In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 and r ...
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
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 and r ...
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 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. The ...
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 (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 seman ...
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 sema ...
, 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 be ...
.
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
*
*
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Monoidal categories