On the functor l^2

TL;DR

This paper investigates the properties of the functor l^2 from partial injections to Hilbert spaces, revealing its preservation of structures and characterizing its image in terms of partial isometries and continuous linear maps.

Contribution

It provides a detailed analysis of the functor l^2, highlighting its structural preservation and the nature of its image within the categories involved.

Findings

The functor l^2 preserves daggers, monoidal structures, enrichment, and (co)limits.

Its direct image consists of partial isometries, up to unitaries.

Its essential image includes all continuous linear maps between Hilbert spaces.

Abstract

We study the functor l^2 from the category of partial injections to the category of Hilbert spaces. The former category is finitely accessible, and its homsets are algebraic domains; the latter category has conditionally algebraic domains for homsets. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous linear maps between Hilbert spaces.