To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

*-autonomous category

From Wikipedia, the free encyclopedia

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

YouTube Encyclopedic

  • 1/3
    91 552
  • Introduction to autonomous differential equations
  • ACT@UCR Seminar: Star-Autonomous Envelopes - Michael Shulman
  • Differential Equations - Intro Video - Linearization of Autonomous Systems



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 .


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.


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 .

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.


  • Barr, Michael (1979). *-autonomous Categories. Lecture Notes in Mathematics. Vol. 752. Springer. doi:10.1007/BFb0064579. ISBN 978-3-540-09563-7.
  • Barr, Michael (1995). "Non-symmetric *-autonomous Categories". Theoretical Computer Science. 139: 115–130. doi:10.1016/0304-3975(94)00089-2. S2CID 14721961.
  • Barr, Michael (1999). "*-autonomous categories: once more around the track" (PDF). Theory and Applications of Categories. 6: 5–24. CiteSeerX
  • Boyarchenko, Mitya; Drinfeld, Vladimir (2013). "A duality formalism in the spirit of Grothendieck and Verdier". Quantum Topology. 4 (4): 447–489. arXiv:1108.6020. doi:10.4171/QT/45. MR 3134025. S2CID 55605535.
This page was last edited on 31 March 2023, at 12:56
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.