On Vector Spaces with Formal Infinite Sums

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 Vect -enriched endofunctor of Vect that is right orthogonal for every cardinal λ , to the cokernel of the canonical inclusion of the λ -th copower in the λ -th power of the identity functor: these form the objects for a universal r.c.s.v.s. I call Σ 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 BΣ Vect and to the r.c.s.v.s. KTVectₛ 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 Ind-(Vectᵒp) . In particular with respect to the problem of closure under the tensor product of Ind-(Vectᵒp) . Most of the technical results apply to a more general class of orthogonal subcategories of Ind-(Vectᵒp) and I work with that generality as it’s cost-free.