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arXiv · 2404.12313

Grothendieck prelopologies: towards a closed monoidal sheaf category

Ana Luiza Tenório·Hugo Luiz Mariano
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Abstract

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.

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Ana Luiza Tenório, Hugo Luiz Mariano. 2024-04-18. Grothendieck prelopologies: towards a closed monoidal sheaf category. https://arxiv.org/abs/2404.12313

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