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\begin{document}

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\section*{monoidal adjunction}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Details}{Details}\dotfill \pageref*{Details} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{monoidal adjunction} is an [[adjunction]] between [[monoidal categories]] which respects the monoidal structure.

Since there are several types of [[monoidal functors]] (lax, colax, and strong) there are several types of ``adjunctions between monoidal categories which respect the monoidal structure.'' Namely, we could have:

\begin{itemize}%
\item An adjunction in the [[2-category]] [[MonCat]] of [[monoidal categories]] and [[strong monoidal functors]]. In this case both the left and right adjoint are strong.

\begin{displaymath}
\itexarray{
    \underoverset
      {\underset{R \, \text{strong monoidal}}{\longrightarrow}}
      {\overset{L \, \text{strong monoidal}}{\longleftarrow}}
      {}
  }
\end{displaymath}
We call this a \textbf{strong monoidal adjunction}.


\item An adjunction in the 2-category [[MonCat]]${}_\ell$ of monoidal categories and [[lax monoidal functors]]. In this case the right adjoint is lax, while the left adjoint is necessarily strong (by [[doctrinal adjunction]]; see \href{/nlab/show/doctrinal+adjunction#strength}{here}).

\begin{displaymath}
\itexarray{
\underoverset
{\underset{R \, \text{lax monoidal}}{\longrightarrow}}
{\overset{L \, \text{strong monoidal}}{\longleftarrow}}
{}
}
\end{displaymath}
In fact, since the [[right adjoint]] of an [[oplax monoidal functor]] is necessarily a [[lax monoidal functor]] (\href{oplax+monoidal+functor#OplaxAdjointToLax}{this prop.}), it is sufficient to demand that $L$ be strong monoidal.

This version, which is one of the most frequently occurring, is often called simply a \textbf{monoidal adjunction}.


\item The dual: an adjunction in the 2-category $MonCat_c$ of monoidal categories and colax monoidal functors, in which case the left adjoint is colax and the right adjoint is strong. One might call this an \textbf{opmonoidal adjunction}.


\item A mixed situation, in which the left adjoint is colax, the right adjoint is lax, and the lax and colax structure maps are [[mates]] under the adjunction. This is a [[conjunction]] in the [[double category]] of monoidal categories and lax and colax monoidal functors, so we may call it a \textbf{monoidal conjunction} or a \textbf{lax/colax monoidal adjunction}. By [[doctrinal adjunction]], given any adjunction between monoidal categories, if the right adjoint is lax monoidal, then the left adjoint automatically acquires a colax monoidal structure making the adjunction into a monoidal conjunction, and dually.



\end{itemize}
\hypertarget{Details}{}\subsection*{{Details}}\label{Details}

As mentioned \hyperlink{MonoidalAdjnctionIdea}{above}, the nature of monoidal adjunctions follows as a special case from generalities of [[doctrinal adjunctions]]. For the record, here is an explicit discussion:

\begin{prop}
\label{MonoidalAdjunctionElementary}\hypertarget{MonoidalAdjunctionElementary}{}
Let

\begin{displaymath}
\mathcal{C}
  \underoverset
    {\underset{R}{\longrightarrow}}
    {\overset{L}{\longleftarrow}}
    {\bot}
  \mathcal{D}
\end{displaymath}
be a pair of [[adjoint functors]] between [[monoidal categories]], such that the [[left adjoint]] $L$ is a [[strong monoidal functor]] by [[natural isomorphisms]]

\begin{displaymath}
\mu_L(X,Y)
  \;\colon\;
  L(X) \otimes L(Y) \overset{\simeq}{\longrightarrow} L(X \otimes Y)
\end{displaymath}
and

\begin{displaymath}
e_L 
    \;\colon\;
  1 \overset{\simeq}{\longrightarrow} L(1)
  \,.
\end{displaymath}
Then

\begin{enumerate}%
\item the [[right adjoint]] $R$ becomes a [[lax monoidal functor]] via natural morphisms

\begin{displaymath}
\mu_R(X,Y)
  \;\colon\;
  R (X) \otimes R(Y)
   \overset{\eta(R(X) \otimes R(Y))}{\longrightarrow}
  R L (R(X) \otimes R(Y))
   \underoverset{}{ R( {\mu_L^{-1}(R(X), R(Y))} ) }{\longrightarrow}
  R ( L  R(X) \otimes L R (Y) )
    \overset{R( \epsilon(X) \otimes \epsilon(Y) )}{\longrightarrow}
  R(X \otimes Y)
\end{displaymath}
and

\begin{displaymath}
e_R
  \;\colon\;
  1 
    \overset{\eta(1)}{\longrightarrow} 
  R L(1) 
    \overset{R(e_L^{-1})}{\longrightarrow} 
  R(1)
  \,,
\end{displaymath}
where $\eta$ denotes the [[adjunction unit]] and $\epsilon$ denotes the [[adjunction counit]], as usual.


\item For any object $A \in \mathcal{D}$ carrying the structure of a [[monoid object]] $(A, \mu_A, e_A)$, then

\begin{enumerate}%
\item the [[unit of the adjunction]] $\eta(A) \;\colon\; A \longrightarrow R L(A)$ is a monoid [[homomorphism]] with respect to the canonically induced monoid structure on $R L(A)$ (\href{monoidal+functor#MonoidsToMonoidsByLaxMonoidal}{this prop.}) given by

\begin{displaymath}
\mu_{R L(A)} 
 \;\colon\; 
  R L(A) \otimes R L(A)
 \overset{\mu_R(L(A))}{\longrightarrow}
  R( L(A) \otimes L(A))
 \overset{R(\mu_L(A))}{\longrightarrow}
  R L(A \otimes A)
 \overset{R L(\mu_A)}{\longrightarrow}
  R L(A)
\end{displaymath}
and

\begin{displaymath}
e_{R L(A)}
 \;\colon\;
  1
 \overset{e_R}{\longrightarrow}  
  R(1)
 \overset{R(e_L)}{\longrightarrow}
  R L(1) 
 \overset{R L(e_A)}{\longrightarrow}
  R L(A)
\end{displaymath}

\item similarly for the [[counit of the adjunction]].



\end{enumerate}


\end{enumerate}
\end{prop}
\begin{proof}
The first statement is discussed at \emph{[[oplax monoidal functor]]}.

For the second statement, we need first need to check that the following [[commuting square|square commutes]]:

\begin{displaymath}
\itexarray{
    A \otimes A 
     &\overset{\eta(A) \otimes \eta(A)}{\longrightarrow}&
    R L (A) \otimes R L (A)
    \\
    {}^{\mathllap{\mu_A}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{R L(A)}}}
    \\
    A &\underset{\eta(A)}{\longrightarrow}& R L(A)
  }
\end{displaymath}
Now by definition, the composite of the top and right morphism here is the total diagonal composite in the following diagram:

\begin{displaymath}
\itexarray{
  && A \otimes A
  &\overset{\eta(A \otimes A)}{\longrightarrow}&
  R L(A \otimes A)
  &\overset{R(\mu_L^{-1})}{\longrightarrow}&
  R( L(A) \otimes L(A) )
  \\
  && 
    {}^{\mathllap{\eta(A) \otimes \eta(A)}}\downarrow && {}^{\mathllap{ R L( \eta(A) \otimes \eta(A) ) }}\downarrow 
  && 
    {}^{\mathllap{R( L(\eta(A))  \otimes L(\eta(A)) )}}\downarrow
  & \searrow^{\mathrlap{R(id \otimes id)}}
  \\
  \mu_R(L(A))
  &\colon&
  R L (A) \otimes R L ( A)
   &\overset{\eta(R L (A) \otimes R L (A))}{\longrightarrow}&
  R L (R L ( A) \otimes R L (A))
   &\underoverset{}{ R( {\mu_L^{-1}(R L(A))} ) }{\longrightarrow}&
  R ( L  R L (A) \otimes L R L (A) )
    &\overset{R( \epsilon(L(A)) \otimes \epsilon(L(A)) )}{\longrightarrow}&
  R(L(A) \otimes L(A))
  \\
  && && && && \downarrow^{\mathrlap{ R( \mu_L(A)) }}
  \\
  && && && && R L( A \otimes A )
  \\
  && && && && \downarrow^{\mathrlap{ R L (\mu_A) }}
  \\
  && && && && R L (A)
 }
\end{displaymath}
Here the top sqaures commute by naturality of $\eta$ and $\mu_L$, and top right diagonal morphism is the [[identity morphism]], as shown, by the [[zig-zag identity]] for the adjunction $(L \dashv R)$. Therfore $R(\mu_L^{-1})$ cancels agains $R(\mu_L)$. so that the composite morphism in question becomes just $A \otimes A \overset{\eta(A \otimes A)}{\longrightarrow} R L(A \otimes A) \overset{ R L(\mu_A) }{\longrightarrow} R L(A)$. Again by the naturality of the [[adjunction unit]] $\eta$

\begin{displaymath}
\itexarray{
    A \otimes A 
      &\overset{\eta(A \otimes A)}{\longrightarrow}&
    R L (A \otimes A)
    \\
    {}^{\mathllap{ \mu_A }}\downarrow 
      &&  
    \downarrow^{\mathrlap{R L(\mu_A)}}
    \\
    A &\underset{\eta(A)}{\longrightarrow}& R L (A)
  }
\end{displaymath}
this equals $\eta(A) \circ \mu_A$, as required.

Finally we need to check that the following diagram commutes:

\begin{displaymath}
\itexarray{
     && 1
     \\
     & {}^{\mathllap{e_A}}\swarrow && \searrow^{\mathrlap{ e_{R L (A)} }}
     \\
     A 
     &&
       \underset{\eta(A)}{\longrightarrow}
     &&
    R L(A)
  }
\end{displaymath}
Now unwinding the above definitions of $e_{R L}(A)$ in terms of the definition of $e_R$ we find that

\begin{displaymath}
e_{R L (A)}
  \;\colon\;
  1 
    \overset{\eta(1)}{\longrightarrow} 
  R L(1) 
    \overset{R(e_L^{-1})}{\longrightarrow} 
  R(1)
  \overset{R(e_L)}{\longrightarrow} R L(1) \overset{R L(e_a)}{\longrightarrow} R L(A)
  \,.
\end{displaymath}
Here the two morphisms in the middle cancel, so that we are left just with

\begin{displaymath}
e_{R L(A)}
  \;\colon\;
  1 
    \overset{\eta(1)}{\longrightarrow}
  R L(1)
   \overset{R L(e_A)}{\longrightarrow}
  R L (A)
  \,.
\end{displaymath}
That this equals $\eta(A)\circ e_A$, as required, follows by the naturality of $\eta$:

\begin{displaymath}
\itexarray{
    1 
      &\overset{\eta(1)}{\longrightarrow}&
    R L(1)
    \\
    {}^{\mathllap{e_A}}\downarrow && \downarrow^{\mathrlap{R L(e_A)}}
    \\
    A &\underset{\eta(A)}{\longrightarrow}& R L(A)
  }
  \,.
\end{displaymath}
The argument for the homomorphism property of the counit should be formally dual to the above.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{Stabilization}\hypertarget{Stabilization}{}
\textbf{([[stabilization]] in [[stable homotopy theory]])}

The [[stabilization]] adjunction

\begin{displaymath}
Ho(Spectra)
     \underoverset
       {\underset{\Omega^\infty}{\longrightarrow}}
       {\overset{\Sigma^\infty(-)}{\longleftarrow}}
       {\bot}
    Ho(Spaces^{\ast/})
       \underoverset
         {\underset{}{\longrightarrow}}
         {\overset{(-)_+}{\longleftarrow}}
         {}
    Ho(Spaces)
\end{displaymath}
between the [[classical homotopy categories]] $Ho(Spaces)$ and $Ho(Spaces^{\ast/})$ of ([[pointed topological space|pointed]]) [[topological spaces]] and the [[stable homotopy category]] $Ho(Spectra)$ is a monoidal adjunction, since the [[left adjoint]] $\Sigma^\infty(-)_+$ (forming the [[suspension spectrum]] of a space after freely [[pointed topological space|adjoining a basepoint]]) is [[strong monoidal functor|strong monoidal]] with respect to forming [[product topological spaces]] and forming [[smash product of spectra]], respectively.

In fact this is the [[derived functors]] of what is even a [[monoidal Quillen adjunction]] between the [[classical model structure on topological spaces]] and the stable [[model structure on orthogonal spectra]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#StableMonoidalQuillenSuspensionSpectrumFunctor}{this cor.}), which implies (strong) monoidality of the derived functors on homotopy categories (\href{Introduction+to+Stable+homotopy+theory+--+1-2#StrongMonoidalDerivedFunctorFromStrongMonoidalQuillenAdjunction}{this prop.}).

In detail, let

\begin{displaymath}
(L \dashv R)
   \;\colon\;
  \mathbb{S}_{orth}Mod_{stable}
     \underoverset
      {\underset{\Omega^\infty_{orth}}{\longrightarrow}}
      {\overset{\Sigma^\infty_{orth}}{\longleftarrow}}
      {\bot}
  Top^{\ast/}_{Quillen}
    \underoverset
       {\underset{}{\longrightarrow}}
       {\overset{(-)_+}{\longleftarrow}}
       {\bot}
  Top_{Quillen}
\end{displaymath}
be the Quillen adjunction on orthogonal spectra (\href{Introduction+to+Stable+homotopy+theory+--+1-2#SuspensionSpectrumStructuredStrictQuillenAdjunction}{here}). The left adjoint $L$ is a strong monoidal functor, and hence so is its [[derived functor]] $\Sigma^\infty(-)_+ \colon Ho(Top) \to Ho(Spectra)$ (by \href{Introduction+to+Stable+homotopy+theory+--+1-2#StrongMonoidalDerivedFunctorFromStrongMonoidalQuillenAdjunction}{this prop.}).

 We want to see that the structure of a lax monoidal functor which is induced on the derived right adjoint $\Omega^\infty(-) \colon Ho(Top) \to Ho(Spectra)$ via prop. \ref{MonoidalAdjunctionElementary} is the expected one, given on [[Omega-spectra]] $X$ and $Y$ by

\begin{displaymath}
\Omega^\infty(X) \wedge \Omega^\infty(X)
    =
   X_0 \wedge Y_0
    \overset{}{\to}
   (X \wedge Y)_0
    =
   \Omega^\infty( X \wedge Y )
  \,.
\end{displaymath}
To see this, observe that if $X$ and $Y$ are [[CW-spectrum|CW-]][[Omega-spectra]] and hence cofibrant and fibrant in $\mathbb{S}_{orth}Mod_{stable}$ then the derived lax monoidal structure is given by the total bottom composite in the following diagram

\begin{displaymath}
\itexarray{     
     R (X) \otimes R(Y)
      &\overset{\eta(R(X) \otimes R(Y))}{\longrightarrow}&
     R L (R(X) \otimes R(Y))
      &\underoverset{}{ R( {\mu_L^{-1}(R(X), R(Y))} ) }{\longrightarrow}&
     R ( L  R(X) \otimes L R (Y) )
       &\overset{R( \epsilon(X) \otimes \epsilon(Y) )}{\longrightarrow}&
     R(X \otimes Y)
     \\
     &{}_{\mathllap{ \text{derived } \atop {\text{adjunction unit}} }}\searrow& 
     \downarrow^{\mathrlap{R j L (R(X) \otimes R(Y))}}
     &&
     \downarrow^{\mathrlap{R j ( L R (X) \otimes L R(Y) )}}
     &&
     \downarrow^{\mathrlap{ R j( X \otimes Y ) }}
     \\
      &&
     R P L (R(X) \otimes R(Y))
      &\underoverset{}{ R Q( {\mu_L^{-1}(R(X), R(Y))} ) }{\longrightarrow}&
     R P ( L  R(X) \otimes L R (Y) )
       &\overset{R Q( \epsilon(X) \otimes \epsilon(Y) )}{\longrightarrow}&
     R P (X \otimes Y)
  }
  \,,
\end{displaymath}
where we write for brevity $(L \dashv R) \coloneqq (\Sigma^\infty_{orth} \dashv \Omega^\infty_{orth})$, and where $j \colon id \longrightarrow P$ denotes [[functorial factorization|functorial]] [[fibrant replacement]] (which exists since the [[small object argument]] applies in $\mathbb{S}_{orth}Mod_{stable}$). By functoriality of the replacement, all the squares commute, so that the derived lax monoidal structure on CW-Omega spectra is seen to be equivalently the underived one.

But that underived top horizontal composite is manifestly just the canonical isomorphism $X_0 \wedge Y_0 \simeq (X \wedge Y)_0$ (since $R$ simply picks the component space in degree-0 and $L$ preserves the component space in degree 0).

\end{example}
\begin{example}
\label{ExponentialModality}\hypertarget{ExponentialModality}{}
\textbf{([[exponential modality]] in [[linear type theory]])}

In [[linear type theory]] (see there for more) the [[exponential modality]] $!$ may have [[categorical semantics]] as the [[comonad]] induced by a monoidal adjunction.

\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[projection formula]]


\item [[enriched adjunction]]



\end{itemize}
[[!redirects monoidal adjunctions]] [[!redirects opmonoidal adjunction]] [[!redirects opmonoidal adjunctions]] [[!redirects comonoidal adjunction]] [[!redirects comonoidal adjunctions]] [[!redirects monoidal conjunction]] [[!redirects monoidal conjunctions]]

[[!redirects strong monoidal adjunction]] [[!redirects strong monoidal adjunctions]]



\end{document}