On Vector Spaces with Formal Infinite Sums
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908872199897088 |
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| author | Freni, Pietro |
| author_facet | Freni, Pietro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_08000 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Vector Spaces with Formal Infinite Sums Freni, Pietro Category Theory Logic Primary 13F25, 13J05, Secondary 08A65, 18B15, 18F60 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. |
| title | On Vector Spaces with Formal Infinite Sums |
| topic | Category Theory Logic Primary 13F25, 13J05, Secondary 08A65, 18B15, 18F60 |
| url | https://arxiv.org/abs/2303.08000 |