Prekosmic Grothendieck/Galois Categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910921884958720 |
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| author | Lee, Jaehyeok |
| author_facet | Lee, Jaehyeok |
| contents | We establish a generalized version of the duality between groups and the categories of their representations on sets. Given an abstract symmetric monoidal category $K$ called Galois prekosmos, we define pre-Galois objects in $K$ and study the categories of their representations internal to $K$. The motivating example of $K$ is the cartesian monoidal category $\textit{Set}$ of sets, and pre-Galois objects in $\textit{Set}$ are groups. We present an axiomatic definition of pre-Galois $K$-categories, which is a complete abstract characterization of the categories of representations of pre-Galois objects in $K$. The category of covering spaces over a well-connected topological space is a prototype of a pre-Galois $\textit{Set}$-category. We establish a perfect correspondence between pre-Galois objects in $K$ and pre-Galois $K$-categories pointed with pre-fiber functors.
We also establish a generalized version of the duality between flat affine group schemes and the categories of their linear representations. Given an abstract symmetric monoidal category $K$ called Grothendieck prekosmos, we define what are pre-Grothendieck objects in $K$ and study the categories of their representations internal to $K$. The motivating example of $K$ is the symmetric monoidal category $\textit{Vec}_k$ of vector spaces over a field $k$, and pre-Grothendieck objects in $\textit{Vec}_k$ are affine group $k$-schemes. We present an axiomatic definition of pre-Grothendieck $K$-categories, which is a complete abstract characterization of the categories of representations of pre-Grothendieck objects in $K$. The indization of a neutral Tannakian category over a field $k$ is a prototype of a pre-Grothendieck $\textit{Vec}_k$-category. We establish a perfect correspondence between pre-Grothendieck objects in $K$ and pre-Grothendieck $K$-categories pointed with pre-fiber functors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prekosmic Grothendieck/Galois Categories Lee, Jaehyeok Algebraic Geometry We establish a generalized version of the duality between groups and the categories of their representations on sets. Given an abstract symmetric monoidal category $K$ called Galois prekosmos, we define pre-Galois objects in $K$ and study the categories of their representations internal to $K$. The motivating example of $K$ is the cartesian monoidal category $\textit{Set}$ of sets, and pre-Galois objects in $\textit{Set}$ are groups. We present an axiomatic definition of pre-Galois $K$-categories, which is a complete abstract characterization of the categories of representations of pre-Galois objects in $K$. The category of covering spaces over a well-connected topological space is a prototype of a pre-Galois $\textit{Set}$-category. We establish a perfect correspondence between pre-Galois objects in $K$ and pre-Galois $K$-categories pointed with pre-fiber functors. We also establish a generalized version of the duality between flat affine group schemes and the categories of their linear representations. Given an abstract symmetric monoidal category $K$ called Grothendieck prekosmos, we define what are pre-Grothendieck objects in $K$ and study the categories of their representations internal to $K$. The motivating example of $K$ is the symmetric monoidal category $\textit{Vec}_k$ of vector spaces over a field $k$, and pre-Grothendieck objects in $\textit{Vec}_k$ are affine group $k$-schemes. We present an axiomatic definition of pre-Grothendieck $K$-categories, which is a complete abstract characterization of the categories of representations of pre-Grothendieck objects in $K$. The indization of a neutral Tannakian category over a field $k$ is a prototype of a pre-Grothendieck $\textit{Vec}_k$-category. We establish a perfect correspondence between pre-Grothendieck objects in $K$ and pre-Grothendieck $K$-categories pointed with pre-fiber functors. |
| title | Prekosmic Grothendieck/Galois Categories |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2504.20949 |