Dualizing Object

In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object $\bot$. The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of Verdier duality.

Definition

Let C be a symmetric monoidal closed category. For any object ''A'' and $\bot$, there exists a morphism
:$\partial_:A\to(A\Rightarrow\bot)\Rightarrow\bot$
defined as the image by the bijection defining the monoidal closure
:$\mathrm((A\Rightarrow\bot)\otimes A,\bot)\cong\mathrm(A,(A\Rightarrow\bot)\Rightarrow\bot)$
of the morphism
:$\mathrm_\circ\gamma_ : (A\Rightarrow\bot)\otimes A\to\bot$
where $\gamma$ is the ''symmetry'' of the tensor product. An object $\bot$ of the category C is called dualizing when the associated morphism $\partial_$ 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 $(-)^*:C^\to C$ such that for every object ''A'' there is a natural isomorphism $A\cong$, and for every three objects ''A'', ''B'' and ''C'' there is a natural bijection
:$\mathrm(A\otimes B,C^*)\cong\mathrm(A,(B\otimes C)^*)$.
The dualizing object of ''C'' is then defined by $\bot=I^*$. The equivalence of the two definitions is shown by identifying $A^*=A\Rightarrow\bot$.

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

:$A^*\otimes B^* \to (B\otimes A)^*$ .

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 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 spaces 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 spaces, which is a *-autonomous category with the dualizing object $$ and the tensor product $\circledast$.

Various models of linear logic form *-autonomous categories, the earliest of which was Jean-Yves Girard'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 (as a partially ordered set) made monoidal using conjunction for the tensor product and taking 0 as the dualizing object.

The formalism of Verdier duality gives further examples of *-autonomous categories. For example, mention that the bounded derived category of constructible l-adic sheaves on an algebraic variety 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 ''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