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Exponentiable functors between quantaloid-enriched categories
Clementino, Maria Manuel; Hofmann, Dirk; Stubbe, IsarExponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity; this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor’s exponentiability follow...
Categorical structures enriched in a quantaloid: tensored and cotensored catego...
Stubbe, IsarOur subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and order-cocomplete. Bearing this in mind, we analyze how Sup-valued homomorphisms on Q are related to Q-categories. With an appendix on action, representation and...
More on Q-modules
Stubbe, IsarA. Joyal and M. Tierney showed that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. Using a totally different technique, I shall show a generalization of this result to the case of (ordered) sheaves on a (small) quantaloid. Then I make a comment on module-equivalence versus sheafequivalence, using a recent observation of B. Mesablishvili and the notion of ‘...
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