{{inline citations needed|date=March 2023}} {{Short description|Symmetric monoidal closed category equipped with a dualizing object}} In mathematics, a '''*-autonomous''' (read "star-autonomous") category is a symmetric monoidal closed category equipped with a dualizing object <math>\bot</math>. The concept is also referred to as '''Grothendieck—Verdier category''' in view of its relation to the notion of Verdier duality.
==Definition==
Let <math>\mathcal C</math> be a symmetric monoidal closed category <math>\langle\mathcal C, \otimes, I, \Rightarrow \rangle</math>. For any pair of objects, in particular ''A'' and <math>\bot</math>, there exists a morphism :<math>\partial_{A,\bot}:A\to(A\Rightarrow\bot)\Rightarrow\bot</math> defined as the image by the bijection defining the monoidal closure :<math>\mathrm{Hom}((A\Rightarrow\bot)\otimes A,\bot)\cong\mathrm{Hom}(A,(A\Rightarrow\bot)\Rightarrow\bot)</math> of the evaluation map: :<math>\mathrm{eval}_{A,A\Rightarrow\bot}\circ\gamma_{A\Rightarrow\bot,A} : (A\Rightarrow\bot)\otimes A\to\bot</math> where <math>\gamma</math> is the ''symmetry'' of the tensor product. An object <math>\bot</math> of the category <math>\mathcal C</math> is called '''dualizing''' when the associated morphism <math>\partial_{A,\bot}</math> is an isomorphism for every object ''A'' of <math>\mathcal C</math>.
Equivalently, a '''*-autonomous category''' is a symmetric monoidal category <math>\mathcal C</math> together with a functor <math>(-)^*:\mathcal C^{\mathrm{op}}\to\mathcal C</math> such that for every object ''A'' there is a natural isomorphism <math>A\cong{A^{**}}</math>, and for every three objects ''A'', ''B'' and ''C'' there is a natural bijection :<math>\mathrm{Hom}(A\otimes B,C^*)\cong\mathrm{Hom}(A,(B\otimes C)^*)</math>. The dualizing object of <math>\mathcal C</math> is then defined by <math>\bot=I^*</math>. The equivalence of the two definitions is shown by identifying <math>A^*=A\Rightarrow\bot</math>.
==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
:<math>A^*\otimes B^* \to (B\otimes A)^*</math> .
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 <math>{\mathbb C}</math> and the tensor product <math>\circledast</math>.
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, {{harvtxt|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 <math>\mathcal C</math>, 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.
==See also== *Autonomous category
==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 }} * {{Cite journal |last1=Boyarchenko|first1=Mitya|last2=Drinfeld|first2=Vladimir|author-link2=Vladimir Drinfeld|title=A duality formalism in the spirit of Grothendieck and Verdier|journal=Quantum Topology|volume=4|year=2013|issue=4|pages=447–489|mr=3134025|doi=10.4171/QT/45|arxiv=1108.6020|s2cid=55605535 }} * {{nlab|id=star-autonomous+category|title=star-autonomous category}}
{{DEFAULTSORT:-autonomous category}} Category:Monoidal categories Category:Closed categories Category:Duality (mathematics)