Abstract
We consider the theory of categories enriched in an involutive quantaloid \(\mathcal{Q}\) : the \(\mathcal{Q}\)-*-categories. After giving an introduction to involutive quantaloids and nuclei, we use matrices with entries in \(\mathcal{Q}\) to define \(\mathcal{Q}\)-*-categories. Then we examine the relations between two kinds of morphisms between them, the functors and the *-maps, to provide a basis to study completeness properties. These results are used to provide a definition of pseudo-presheaves, presheaves and sheaves on involutive quantaloids in order to get a generalization of presheaves and sheaves on sites. Finally a characterization of these sheaves in terms of covers and compatible families is presented.
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Heymans, H. \(\boldsymbol{\mathcal{Q}}\)-*-Categories. Appl Categor Struct 17, 1–28 (2009). https://doi.org/10.1007/s10485-008-9149-0
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DOI: https://doi.org/10.1007/s10485-008-9149-0