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Categories with duals
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Monoidal categories
monoidal categories enriched monoidal category, tensor category string diagram, tensor network With braiding braided monoidal category balanced monoidal category twistsymmetric monoidal category With duals for objects category with duals (list of them) dualizable object (what they have) rigid monoidal category, a.k.a. autonomous category pivotal category spherical category ribbon category, a.k.a. tortile category compact closed category With duals for morphisms monoidal dagger-category? symmetric monoidal dagger-category dagger compact category With traces trace traced monoidal category Closed structure closed monoidal category cartesian closed category closed category star-autonomous category Special sorts of products cartesian monoidal category semicartesian monoidal category monoidal category with diagonals multicategory Semisimplicity semisimple category fusion category modular tensor category Morphisms monoidal functor (lax, oplax, strong bilax, Frobenius) braided monoidal functor symmetric monoidal functor Internal monoids monoid in a monoidal category commutative monoid in a symmetric monoidal category module over a monoid Examples tensor product closed monoidal structure on presheaves Day convolution Theorems coherence theorem for monoidal categories monoidal Dold-Kan correspondence In higher category theory monoidal 2-category braided monoidal 2-categorymonoidal bicategory cartesian bicategoryk-tuply monoidal n-category little cubes operadmonoidal (∞,1)-category symmetric monoidal (∞,1)-categorycompact double category Categories with duals Idea Categories with duals for objects Categories with duals for morphisms References IdeaA category with duals is a category where objects and/or morphisms have duals. This exists in several flavours; this list is mostly taken from a recent categories list post from Peter Selinger. Categories with duals for objectsA left autonomous category is a monoidal category in which every object is dualisable on the left. A right autonomous category is a monoidal category in which every object is dualisable on the right. An autonomous category, or rigid category is a monoidal category that is both left and right autonomous. Note that any braided monoidal category is autonomous on both sides if it is autonomous on either side. A pivotal category is an autonomous category equipped with a monoidal natural isomorphism from the identity functor to the double dual? functor. A one-sided autonomous category with such an isomorphism is automatically two-sided autonomous. Although each braided autonomous category has an isomorphism from AA to A **A^{**}, such a category is not necessarily pivotal because this isomorphism is not in general monoidal. On the other hand, every balanced autonomous category is pivotal. A spherical category is a pivotal category where the left and right trace operations coincide on all objects. A tortile category, or ribbon category, is a balanced autonomous (therefore pivotal) category in which the twist on A *A^* is the dual of the twist on AA. A compact closed category is a symmetric tortile category, or equivalently, a symmetric autonomous category. The **-autonomous categories do not really belong on this list; being **-autonomous is logically independent of being autonomous, and while **-autonomous categories have duals, these are not in general duals in the sense of a dualisable object. However, any compact closed category is **-autonomous. Likewise, closed categories or closed monoidal categories do not really belong on this list, but there is a sense of dual there which should be carefully distinguished from the primary sense here, which is generally stronger. See dual object in a closed category. Categories with duals for morphismsOne might write something about these too, or put them on a separate page. In the meantime, see the table of contents to the right. There at least two commonspread kinds of categories with duals for morphisms: dagger categories where each morphism f:X→Yf:X \to Y has a †\dagger-dual f †:Y→Xf^\dagger : Y \to X, without any extra property. groupoids, where each morphism f:X→Yf:X \to Y has an inverse f −1:Y→Xf^{-1} :Y \to X defined by the properties ff −1=1 Yf f^{-1} = 1_Y, f −1f=1 Xf^{-1}f = 1_X.Moreover, every category enriched in one of the kind of categories listed above will have a notion of ‘dual’ for its morphisms. References Peter Selinger, categories post of 2010-05-15; Peter Selinger, A survey of graphical languages for monoidal categoriesLast revised on October 21, 2021 at 09:19:05. See the history of this page for a list of all contributions to it. EditDiscussPrevious revisionChanges from previous revisionHistory (9 revisions) Cite Print Source |
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