# The universal property of infinite direct sums in C$^*$-categories and W$^*$-categories

## TL;DR

The paper establishes a universal property for infinite direct sums in C$^*$-categories, showing its equivalence to existing definitions in W$^*$-categories and connecting to classical Hilbert space cases.

## Contribution

It formulates a universal property for infinite direct sums in C$^*$-categories and proves its equivalence with known definitions in W$^*$-categories.

## Key findings

- Universal property for infinite direct sums in C$^*$-categories.
- Equivalence with Ghez, Lima, and Roberts' definition in W$^*$-categories.
- Specializes to classical Hilbert space direct sums.

## Abstract

When formulating universal properties for objects in a dagger category, one usually expects a universal property to characterize the universal object up to unique unitary isomorphism. We observe that this is automatically the case in the important special case of C$^*$-categories, provided that one uses enrichment in Banach spaces. We then formulate such a universal property for infinite direct sums in C$^*$-categories, and prove the equivalence with the existing definition due to Ghez, Lima and Roberts in the case of W$^*$-categories. These infinite direct sums specialize to the usual ones in the category of Hilbert spaces, and more generally in any W$^*$-category of normal representations of a W$^*$-algebra. Finding a universal property for the more general case of direct integrals remains an open problem.