Pietro Freni (University of Leeds)

Summability in Algebras of Generalized Power Series

Spaces of generalized power series have been important objects in asymptotic analysis and in the algebra and model theory of valued structures ever since the introduction of the first instances of them by Levi-Civita and Hahn. A space of generalized series can be understood to be built up from a triple (Γ , < , F), where (Γ, <) is an ordered set and F is an ideal of Noetherian subsets of Γ, as the k-vector space k(Γ,F) of functions f : Γ → k whose support Supp f = {γ ∈ Γ : f(γ)= 0} lies in F.
A prominent feature of these objects is the notion of formal infinite sum, or rather more appropriately, of infinite k-linear combination: a family (f_i)_{i \in I} such that the union fo the Supp(f_i) is yet in F and for every γ ∈ Γ the set {i ∈ I : γ ∈ Supp(f_i)} is finite, is said to be summable. Given such a family it is possible to associate to any I-indexed family of scalars (k_i)_{i \in I} the infinite linear combination of the f_i with the coefficienti  k_i in the obvious way. This feature is referred to, throughout literature, as a strong linear structure on the vector space k(Γ,F) and maps F : k(Γ,F) → k(∆,G) preserving it, are referred to as strong(ly) linear maps. In this talk, after some history and motivation, I will argue about a suitable category-theoretical framework for the study of the above outlined notion of strong linearity, in particular I will justify a notion of reasonable category of strong k-vector spaces generalizing the above setting and prove that up to equivalence there is a unique universal category ΣVect with the property that every reasonable category of strong vector spaces has a fully faithful functor to into it.
ΣVect can be defined as an orthogonal subcategory of Ind-(Vect^op) and the canonical monoidal closed struture on Ind-(Vect^op) restricts to ΣVect.
Time permitting, the relation ΣVect has with another orthogonal subcategory of Ind-(Vectop) equivalent to the category of linearly topologized vector spaces that are colimits of linearly compact spaces will  be also described. Finally I will present some open questions of  combinatorial nature in this setting.