词条 | Monoidal monad |
释义 |
In category theory, a monoidal monad is a monad on a monoidal category such that the functor is a lax monoidal functor and the natural transformations and are monoidal natural transformations. In other words, is equipped with coherence maps and satisfying certain properties, and the unit and multiplication are monoidal natural transformations. By monoidality of , the morphisms and are necessarily equal. This is equivalent to say that a monoidal monad is a monad in the 2-category of monoidal categories, lax monoidal functors, and monoidal natural transformations. Opmonoidal MonadsOpmonoidal monads have been studied under various names. Ieke Moerdijk introduced them as "Hopf monads",[1] while in works of Bruguières and Virelizier they are called "bimonads", by analogy to "bialgebra",[2] reserving the term "Hopf monad" for opmonoidal monads with an antipode, in analogy to "Hopf algebras". An opmonoidal monad is a monad in the 2-category of monoidal categories, oplax monoidal functors and monoidal natural transformations. That means a monad on a monoidal category together with coherence maps and satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit and the multiplication into opmonoidal natural transformations. Alternatively, an opmonoidal monad is a monad on a monoidal category such that the category of Eilenberg-Moore algebras has a monoidal structure for which the forgetful functor is strong monoidal.[1][3] An easy example for the monoidal category of vector spaces is the monad , where is a bialgebra.[2] The multiplication and unit of define the multiplication and unit of the monad, while the comultiplication and counit of give rise to the opmonoidal structure. The algebras of this monad are right -modules, which one may tensor in the same way as their underlying vector spaces. Properties
ExamplesThe following monads on the category of sets, with its cartesian monoidal structure, are monoidal monads:
The following monads on the category of sets, with its cartesian monoidal structure, are not monoidal monads
References1. ^1 2 {{cite journal|last=Moerdijk|first=Ieke|title=Monads on tensor categories|journal=Journal of Pure and Applied Algebra|date=23 March 2002|volume=168|issue=2–3|pages=189–208|doi=10.1016/S0022-4049(01)00096-2|url=http://www.sciencedirect.com/science/article/pii/S0022404901000962|accessdate=23 May 2014}} {{DEFAULTSORT:Monoidal Monad}}2. ^1 {{cite journal|last=Bruguières|first=Alain|author2=Alexis Virelizier|title=Hopf monads|journal=Advances in Mathematics|date=10 November 2007|volume=215|issue=2|pages=679–733|doi=10.1016/j.aim.2007.04.011|url=http://www.sciencedirect.com/science/article/pii/S0001870807001430|accessdate=23 May 2014}} 3. ^1 {{Cite journal|last=McCrudden|first=Paddy|year=2002|title=Opmonoidal monads|url=http://www.tac.mta.ca/tac/volumes/10/19/10-19abs.html|journal=Theory and Applications of Categories|volume= 10 No. 19|issue= 19|pages=469–485|via=}} 4. ^1 {{Cite journal|last=Zawadowski|first=Marek|year=2011|title=The Formal Theory of Monoidal Monads The Kleisli and Eilenberg-Moore objects|url=http://linkinghub.elsevier.com/retrieve/pii/S0022404912000692|journal=Journal of Pure and Applied Algebra|volume=216|issue=8–9|pages=1932–1942|doi=10.1016/j.jpaa.2012.02.030|via=|arxiv=1012.0547}} 1 : Monoidal categories |
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