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The Category of Operator Spaces and Complete Contractions

Bert Lindenhovius, Vladimir Zamdzhiev

TL;DR

This work establishes that the category $ extbf{OS}$ of operator spaces with complete contractions is locally countably presentable, mirroring the Banach-space case and leveraging a symmetric monoidal closed structure under the operator space projective tensor product. By identifying countably-presentable objects as precisely the separable operator spaces and proving the existence of strong generators (notably $igl\{T_n\bigr\}$ and $T(\ell_2)$), the authors derive the existence of cofree (cocommutative) coalgebras via adjoint functors, thereby providing a categorical model of Intuitionistic Linear Logic in the sense of Lafont. The results connect operator-space theory to powerful categorical machinery and offer a robust framework for coalgebraic semantics in noncommutative settings. Overall, the paper highlights how local presentability and monoidal closure yield structural tools for constructing cofree coalgebras and for interpreting logical systems within operator spaces, with implications for noncommutative geometry and quantum information theory.

Abstract

We show that the category OS of operator spaces, with complete contractions as morphisms, is locally countably presentable. This result, together with its symmetric monoidal closed structure with respect to the projective tensor product of operator spaces, implies the existence of cofree (cocommutative) coalgebras with respect to the projective tensor product and therefore provides a mathematical model of Intuitionistic Linear Logic in the sense of Lafont.

The Category of Operator Spaces and Complete Contractions

TL;DR

This work establishes that the category of operator spaces with complete contractions is locally countably presentable, mirroring the Banach-space case and leveraging a symmetric monoidal closed structure under the operator space projective tensor product. By identifying countably-presentable objects as precisely the separable operator spaces and proving the existence of strong generators (notably and ), the authors derive the existence of cofree (cocommutative) coalgebras via adjoint functors, thereby providing a categorical model of Intuitionistic Linear Logic in the sense of Lafont. The results connect operator-space theory to powerful categorical machinery and offer a robust framework for coalgebraic semantics in noncommutative settings. Overall, the paper highlights how local presentability and monoidal closure yield structural tools for constructing cofree coalgebras and for interpreting logical systems within operator spaces, with implications for noncommutative geometry and quantum information theory.

Abstract

We show that the category OS of operator spaces, with complete contractions as morphisms, is locally countably presentable. This result, together with its symmetric monoidal closed structure with respect to the projective tensor product of operator spaces, implies the existence of cofree (cocommutative) coalgebras with respect to the projective tensor product and therefore provides a mathematical model of Intuitionistic Linear Logic in the sense of Lafont.
Paper Structure (11 sections, 39 theorems, 77 equations)

This paper contains 11 sections, 39 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Categorical Preliminaries
  3. Local Presentability of Banach Spaces
  4. Local Presentability of Operator Spaces
  5. Preliminaries on Operator Spaces
  6. Limits and Colimits in $\mathbf{OS}$
  7. Strong generators in $\mathbf{OS}$
  8. Countably-presentable objects in $\mathbf{OS}$
  9. Local Presentability of $\mathbf{OS}$
  10. Cofree (Cocommutative) Coalgebras
  11. Conclusion

Key Result

Proposition 2.6

An object $A$ of a category $\mathbf{C}$ is $\alpha-$presentable iff for every $\alpha$-directed diagram $D \colon \Lambda \to \mathbf{C}$, for every colimiting cocone $(C, \{ c_\lambda \colon D_\lambda \to C\}_{\lambda \in \Lambda})$ of $D$, and for every morphism $f \colon A \to C,$ there must exi

Theorems & Definitions (103)

  • Definition 2.1: $\alpha$-directed Poset
  • Example 2.2
  • Definition 2.3: $\alpha$-directed Diagram
  • Definition 2.4
  • Example 2.5
  • Proposition 2.6
  • Definition 2.7: Locally $\alpha$-Presentable Category
  • Definition 2.8
  • Example 2.9
  • Proposition 2.10
  • ...and 93 more