Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 24th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItex## Idea A __W\*-category__ is a [[horizontal categorification]] of a [[von Neumann algebra]]. W\*-categories are in the same relation to [[C*-categories]] as [[von Neumann algebras]] are to [[C*-algebras]]. ## Definition A __W\*-category__ is a [[C*-category]] $C$ such that for any objects $A,B\in C$, the [[hom-object]] $Hom(A,B)$ admits a [[predual]] as a [[Banach space]]. That is, there is a [[Banach space]] $Hom(A,B)_*$ such that $(Hom(A,B)_*)^*$ is isomorphic to $Hom(A,B)$ in the category of [[Banach spaces]] and [[contractive maps]] (alias _short maps_). ## W\*-functors A __W\*-functor__ is a [[functor]] $F\colon C\to D$ such that $F(f^*)=F(f)^*$ for any [[morphism]] $f$ in $C$ and the map of [[Banach spaces]] $Hom(A,B)\to Hom(F(A),F(B))$ admits a [[predual]] for any objects $A$ and $B$ in $C$. ## Bounded natural transformation The good notion of natural transformations between W\*-categories is given by __bounded natural transformations__: a [[natural transformation]] $t\colon F\to G$ between W\*-functors is bounded if the norm of $t_X\colon F(X)\to G(X)$ is bounded with respect to the object $X$ of $C$. ## The bicategory of W\*-categories W\*-categories, W\*-functors, and bounded natural transformations form a [[bicategory]]. This [[bicategory]] is a good setting to work with objects like [[Hilbert spaces]], [[Hilbert W*-modules]] over [[von Neumann algebras]], [[W*-representations]] of [[von Neumann algebras]], etc. In particular, in this [[bicategory]], the [[category of Hilbert spaces]] is both [[complete]] and [[cocomplete]], unlike in the usual [[bicategory]] of [[categories]], [[functors]], and [[natural transformations]], where it only has [[finite limits]] and [[finite colimits]]. The same is true for [[Hilbert W*-modules]] over [[von Neumann algebras]], [[W*-representations]] of [[von Neumann algebras]]. ## References * [[P. Ghez]], [[Ricardo Lima]], [[John E. Roberts]], _W\*-categories_, Pacific Journal of Mathematics 120:1 (1985), 79–109, [doi:10.2140/pjm.1985.120.79](https://doi.org/10.2140/pjm.1985.120.79). <a href="https://ncatlab.org/nlab/revision/W%2A-category/1">v1</a>, <a href="https://ncatlab.org/nlab/show/W%2A-category">current</a>

   Idea

   A W*-category is a horizontal categorification of a von Neumann algebra.

   W*-categories are in the same relation to C*-categories as von Neumann algebras are to C*-algebras.

   Definition

   A W*-category is a C*-category $C$ such that for any objects $A,B\in C$, the hom-object $Hom(A,B)$ admits a predual as a Banach space. That is, there is a Banach space $Hom(A,B)_*$ such that $(Hom(A,B)_*)^*$ is isomorphic to $Hom(A,B)$ in the category of Banach spaces and contractive maps (alias short maps).

   W*-functors

   A W*-functor is a functor $F\colon C\to D$ such that $F(f^*)=F(f)^*$ for any morphism $f$ in $C$ and the map of Banach spaces $Hom(A,B)\to Hom(F(A),F(B))$ admits a predual for any objects $A$ and $B$ in $C$.

   Bounded natural transformation

   The good notion of natural transformations between W*-categories is given by bounded natural transformations: a natural transformation $t\colon F\to G$ between W*-functors is bounded if the norm of $t_X\colon F(X)\to G(X)$ is bounded with respect to the object $X$ of $C$.

   The bicategory of W*-categories

   W*-categories, W*-functors, and bounded natural transformations form a bicategory.

   This bicategory is a good setting to work with objects like Hilbert spaces, Hilbert W*-modules over von Neumann algebras, W*-representations of von Neumann algebras, etc.

   In particular, in this bicategory, the category of Hilbert spaces is both complete and cocomplete, unlike in the usual bicategory of categories, functors, and natural transformations, where it only has finite limits and finite colimits.

   The same is true for Hilbert W*-modules over von Neumann algebras, W*-representations of von Neumann algebras.

   References

   • P. Ghez, Ricardo Lima, John E. Roberts, W*-categories, Pacific Journal of Mathematics 120:1 (1985), 79–109, doi:10.2140/pjm.1985.120.79.

   v1, current