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*-autonomous category

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Symmetric monoidal closed category equipped with a dualizing object

In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object ⊥ {\displaystyle \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 ⊥ {\displaystyle \bot } , there exists a morphism

∂ A , ⊥ : A → ( A ⇒ ⊥ ) ⇒ ⊥ {\displaystyle \partial _{A,\bot }:A\to (A\Rightarrow \bot )\Rightarrow \bot }

defined as the image by the bijection defining the monoidal closure

H o m ( ( A ⇒ ⊥ ) ⊗ A , ⊥ ) ≅ H o m ( A , ( A ⇒ ⊥ ) ⇒ ⊥ ) {\displaystyle \mathrm {Hom} ((A\Rightarrow \bot )\otimes A,\bot )\cong \mathrm {Hom} (A,(A\Rightarrow \bot )\Rightarrow \bot )}

of the morphism

e v a l A , A ⇒ ⊥ ∘ γ A ⇒ ⊥ , A : ( A ⇒ ⊥ ) ⊗ A → ⊥ {\displaystyle \mathrm {eval} _{A,A\Rightarrow \bot }\circ \gamma _{A\Rightarrow \bot ,A}:(A\Rightarrow \bot )\otimes A\to \bot }

where γ {\displaystyle \gamma } is the symmetry of the tensor product. An object ⊥ {\displaystyle \bot } of the category C is called dualizing when the associated morphism ∂ A , ⊥ {\displaystyle \partial _{A,\bot }} 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 o p → C {\displaystyle (-)^{*}:C^{\mathrm {op} }\to C} such that for every object A there is a natural isomorphism A ≅ A ∗ ∗ {\displaystyle A\cong {A^{**}}} , and for every three objects A, B and C there is a natural bijection

H o m ( A ⊗ B , C ∗ ) ≅ H o m ( A , ( B ⊗ C ) ∗ ) {\displaystyle \mathrm {Hom} (A\otimes B,C^{*})\cong \mathrm {Hom} (A,(B\otimes C)^{*})} .

The dualizing object of C is then defined by ⊥ = I ∗ {\displaystyle \bot =I^{*}} . The equivalence of the two definitions is shown by identifying A ∗ = A ⇒ ⊥ {\displaystyle 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 ∗ ⊗ B ∗ → ( B ⊗ A ) ∗ {\displaystyle 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 C {\displaystyle {\mathbb {C} }} and the tensor product ⊛ {\displaystyle \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, Boyarchenko & Drinfeld (2013) 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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