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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