*-autonomous category

In mathematics, a *-autonomous (read "star-autonomous") category 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.

Let ${\displaystyle {\mathcal {C}}}$ be a symmetric monoidal closed category ${\displaystyle \langle {\mathcal {C}},\otimes ,I,\Rightarrow \rangle }$. For any pair of objects, in particular A and ${\displaystyle \bot }$, there exists a morphism

${\displaystyle \partial _{A,\bot }:A\to (A\Rightarrow \bot )\Rightarrow \bot }$

defined as the image by the bijection defining the monoidal closure

${\displaystyle \mathrm {Hom} ((A\Rightarrow \bot )\otimes A,\bot )\cong \mathrm {Hom} (A,(A\Rightarrow \bot )\Rightarrow \bot )}$

of the evaluation map:

${\displaystyle \mathrm {eval} _{A,A\Rightarrow \bot }\circ \gamma _{A\Rightarrow \bot ,A}:(A\Rightarrow \bot )\otimes A\to \bot }$

where ${\displaystyle \gamma }$ is the symmetry of the tensor product. An object ${\displaystyle \bot }$ of the category ${\displaystyle {\mathcal {C}}}$ is called dualizing when the associated morphism ${\displaystyle \partial _{A,\bot }}$ is an isomorphism for every object A of ${\displaystyle {\mathcal {C}}}$.

Equivalently, a *-autonomous category is a symmetric monoidal category ${\displaystyle {\mathcal {C}}}$ together with a functor ${\displaystyle (-)^{*}:{\mathcal {C}}^{\mathrm {op} }\to {\mathcal {C}}}$ such that for every object A there is a natural isomorphism ${\displaystyle A\cong {A^{**}}}$, and for every three objects A, B and C there is a natural bijection

${\displaystyle \mathrm {Hom} (A\otimes B,C^{*})\cong \mathrm {Hom} (A,(B\otimes C)^{*})}$.

The dualizing object of ${\displaystyle {\mathcal {C}}}$ is then defined by ${\displaystyle \bot =I^{*}}$. The equivalence of the two definitions is shown by identifying ${\displaystyle A^{*}=A\Rightarrow \bot }$.

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

${\displaystyle A^{*}\otimes B^{*}\to (B\otimes A)^{*}}$ .

These are all isomorphisms if and only if the *-autonomous category is compact closed.

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 ${\displaystyle {\mathbb {C} }}$ 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.

In a biclosed monoidal category ${\displaystyle {\mathcal {C}}}$, 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.