On Vector Spaces with Formal Infinite Sums

Abstract

I discuss possible definitions of categories of vector spaces enriched with a notion of formal infinite linear combination in the likes of the formal infinite linear combinations one has in the context of generalized power series, I call these reasonable categories of strong vector spaces (r.c.s.v.s.). I show that, in a precise sense, the more general possible definition for a strong vector space is that of a small \(\textrm{Vect}\)-enriched endofunctor of \(\textrm{Vect}\) that is right orthogonal for every cardinal \(\lambda \), to the cokernel of the canonical inclusion of the \(\lambda \)-th copower in the \(\lambda \)-th power of the identity functor: these form the objects for a universal r.c.s.v.s. I call \(\Sigma \textrm{Vect}\). I show this is equivalent to the category of ultrafinite summability spaces defined independently in Bagayoko et al. (Automorphisms and derivations on algebras endowed with formal infinite sums, 2024). I relate this category to what could be understood to be the obvious category of strong vector spaces \(B\Sigma \textrm{Vect}\) and to the r.c.s.v.s. \(K\textrm{TVect}_s\) of separated linearly topologized spaces that are generated by linearly compact spaces. I analyze the monoidal closed structures on various r.c.s.v.s. induced by the natural one on \(\textrm{Ind}{\text {-}}(\textrm{Vect}^\textrm{op})\). In particular with respect to the problem of closure under the tensor product of \(\textrm{Ind}{\text {-}}(\textrm{Vect}^\textrm{op})\). Most of the technical results apply to a more general class of orthogonal subcategories of \(\textrm{Ind}{\text {-}}(\textrm{Vect}^\textrm{op})\) and I work with that generality as it’s cost-free.