Cauchy convergence in V-normed categories

Abstract

Building on the notion of normed category as suggested by Lawvere, we introduce notions of Cauchy convergence and cocompleteness for such categories that differ from proposals in previous works. Key to our approach is to treat them consequentially as categories enriched in the monoidal-closed category of normed sets, i.e., of sets which come with a norm function. Our notions lead to the anticipated outcomes for individual metric spaces or the additive groups of normed vector spaces considered as small normed categories. But they quickly become more challenging when we consider large categories, such as the categories of all semi-normed or normed vector spaces and all linear maps as morphisms, not just because norms of vectors need to be allowed to have value ∞ in order to guarantee the existence of colimits of (sufficiently many) infinite sequences. These categories, along with categories of generalized metric spaces, are the key example categories discussed in detail in this paper.

Working with a general commutative quantale $V$ as a value recipient for norms, rather than only with Lawvere's quantale $R_{+}$ of the extended real half-line, we observe that the categorically atypical, and perhaps even irritating, structure gap between objects and morphisms in the example categories is already present in the underlying normed category of the enriching category of $V$-normed sets. To show that the normed category and, in fact, all presheaf categories over it, are Cauchy cocomplete, we assume the quantale $V$ to satisfy a couple of light alternative extra properties which, however, are satisfied in all instances of interest to us. Of utmost importance to the general theory is the fact that our notion of Cauchy convergence is subsumed by the notion of weighted colimit of enriched category theory. With this theory and, in particular, with results of Albert, Kelly and Schmitt, we are able to prove that all $V$-normed categories have correct-size Cauchy cocompletions, for $V$ satisfying our light alternative assumptions.

We also show that our notions are suitable to prove a Banach Fixed Point Theorem for contractive endofunctors of Cauchy cocomplete normed categories.