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On Vector Spaces with Formal Infinite Sums

Pietro Freni

TL;DR

The paper develops a category-theoretic framework for vector spaces equipped with a formal notion of infinite linear sums, culminating in the universal category $Σ\mathrm{Vect}$ of strong vector spaces. It connects this framework to ultrafinite summability spaces, based variants, and linearly topologized spaces, and provides a reflective embedding into $Ind(\mathrm{Vect}^{op})$ with a robust torsion-theoretic structure. A monoidal closed structure is established via the indized tensor product, and strong linear algebras and Kähler differentials are defined within this enriched setting, enabling a coherent treatment of infinite-sum operations in algebraic contexts. The results give a principled basis for extending familiar algebraic constructions (tensors, Hom, algebras, modules, and differentials) to categories that faithfully encode formal infinite sums, with potential applications to generalized power series and valued-field theories.

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 \emph{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 $\mathrm{Vect}$-enriched endofunctor of $\mathrm{Vect}$ 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 $Σ\mathrm{Vect}$. I show this is equivalent to the category of \emph{ultrafinite summability spaces} defined independently in arXiv:2403.05827. I relate this category to what could be understood to be the obvious category of strong vector spaces $BΣ\mathrm{Vect}$ and to the r.c.s.v.s. $K\mathrm{TVect}_s$ of separated linearly topologized spaces that are generated by linearly compact spaces. I analyze the monoidal closed structures on various r.s.v.s. induced by the natural one on $\mathrm{Ind}(\mathrm{Vect}^{\mathrm{op}})$. In particular with respect to the problem of closure under the tensor product of $\mathrm{Ind}(\mathrm{Vect}^{\mathrm{op}})$. Most of the technical results apply to a more general class of orthogonal subcategories of $\mathrm{Ind}(\mathrm{Vect}^{\mathrm{op}})$ and we work with that generality as it's cost-free.

On Vector Spaces with Formal Infinite Sums

TL;DR

The paper develops a category-theoretic framework for vector spaces equipped with a formal notion of infinite linear sums, culminating in the universal category of strong vector spaces. It connects this framework to ultrafinite summability spaces, based variants, and linearly topologized spaces, and provides a reflective embedding into with a robust torsion-theoretic structure. A monoidal closed structure is established via the indized tensor product, and strong linear algebras and Kähler differentials are defined within this enriched setting, enabling a coherent treatment of infinite-sum operations in algebraic contexts. The results give a principled basis for extending familiar algebraic constructions (tensors, Hom, algebras, modules, and differentials) to categories that faithfully encode formal infinite sums, with potential applications to generalized power series and valued-field theories.

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 \emph{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 -enriched endofunctor of that is right orthogonal for every cardinal , to the cokernel of the canonical inclusion of the -th copower in the -th power of the identity functor: these form the objects for a universal r.c.s.v.s. I call . I show this is equivalent to the category of \emph{ultrafinite summability spaces} defined independently in arXiv:2403.05827. I relate this category to what could be understood to be the obvious category of strong vector spaces and to the r.c.s.v.s. of separated linearly topologized spaces that are generated by linearly compact spaces. I analyze the monoidal closed structures on various r.s.v.s. induced by the natural one on . In particular with respect to the problem of closure under the tensor product of . Most of the technical results apply to a more general class of orthogonal subcategories of and we work with that generality as it's cost-free.
Paper Structure (26 sections, 60 theorems, 118 equations)

This paper contains 26 sections, 60 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Overview
  4. Addendum
  5. Acknowledgments
  6. Based strong vector spaces
  7. Some duality reductions
  8. General strong vector spaces
  9. Reasonable categories of strong vector spaces
  10. Consequences of (A1) and (A2)
  11. (A3) and uniqueness of infinite sums
  12. Strong Vector Spaces as structures in a signature
  13. Orthogonality, Smallness, and Reflectivity
  14. Factorization systems and Torsion Theories
  15. Some special torsion theories in [Vect,Vect]ₖ
  16. ...and 11 more sections

Key Result

Lemma 2.7

Every summable family $(f_i)_{i \in I}$ in some $\mathbf{k}(\Gamma, \mathcal{F})$ determines a strong linear map $\mathbf{k}(I; \mathcal{P}(I)) \to \mathbf{k}(\Gamma; \mathcal{F})$ given by Conversely every strong linear map $\bar{f}: \mathbf{k}(I; \mathcal{P}(I)) \to \mathbf{k}(\Gamma; \mathcal{F})$ arises this way from the summable family $(f_i)_{I \in I}:=(\bar{f} (\delta_i))_{i \in I}$.

Theorems & Definitions (217)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Definition 2.9
  • ...and 207 more