In
mathematics, a *-autonomous (read "star-autonomous") category C is a
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
monoidal closed category equipped with a
dualizing object
. The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of
Verdier duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of
Alexander Groth ...
.
Definition
Let C be a symmetric monoidal closed category. For any object ''A'' and
, there exists a morphism
:
defined as the image by the bijection defining the monoidal closure
:
of the morphism
:
where
is the ''symmetry'' of the tensor product. An object
of the category C is called dualizing when the associated morphism
is an isomorphism for every object ''A'' of the category C.
Equivalently, a *-autonomous category is a symmetric monoidal category ''C'' together with a functor
such that for every object ''A'' there is a natural isomorphism
, and for every three objects ''A'', ''B'' and ''C'' there is a natural bijection
:
.
The dualizing object of ''C'' is then defined by
. The equivalence of the two definitions is shown by identifying
.
Properties
Compact closed categories are *-autonomous, with the monoidal unit as the dualizing object. Conversely, if the unit of a *-autonomous category is a dualizing object then there is a canonical family of maps
:
.
These are all isomorphisms if and only if the *-autonomous category is compact closed.
Examples
A familiar example is the category of finite-dimensional vector spaces over any field ''k'' made monoidal with the usual
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
of vector spaces. The dualizing object is ''k'', the one-dimensional vector space, and dualization corresponds to transposition. Although the category of all vector spaces over ''k'' is not *-autonomous, suitable extensions to categories of
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
s can be made *-autonomous.
On the other hand, the category of topological vector spaces contains an extremely wide full subcategory, the category Ste of
stereotype space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an is ...
s, which is a *-autonomous category with the dualizing object
and the tensor product
.
Various models 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 ...
form *-autonomous categories, the earliest of which was
Jean-Yves Girard
Jean-Yves Girard (; born 1947) is a French logician working in proof theory. He is the research director (emeritus) at the mathematical institute of the University of Aix-Marseille, at Luminy.
Biography
Jean-Yves Girard is an alumnus of th ...
's category of coherence spaces.
The category of
complete semilattices with morphisms preserving all joins but not necessarily meets is *-autonomous with dualizer the chain of two elements. A degenerate example (all homsets of cardinality at most one) is given by any
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...
(as a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...
) made monoidal using conjunction for the tensor product and taking 0 as the dualizing object.
The formalism of
Verdier duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of
Alexander Groth ...
gives further examples of *-autonomous categories. For example, mention that the bounded derived category of constructible
l-adic sheaves on an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
has this property. Further examples include derived categories of constructible sheaves on various kinds of topological spaces.
An example of a self-dual category that is not *-autonomous is finite linear orders and continuous functions, which has * but is not autonomous: its dualizing object is the two-element chain but there is no tensor product.
The category of sets and their partial injections is self-dual because the converse of the latter is again a partial injection.
The concept of *-autonomous category was introduced by
Michael Barr in 1979 in a monograph with that title. Barr defined the notion for the more general situation of ''V''-categories, categories enriched in a symmetric monoidal or autonomous category ''V''. The definition above specializes Barr's definition to the case ''V'' = Set of ordinary categories, those whose homobjects form sets (of morphisms). Barr's monograph includes an appendix by his student Po-Hsiang Chu that develops the details of a construction due to Barr showing the existence of nontrivial *-autonomous ''V''-categories for all symmetric monoidal categories ''V'' with pullbacks, whose objects became known a decade later as
Chu spaces.
Non symmetric case
In a
biclosed monoidal category
In mathematics, especially in category theory, a closed monoidal category (or a ''monoidal closed category'') is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.
A classic examp ...
''C'', not necessarily symmetric, it is still possible to define a dualizing object and then define a *-autonomous category as a biclosed monoidal category with a dualizing object. They are equivalent definitions, as in the symmetric case.
References
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{{DEFAULTSORT:-autonomous category
Monoidal categories
Closed categories