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\begin{document}

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\section*{monoidal functor}

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

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

[[!include monoidal categories - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{string_diagrams}{String diagrams}\dotfill \pageref*{string_diagrams} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{monoidal functor} is a [[functor]] between [[monoidal categories]] that preserves the monoidal structure: a [[homomorphism]] of monoidal categories.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{LaxMonoidalFunctor}\hypertarget{LaxMonoidalFunctor}{}
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two [[monoidal categories]]. A \textbf{lax monoidal functor} between them is

\begin{enumerate}%
\item a [[functor]]

\begin{displaymath}
F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
  \,,
\end{displaymath}

\item a morphism

\begin{displaymath}
\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
\end{displaymath}

\item a [[natural transformation]]

\begin{displaymath}
\mu_{x,y} 
    \;\colon\; 
  F(x) \otimes_{\mathcal{D}} F(y) 
    \longrightarrow 
  F(x \otimes_{\mathcal{C}} y)
\end{displaymath}
for all $x,y \in \mathcal{C}$



\end{enumerate}
satisfying the following conditions:

\begin{enumerate}%
\item \textbf{([[associativity]])} For all objects $x,y,z \in \mathcal{C}$ the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z)
      &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}&
    F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) )
    \\
    {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow 
      && 
    \downarrow^{\mathrlap{id\otimes \mu_{y,z}}}
    \\
    F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z)
     &&
    F(x) \otimes_{\mathcal{D}} F(y \otimes_{\mathcal{C}} z)
    \\
    {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z  }}}
    \\
    F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z  )
      &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}&
    F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) )
  }
  \,,
\end{displaymath}
where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the [[associators]] of the monoidal categories;


\item \textbf{([[unitality]])} For all $x \in \mathcal{C}$ the following [[commuting diagram|diagrams commute]]

\begin{displaymath}
\itexarray{
    1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x)
      &\overset{\epsilon \otimes id}{\longrightarrow}&
    F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x)
    \\
    {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}}
    \\
    F(x) 
      &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}&
    F(1 \otimes_{\mathcal{C}} x  )
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    F(x) \otimes_{\mathcal{D}}  1_{\mathcal{D}}
      &\overset{id \otimes \epsilon }{\longrightarrow}&
    F(x) \otimes_{\mathcal{D}}  F(1_{\mathcal{C}}) 
    \\
    {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}}
    \\
    F(x) 
      &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}&
    F(x \otimes_{\mathcal{C}} 1  )
  }
  \,,
\end{displaymath}
where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right [[unitors]] of the two monoidal categories, respectively.



\end{enumerate}
If $\epsilon$ and all $\mu_{x,y}$ are [[isomorphisms]], then $F$ is called a \textbf{strong monoidal functor}. (Note that `strong' is also sometimes applied to `monoidal functor' to indicate possession of a [[tensorial strength]].)

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In the literature often the term ``monoidal functor'' refers by default to what in def. \ref{LaxMonoidalFunctor} is called a strong monoidal functor. With that convention then what def. \ref{LaxMonoidalFunctor} calls a lax monoidal functor is called a \textbf{weak monoidal functor}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Lax monoidal functors are the [[lax morphisms]] for an appropriate [[2-monad]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
An \textbf{[[oplax monoidal functor]]} (with various alternative names including \textbf{comonoidal}), is a monoidal functor from the [[opposite categories]] $C^{op}$ to $D^{op}$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
A \emph{[[monoidal transformation]]} between monoidal functors is a [[natural transformation]] that respects the extra structure in an obvious way.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{MonoidsToMonoidsByLaxMonoidal}\hypertarget{MonoidsToMonoidsByLaxMonoidal}{}
\textbf{(Lax monoidal functors send [[monoid]]s to monoids)}

If $F : (C,\otimes) \to (D,\otimes)$ is a lax monoidal functor and

\begin{displaymath}
(A \in C,\;\; \mu_A : A \otimes A \to A, \; i_A : I \to A)
\end{displaymath}
is a [[monoid object]] in $C$, then the object $F(A)$ is naturally equipped with the structure of a monoid in $D$ by setting

\begin{displaymath}
i_{F(A)} : I_D \stackrel{}{\to} F(I_C) \stackrel{F(i_A)}{\to} F(A)
\end{displaymath}
and

\begin{displaymath}
\mu_{F(A)} : 
  F(A) \otimes F(A)
  \stackrel{\nabla_{F(A), F(A)}}{\to} 
  F(A \otimes A)
  \stackrel{F(\mu_A)}{\to}
  F(A)
  \,.
\end{displaymath}
This construction defines a functor

\begin{displaymath}
Mon(f) : Mon(C) \to Mon(D)
\end{displaymath}
between the [[categories of monoids]] in $C$ and $D$, respectively.

\end{prop}
Similarly:

\begin{uprop}
\textbf{([[oplax monoidal functors]] sends [[comonoids]] to comonoids)}

For $(C,\otimes)$ a [[monoidal category]] write $\mathbf{B}C$ for the corresponding [[delooping]] [[2-category]].

Lax monoidal functor $f : C \to D$ correspond to [[lax 2-functor]]

\begin{displaymath}
\mathbf{B}F : \mathbf{B}C \to \mathbf{B}D
  \,.
\end{displaymath}
If $F$ is strong monoidal then this is an ordinary [[2-functor]]. If it is strict monoidal, then this is a [[strict 2-functor]].

\end{uprop}
\hypertarget{string_diagrams}{}\subsection*{{String diagrams}}\label{string_diagrams}

Just like monoidal categories, monoidal functors have a [[string diagram]] calculus; see \href{http://web.science.mq.edu.au/~mmccurdy/cms2010talk.pdf}{these slides} for some examples.

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

\begin{itemize}%
\item \textbf{monoidal functor}, \textbf{strong monoidal functor}


\item [[multifunctor]]


\item [[module over a monoidal functor]]


\item [[monoidal adjunction]]


\item [[indexed monoidal category]]


\item [[cartesian functor]]


\item \textbf{lax monoidal functor}

\begin{itemize}%
\item [[functor with smash products]]

\end{itemize}

\item [[oplax monoidal functor]]


\item [[bilax monoidal functor]]


\item [[Frobenius monoidal functor]]


\item [[braided monoidal functor]]


\item [[symmetric monoidal functor]]


\item [[monoidal (∞,1)-functor]]


\item [[monoidal (∞,n)-functor]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Exposition of basics of [[monoidal categories]] and [[categorical algebra]]:

\begin{itemize}%
\item \emph{[[geometry of physics -- categories and toposes]]}, Section 2: \emph{\href{geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra}{Basic notions of categorical algebra}}

\end{itemize}
[[!redirects lax monoidal functor]] [[!redirects strict monoidal functor]] [[!redirects strong monoidal functor]] [[!redirects weak monoidal functor]] [[!redirects monoidal functors]] [[!redirects lax monoidal functors]] [[!redirects strict monoidal functors]] [[!redirects strong monoidal functors]] [[!redirects weak monoidal functors]] [[!redirects strength]]



\end{document}