{{technical|date=April 2022}}
In category theory, a '''strong monad''' is a monad on a monoidal category with an additional natural transformation, called the '''strength''', which governs how the monad interacts with the monoidal product.
Strong monads play an important role in theoretical computer science where they are used to model computation with side effects.<ref name="Moggi91">{{cite journal|last1=Moggi|first1=Eugenio|title=Notions of computation and monads|journal=Information and Computation|date=July 1991|volume=93|issue=1|pages=55–92|doi=10.1016/0890-5401(91)90052-4|url = http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf|doi-access=free}}</ref>
== Definition ==
A (left) '''strong monad''' is a monad (''T'', η, μ) over a monoidal category (''C'', ⊗, I) together with a natural transformation ''t''<sub>''A,B''</sub> : ''A'' ⊗ ''TB'' → ''T''(''A'' ⊗ ''B''), called (''tensorial'') ''left strength'', such that the diagrams :Image:Strong monad left unit.svg, Image:Strong monad associative.svg, :Image:Strong monad unit.svg, and Image:Strong monad multiplication.svg commute for every object ''A'', ''B'' and ''C''.
== Commutative strong monads ==
For every strong monad ''T'' on a symmetric monoidal category, a ''right strength'' natural transformation can be defined by
<math display="block">t'_{A,B}=T(\gamma_{B,A})\circ t_{B,A}\circ\gamma_{TA,B} : TA\otimes B\to T(A\otimes B).</math>
A strong monad ''T'' is said to be '''commutative''' when the diagram :Image:Strong monad commutation.svg commutes for all objects <math>A</math> and <math>B</math>.
== Properties ==
The Kleisli category of a commutative monad is symmetric monoidal in a canonical way, see corollary 7 in Guitart<ref>{{Cite journal |last=Guitart |first=René |date=1980 |title=Tenseurs et machines |url=http://www.numdam.org/item/?id=CTGDC_1980__21_1_5_0 |journal=Cahiers de topologie et géométrie différentielle |language=en |volume=21 |issue=1 |pages=5–62 |issn=2681-2398}}</ref> and corollary 4.3 in Power & Robison.<ref>{{Cite journal |last1=Power |first1=John |last2=Robinson |first2=Edmund |date=October 1997 |title=Premonoidal categories and notions of computation |url=https://www.cambridge.org/core/product/identifier/S0960129597002375/type/journal_article |journal=Mathematical Structures in Computer Science |language=en |volume=7 |issue=5 |pages=453–468 |doi=10.1017/S0960129597002375 |issn=0960-1295|url-access=subscription }}</ref> When a monad is strong but not necessarily commutative, its Kleisli category is a premonoidal category.
One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads.<ref>{{Cite journal |last=Kock |first=Anders |date=1972-12-01 |title=Strong functors and monoidal monads |url=https://link.springer.com/article/10.1007/BF01304852 |journal=Archiv der Mathematik |language=en |volume=23 |issue=1 |pages=113–120 |doi=10.1007/BF01304852 |issn=1420-8938|url-access=subscription }}</ref> More explicitly, * a commutative strong monad <math>(T,\eta,\mu,t)</math> defines a symmetric monoidal monad <math>(T,\eta,\mu,m)</math> by<math display="block">m_{A,B}=\mu_{A\otimes B}\circ Tt'_{A,B}\circ t_{TA,B}:TA\otimes TB\to T(A\otimes B)</math> * and conversely a symmetric monoidal monad <math>(T,\eta,\mu,m)</math> defines a commutative strong monad <math>(T,\eta,\mu,t)</math> by<math display="block">t_{A,B}=m_{A,B}\circ(\eta_A\otimes 1_{TB}):A\otimes TB\to T(A\otimes B)</math> and the conversion between one and the other presentation is bijective.
== References == {{Reflist}} ==External links== * [https://ncatlab.org/nlab/show/strong+monad Strong monad] at the nLab
Category:Adjoint functors Category:Monoidal categories