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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.12313 |
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| _version_ | 1866910414720204800 |
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| author | Tenório, Ana Luiza Mariano, Hugo Luiz |
| author_facet | Tenório, Ana Luiza Mariano, Hugo Luiz |
| contents | In this paper, we present a generalization of Grothendieck pretopologies -- suited for semicartesian categories with equalizers $C$ -- leading to a closed monoidal category of sheaves, instead of closed cartesian category. This is proved through a different sheafification process, which is the left adjoint functor of the suitable inclusion functor but does not preserve all finite limits. If the monoidal structure in $C$ is given by the categorical product, all constructions coincide with those for Grothendieck toposes. The motivation for such generalization stems from a certain notion of sheaves on quantales that does not form a topos. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12313 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Grothendieck prelopologies: towards a closed monoidal sheaf category Tenório, Ana Luiza Mariano, Hugo Luiz Category Theory 18F10, 06F07, 18A40 In this paper, we present a generalization of Grothendieck pretopologies -- suited for semicartesian categories with equalizers $C$ -- leading to a closed monoidal category of sheaves, instead of closed cartesian category. This is proved through a different sheafification process, which is the left adjoint functor of the suitable inclusion functor but does not preserve all finite limits. If the monoidal structure in $C$ is given by the categorical product, all constructions coincide with those for Grothendieck toposes. The motivation for such generalization stems from a certain notion of sheaves on quantales that does not form a topos. |
| title | Grothendieck prelopologies: towards a closed monoidal sheaf category |
| topic | Category Theory 18F10, 06F07, 18A40 |
| url | https://arxiv.org/abs/2404.12313 |