Finite-dimensional diffeological vector spaces being and not being coproducts

TL;DR

This paper investigates conditions under which finite-dimensional diffeological vector spaces are coproducts of their subspaces, contrasting with infinite-dimensional cases in other categories, to clarify their categorical structure.

Contribution

It provides a detailed analysis of when finite-dimensional diffeological vector spaces are coproducts of subspaces, highlighting differences from infinite-dimensional cases in other categories.

Findings

Finite-dimensional diffeological vector spaces can or cannot be coproducts of their subspaces.

Differences between finite and infinite-dimensional cases are clarified.

Categorical properties of diffeological vector spaces are better understood.

Abstract

We discuss the question when a finite-dimensional diffeological vector space is, or turns out not to be, the coproduct of its subspaces in the category of diffeological vector spaces, after reviewing the same question in some other categories, where however it is limited to infinite dimension.