In this talk we investigate the notion of normed category: a category  where each hom-set \( X(A,B) \) comes equipped with a norm \( X(A,B)\to [0, \infty] \), subject to suitable axioms. As suggested by Lawvere (1973), we treat normed categories consequentially as categories enriched in the monoidal-closed category of normed sets, i.e., of sets which come with a norm function (Betti and Galuzzi 1975). We discuss various examples of (large) normed categories, introduce in particular the notion of normed colimit and Cauchy cocompleteness, relate this approach to the notion of weighted colimit, and discuss a Banach Fixed Point Theorem for contractive endofunctors of Cauchy cocomplete normed categories. 

In this talk we report on joint work with Maria Manuel Clementino and Walter Tholen.

References

Betti, Renato, and Massimo Galuzzi. 1975. "Categorie Normate." Bollettino Dell'Unione Matematica Italiana 4 (11): 66-75. 

Clementino, Maria Manuel, Hofmann, Dirk and Tholen, Walter. 2024. "Cauchy Convergence in V-Normed Categories." https://doi.org/10.48550/arxiv.2404.09032.

Lawvere, F. William. 1973. "Metric Spaces, Generalized Logic, and Closed Categories." Rendiconti Del Seminario Matemàtico e Fisico Di Milano 43 (1): 135-66. https://doi.org/10.1007/bf02924844.