Rigid category
Encyclopedia
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 category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined by Dold and Puppe in 1978.
are identities. A right rigid object is defined similarly.
are identities. The definition of a right rigid object holds mutatis mutandis.
An inverse is an object X-1 such that both X ⊗ X-1 and X-1 ⊗ X are isomorphic to 1, the one object of the monoidal category. If an object X has a left (resp. right) inverse X-1 with respect to the tensor product then it is left (resp. right) rigid, and X* = X-1.
The operation of taking duals gives a contravariant functor on a rigid category.
.
Then for any endomorphism
, the trace is of f is defined as the composition
,
so
. We may continue further and define the dimension of a rigid object to be
.
Rigidity is also importance because of its relation internal Hom's. If X is a left rigid object, then every internal Hom of the form [X, Z] exists and is isomorphic to Z ⊗ Y. In particular, in a rigid category, all internal Hom's exist.
. An autonomous category that is also symmetric is called a compact closed category
.
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...
, a branch of mathematics
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 rigid 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 every object is rigid, that is, has a 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:...
X* (the internal Hom [X, 1]) 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 left duals. They were first defined by Dold and Puppe in 1978.
Definition
There are at least two equivalent definitions of a rigidity.- An object X of a monoidal category is called left rigid if there is an object Y and morphisms
and 
such that both compositions
are identities. A right rigid object is defined similarly.
- An object X of a monoidal category is called left rigid if it has a left dual X* := [X, 1] and a morphism
such that the compositions
are identities. The definition of a right rigid object holds mutatis mutandis.
An inverse is an object X-1 such that both X ⊗ X-1 and X-1 ⊗ X are isomorphic to 1, the one object of the monoidal category. If an object X has a left (resp. right) inverse X-1 with respect to the tensor product then it is left (resp. right) rigid, and X* = X-1.
The operation of taking duals gives a contravariant functor on a rigid category.
Uses
One important application of rigidity is in the definition of the trace of an endomorphism of a rigid object. The trace can be defined for any rigid category such that taking the ( )**, the functor of taking the dual twice repeated, is isomorphic to the identity functor. Then for any right rigid object X, and any other object Y, we may define the isomorphism.Then for any endomorphism
, the trace is of f is defined as the composition,so
. We may continue further and define the dimension of a rigid object to be.Rigidity is also importance because of its relation internal Hom's. If X is a left rigid object, then every internal Hom of the form [X, Z] exists and is isomorphic to Z ⊗ Y. In particular, in a rigid category, all internal Hom's exist.
Alternative Terminology
A monoidal category where every object has a left (resp. right) dual is also sometimes called a left (resp. right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous categoryAutonomous category
In mathematics, an autonomous category is a monoidal category where dual objects exist.-Definition:A left autonomous category is a monoidal category where every object has a left dual. An autonomous category is a monoidal category where every object has both a left and a right dual...
. An autonomous category that is also symmetric is called a compact closed category
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...
.
See also
- A monoidal categoryMonoidal categoryIn 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...
is a category with a tensor product, precisely the sort of category for which rigidity makes sense. - The category of pure motivesMotive (algebraic geometry)In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
is formed by rigidifying the category of effective pure motives.