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Lax monads. Indexed monoidal monads. (English) Zbl 0749.18002

As part of their goal of internalizing enriched category theory in a topos, the authors consider the notion of a lax monad \(T\) on a bicategory \({\mathcal B}\) and construct a bicategory \({\mathcal B}^ T\) of algebras which appears as part of a factorization \({\mathcal B}\to{\mathcal B}^ T\to{\mathcal B}\), with the two morphisms to be thought of as change of base for bicategories. In case \({\mathcal B}=\text{Span}({\mathcal E})\) (for a topos \({\mathcal E}\)), \({\mathcal B}^ T\) is a bicategory of matrices (indexed by objects of \({\mathcal E}\)) of \(T\)-algebras.

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18C20 Eilenberg-Moore and Kleisli constructions for monads
18D20 Enriched categories (over closed or monoidal categories)
18B25 Topoi
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