A Note on Actions of a Monoidal Category

By I. I, Áµ Òò Î ¼ ´î ¼ ª ¼ Á ¼ Μ Óò×××ø, Óó Øööôðð´ae. Theory and Applications of Categories, 2001

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An action ∗ : V× A−→ Aof a monoidal category V on a category A corresponds to a strong monoidal functor F : V−→ (A, A)into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V−→ (A, A) underlying the monoidal F has a right adjoint g; and when this is so, F itself has a right adjoint G as a monoidal functor—so that, passing to the categories of monoids (also called "algebras") in V and in (A, A), we have an adjunction Mon FMon G between the category Mon V of monoids in V and the category Mon(A, A )=M ndA of monads on A. We give sufficient conditions for the existence of the right adjoint g, which involve the existence of right adjoints for the functors X∗ - and - ∗A, and make A (at least when V is symmetric and closed) into a tensored and cotensored V-category A.W e give explicit formulae, as large ends, for the right adjoints g and Mon G, and also for some related right adjoints, when they exist; as well as another explicit expression for Mon G as a large limit, which uses a new representation of any monad as a (large) limit of monads of two special kinds, and an analogous result for general endofunctors.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5f071a5ba1fb9cd508e0ca5619393068acae3654

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

Theory of interleavings on categories with a flow (2017, unknown): “Given a monoidal category V = (V ,⊗, I) a coherent action of V on C is a lax monoidal functor V → End(C) of monoidal categories [15, 14].”
Second-Order and Dependently-Sorted Abstract Syntax (2008, background): “Further, an important class of V -actions arises from V -enriched categories, for V symmetric monoidal closed, with tensors and powers (see [14]).”
The Picard crossed module of a braided tensor category (2012, background): “Analogues of (bi-)modules over algebras are (bi-@)module categories over tensor categories [Quillen 1973; Janelidze and Kelly 2001; Ostrik 2003b].”