*-autonomous category

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title: "-autonomous category" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["monoidal-categories", "closed-categories"] description: "Symmetric monoidal closed category equipped with a dualizing object" topic_path: "general/monoidal-categories" source: "https://en.wikipedia.org/wiki/-autonomous_category" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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 \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 \langle C, \otimes, I, \Rightarrow \rangle. For any pair of objects, in particular A and \bot, there exists a morphism :\partial_{A,\bot}:A\to(A\Rightarrow\bot)\Rightarrow\bot defined as the image by the bijection defining the monoidal closure :\mathrm{Hom}((A\Rightarrow\bot)\otimes A,\bot)\cong\mathrm{Hom}(A,(A\Rightarrow\bot)\Rightarrow\bot) of the evaluation map: :\mathrm{eval}{A,A\Rightarrow\bot}\circ\gamma{A\Rightarrow\bot,A} : (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_{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^{\mathrm{op}}\to C such that for every object A there is a natural isomorphism A\cong{A^{**}}, and for every three objects A, B and C there is a natural bijection :\mathrm{Hom}(A\otimes B,C^)\cong\mathrm{Hom}(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 {\mathbb C} 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

• {{cite book | author-link = Michael Barr (mathematician) |first= Michael |last=Barr | year = 1979 | title = *-Autonomous Categories | series = Lecture Notes in Mathematics | volume = 752 | publisher = Springer | doi = 10.1007/BFb0064579 | isbn = 978-3-540-09563-7
• {{cite journal | first= Michael |last=Barr | s2cid = 14721961 | year = 1995 | title = Non-symmetric *-autonomous Categories | journal = Theoretical Computer Science | volume = 139 | pages = 115–130 | doi = 10.1016/0304-3975(94)00089-2 | doi-access =
• {{cite journal | first= Michael |last=Barr | year = 1999 | title = *-autonomous categories: once more around the track | journal = Theory and Applications of Categories | volume = 6 | url = http://www.tac.mta.ca/tac/volumes/6/n1/n1.pdf | pages = 5–24 |citeseerx=10.1.1.39.881

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