Monoidal monad

<h2 id="opmonoidal-monads">Opmonoidal monads</h2>
Opmonoidal monads have been studied under various names. <a href="/facts/Ieke_Moerdijk/8gX1AMMl">Ieke Moerdijk</a> introduced them as "Hopf monads",<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> while in works of Bruguières and Virelizier they are called "bimonads", by analogy to "<a href="/facts/Bialgebra/xNdlowKl">bialgebra</a>",<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> reserving the term "Hopf monad" for opmonoidal monads with an antipode, in analogy to "<a href="/facts/Hopf_algebra/3QF6g0tT">Hopf algebras</a>".
An opmonoidal monad is a monad 
 
 
 
 (
 T
 ,
 η
 ,
 μ
 )
 
 
 {\displaystyle (T,\eta ,\mu )}
 
 in the 2-category of 
 
 
 
 
 
 O
 p
 M
 o
 n
 C
 a
 t
 
 
 
 
 {\displaystyle {\mathsf {OpMonCat}}}
 
 monoidal categories, oplax monoidal functors and monoidal natural transformations. That means a monad 
 
 
 
 (
 T
 ,
 η
 ,
 μ
 )
 
 
 {\displaystyle (T,\eta ,\mu )}
 
 on a monoidal category 
 
 
 
 (
 C
 ,
 ⊗
 ,
 I
 )
 
 
 {\displaystyle (C,\otimes ,I)}
 
 together with coherence maps 
 
 
 
 
 T
 
 A
 ,
 B
 
 
 :
 T
 (
 A
 ⊗
 B
 )
 →
 T
 A
 ⊗
 T
 B
 
 
 {\displaystyle T^{A,B}:T(A\otimes B)\to TA\otimes TB}
 
 and 
 
 
 
 
 T
 
 0
 
 
 :
 T
 I
 →
 I
 
 
 {\displaystyle T^{0}:TI\to I}
 
 satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit 
 
 
 
 η
 
 
 {\displaystyle \eta }
 
 and the multiplication 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 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 <a href="/facts/Forgetful_functor/onEwu6HA">forgetful functor</a> is strong monoidal.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
An easy example for the monoidal category 
 
 
 
 Vect
 
 
 {\displaystyle \operatorname {Vect} }
 
 of vector spaces is the monad 
 
 
 
 −
 ⊗
 A
 
 
 {\displaystyle -\otimes A}
 
, where 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a <a href="/facts/Bialgebra/xNdlowKl">bialgebra</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> The multiplication and unit of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 define the multiplication and unit of the monad, while the comultiplication and counit of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 give rise to the opmonoidal structure. The algebras of this monad are right 
 
 
 
 A
 
 
 {\displaystyle A}
 
-modules, which one may tensor in the same way as their underlying vector spaces.

<h2 id="properties">Properties</h2>
<ul><li>The <a href="/facts/Kleisli_category/wIQpcPgo">Kleisli category</a> of a monoidal monad has a canonical monoidal structure, induced by the monoidal structure of the monad, and such that the <a href="/facts/Free_object/VxL28y8B">free functor</a> is strong monoidal. The canonical adjunction between 
 
 
 
 C
 
 
 {\displaystyle C}
 
 and the Kleisli category is a <a href="/facts/Monoidal_adjunction/363J0LzN">monoidal adjunction</a> with respect to this monoidal structure, this means that the 2-category 
 
 
 
 
 
 M
 o
 n
 C
 a
 t
 
 
 
 
 {\displaystyle {\mathsf {MonCat}}}
 
 has Kleisli objects for monads.</li>
<li>The 2-category of monads in 
 
 
 
 
 
 M
 o
 n
 C
 a
 t
 
 
 
 
 {\displaystyle {\mathsf {MonCat}}}
 
 is the 2-category of monoidal monads 
 
 
 
 
 
 M
 n
 d
 (
 M
 o
 n
 C
 a
 t
 )
 
 
 
 
 {\displaystyle {\mathsf {Mnd(MonCat)}}}
 
 and it is isomorphic to the 2-category 
 
 
 
 
 
 M
 o
 n
 (
 M
 n
 d
 (
 C
 a
 t
 )
 )
 
 
 
 
 {\displaystyle {\mathsf {Mon(Mnd(Cat))}}}
 
 of monoidales (or pseudomonoids) in the category of monads 
 
 
 
 
 
 M
 n
 d
 (
 C
 a
 t
 )
 
 
 
 
 {\displaystyle {\mathsf {Mnd(Cat)}}}
 
, (lax) monoidal arrows between them and monoidal cells between them.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>The <a href="/facts/Monad_(category_theory)/6fX10Qrx">Eilenberg-Moore category</a> of an opmonoidal monad has a canonical monoidal structure such that the forgetful functor is strong monoidal.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Thus, the 2-category 
 
 
 
 
 
 O
 p
 m
 o
 n
 C
 a
 t
 
 
 
 
 {\displaystyle {\mathsf {OpmonCat}}}
 
 has Eilenberg-Moore objects for monads.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>The 2-category of monads in 
 
 
 
 
 
 O
 p
 m
 o
 n
 C
 a
 t
 
 
 
 
 {\displaystyle {\mathsf {OpmonCat}}}
 
 is the 2-category of monoidal monads 
 
 
 
 
 
 M
 n
 d
 (
 O
 p
 m
 o
 n
 C
 a
 t
 )
 
 
 
 
 {\displaystyle {\mathsf {Mnd(OpmonCat)}}}
 
 and it is isomorphic to the 2-category 
 
 
 
 
 
 O
 p
 m
 o
 n
 (
 M
 n
 d
 (
 C
 a
 t
 )
 )
 
 
 
 
 {\displaystyle {\mathsf {Opmon(Mnd(Cat))}}}
 
 of monoidales (or pseudomonoids) in the category of monads 
 
 
 
 
 
 M
 n
 d
 (
 C
 a
 t
 )
 
 
 
 
 {\displaystyle {\mathsf {Mnd(Cat)}}}
 
 opmonoidal arrows between them and opmonoidal cells between them.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li></ul>
<h2 id="examples">Examples</h2>
The following monads on the <a href="/facts/Category_of_sets/LF5ykYl0">category of sets</a>, with its <a href="/facts/Cartesian_monoidal_category/gqyQs8vz">cartesian monoidal</a> structure, are monoidal monads:

<ul><li>The <a href="/facts/Power_set/FVuf8PDD">power set</a> monad 
 
 
 
 (
 
 P
 
 ,
 ∅
 ,
 ∪
 )
 
 
 {\displaystyle (\mathbb {P} ,\varnothing ,\cup )}
 
. Indeed, there is a function 
 
 
 
 
 P
 
 (
 X
 )
 ×
 
 P
 
 (
 Y
 )
 →
 
 P
 
 (
 X
 ×
 Y
 )
 
 
 {\displaystyle \mathbb {P} (X)\times \mathbb {P} (Y)\to \mathbb {P} (X\times Y)}
 
, sending a pair 
 
 
 
 (
 
 X
 ′
 
 ⊆
 X
 ,
 
 Y
 ′
 
 ⊆
 Y
 )
 
 
 {\displaystyle (X'\subseteq X,Y'\subseteq Y)}
 
 of subsets to the subset 
 
 
 
 {
 (
 x
 ,
 y
 )
 ∣
 x
 ∈
 
 X
 ′
 
 
  and 
 
 y
 ∈
 
 Y
 ′
 
 }
 ⊆
 X
 ×
 Y
 
 
 {\displaystyle \{(x,y)\mid x\in X'{\text{ and }}y\in Y'\}\subseteq X\times Y}
 
. This function is natural in X and Y. Together with the unique function 
 
 
 
 {
 1
 }
 →
 
 P
 
 (
 ∅
 )
 
 
 {\displaystyle \{1\}\to \mathbb {P} (\varnothing )}
 
 as well as the fact that 
 
 
 
 μ
 ,
 η
 
 
 {\displaystyle \mu ,\eta }
 
 are monoidal natural transformations, 
 
 
 
 (
 
 P
 
 
 
 {\displaystyle (\mathbb {P} }
 
 is established as a monoidal monad.</li>
<li>The <a href="/facts/Giry_monad/YX2LEHrY">probability distribution (Giry) monad</a>.</li></ul>
The following monads on the category of sets, with its cartesian monoidal structure, are not monoidal monads

<ul><li>If 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a monoid, then 
 
 
 
 X
 ↦
 X
 ×
 M
 
 
 {\displaystyle X\mapsto X\times M}
 
 is a monad, but in general there is no reason to expect a monoidal structure on it (unless 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is commutative).</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Moerdijk, Ieke (23 March 2002). "Monads on tensor categories". Journal of Pure and Applied Algebra. 168 (2–3): 189–208. doi:10.1016/S0022-4049(01)00096-2. <a href="https://doi.org/10.1016%2FS0022-4049%2801%2900096-2" target="_blank">https://doi.org/10.1016%2FS0022-4049%2801%2900096-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Bruguières, Alain; Alexis Virelizier (2007). "Hopf monads". Advances in Mathematics. 215 (2): 679–733. doi:10.1016/j.aim.2007.04.011. <a href="https://doi.org/10.1016%2Fj.aim.2007.04.011" target="_blank">https://doi.org/10.1016%2Fj.aim.2007.04.011</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Moerdijk, Ieke (23 March 2002). "Monads on tensor categories". Journal of Pure and Applied Algebra. 168 (2–3): 189–208. doi:10.1016/S0022-4049(01)00096-2. <a href="https://doi.org/10.1016%2FS0022-4049%2801%2900096-2" target="_blank">https://doi.org/10.1016%2FS0022-4049%2801%2900096-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">McCrudden, Paddy (2002). "Opmonoidal monads". Theory and Applications of Categories. 10 (19): 469–485. CiteSeerX 10.1.1.13.4385. <a href="http://www.tac.mta.ca/tac/volumes/10/19/10-19abs.html" target="_blank">http://www.tac.mta.ca/tac/volumes/10/19/10-19abs.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Bruguières, Alain; Alexis Virelizier (2007). "Hopf monads". Advances in Mathematics. 215 (2): 679–733. doi:10.1016/j.aim.2007.04.011. <a href="https://doi.org/10.1016%2Fj.aim.2007.04.011" target="_blank">https://doi.org/10.1016%2Fj.aim.2007.04.011</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Zawadowski, Marek (2011). "The Formal Theory of Monoidal Monads The Kleisli and Eilenberg-Moore objects". Journal of Pure and Applied Algebra. 216 (8–9): 1932–1942. arXiv:1012.0547. doi:10.1016/j.jpaa.2012.02.030. S2CID 119301321. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Moerdijk, Ieke (23 March 2002). "Monads on tensor categories". Journal of Pure and Applied Algebra. 168 (2–3): 189–208. doi:10.1016/S0022-4049(01)00096-2. <a href="https://doi.org/10.1016%2FS0022-4049%2801%2900096-2" target="_blank">https://doi.org/10.1016%2FS0022-4049%2801%2900096-2</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">McCrudden, Paddy (2002). "Opmonoidal monads". Theory and Applications of Categories. 10 (19): 469–485. CiteSeerX 10.1.1.13.4385. <a href="http://www.tac.mta.ca/tac/volumes/10/19/10-19abs.html" target="_blank">http://www.tac.mta.ca/tac/volumes/10/19/10-19abs.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Zawadowski, Marek (2011). "The Formal Theory of Monoidal Monads The Kleisli and Eilenberg-Moore objects". Journal of Pure and Applied Algebra. 216 (8–9): 1932–1942. arXiv:1012.0547. doi:10.1016/j.jpaa.2012.02.030. S2CID 119301321. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Monoidal monad open-in-new

Monoidal monad