Functors on the category of finite sets revisited
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911151678291968 |
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| author | Powell, Geoffrey |
| author_facet | Powell, Geoffrey |
| contents | We study the structure of the category of representations of $\mathbf{FA}$, the category of finite sets and all maps, mostly working over a field of characteristic zero. This category is not semi-simple and exhibits interesting features.
We first construct the simple representations, recovering the classification given by Wiltshire-Gordon. The construction given here also yields explicit descriptions of the indecomposable projectives.
These results are used to give a convenient set of projective generators of the category of representations of $\mathbf{FA}$ and hence a Morita equivalence result. This is used to explain how to calculate the multiplicities of the composition factors of an arbitrary object, based only on its underlying $\mathbf{FB}$-representation, where $\mathbf{FB}$ is the category of finite sets and bijections.
This is applied to show how to calculate the morphism spaces between projectives in our chosen set of generators, as well as for a closely related family of objects (the significance of which can be shown by relative nonhomogeneous Koszul duality theory). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Functors on the category of finite sets revisited Powell, Geoffrey Representation Theory Algebraic Topology We study the structure of the category of representations of $\mathbf{FA}$, the category of finite sets and all maps, mostly working over a field of characteristic zero. This category is not semi-simple and exhibits interesting features. We first construct the simple representations, recovering the classification given by Wiltshire-Gordon. The construction given here also yields explicit descriptions of the indecomposable projectives. These results are used to give a convenient set of projective generators of the category of representations of $\mathbf{FA}$ and hence a Morita equivalence result. This is used to explain how to calculate the multiplicities of the composition factors of an arbitrary object, based only on its underlying $\mathbf{FB}$-representation, where $\mathbf{FB}$ is the category of finite sets and bijections. This is applied to show how to calculate the morphism spaces between projectives in our chosen set of generators, as well as for a closely related family of objects (the significance of which can be shown by relative nonhomogeneous Koszul duality theory). |
| title | Functors on the category of finite sets revisited |
| topic | Representation Theory Algebraic Topology |
| url | https://arxiv.org/abs/2407.11623 |