of dual ...

s exist.

Definition

A ''left'' (resp. ''right'') ''autonomous category'' is a

where every object has a left (resp. right)

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

. An ''autonomous category'' is a monoidal category where every object has both a left and a right

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''* (the internal Hom 'X'', 1 and a morphism 1 → ''X'' ⊗ ''X''* satisfying natural conditions. ...

is a synonym for autonomous category. In a

symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict s ...

, the existence of left duals is equivalent to the existence of right duals, categories of this kind are called (symmetric) compact closed categories. In

categorial grammar Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and sema ...

s, categories which are both left and right rigid are often called pregroups, and are employed in

Lambek calculus Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and seman ...

, a non-symmetric extension of

linear logic Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also ...

. 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.

Notes and references

Sources

* * {{categorytheory-stub Monoidal categories