From infinite series in a normed vector space to infinite composites in a normed category.

We use elementary examples of sequential convergence from Calculus, Linear Algebra and Functional Analysis to motivate our notion of Cauchy sequence and its convergence in a category whose morphisms come with a norm in the sense of Lawvere [1973]. This notion, as well as the naturally ensuing notion of Cauchy cocompleteness of a normed category, provide a common framework for the given examples, and beyond. For instance, the additive group of a normed vector space, viewed as a one-object (small) category, is Cauchy cocomplete precisely when the space is Banach. Much harder to show is the result that also the (large) category of all normed vector spaces (with norms allowed to be infinite) and their linear operators as morphisms, normed by the logarithm of its usual operator norm, is Cauchy cocomplete. Furthermore, as first indicated by Kubis [2017], for a contractive endofunctor of a Cauchy cocomplete normed category, we prove a Banach Fixed Point Theorem that directly generalizes the classical metric result.

Our norms are allowed to take values in a (commutative and unital) quantale, rather than just in Lawvere’s extended real half-line. This generalization not only increases the range of examples and applications, but also makes it easier to see the theory of normed categories as part of enriched category theory. In this way we can apply some powerful results from that theory in order to characterize Cauchy cocompleteness and establish Cauchy cocompletions of normed categories.

This talk is based on joint work with Maria Manuel Clementino and Dirk Hofmann [2025].

References

[1973] F.W. Lawvere. Metric spaces, generalized logic, and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano 43:135—166.
Republished in: Reprints in Theory and Applications of Categories 1.

[2017] W. Kubis. Categories with norms. arXiv:1705.10189v1 [math.CT].

[2025] M.M. Clementino, D. Hofmann, W. Tholen.  Cauchy convergence in -normed categories. Advances in Mathematics 470, 110247.