  Building on the notion of normed category as suggested by Lawvere, we
introduce notions of Cauchy convergence and cocompleteness which 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. Our notions largely lead to the anticipated outcomes when
considering individual metric spaces as small normed categories, but they can
be challenging when considering some large categories, like those of
semi-normed or normed vector spaces and all linear maps, or of generalized
metric spaces and all mappings. These 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 of the
extended real half-line, we observe that the categorically atypical 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 this 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. Of utmost importance to the general theory
is the fact that our notion of normed colimit is subsumed by the notion of
weighted colimit of enriched category theory. With this theory we are able to
prove that all V-normed categories have correct-size Cauchy cocompletions. We
also prove a Banach Fixed Point Theorem for contractive endofunctors of Cauchy
cocomplete normed categories.
