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\begin{document}

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\section*{monoid in a monoidal category}

\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{morphism_of_monoids}{Morphism of monoids}\dotfill \pageref*{morphism_of_monoids} \linebreak
\noindent\hyperlink{as_categories_with_one_object}{As categories with one object}\dotfill \pageref*{as_categories_with_one_object} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{preservation_by_lax_monoidal_functors}{Preservation by lax monoidal functors}\dotfill \pageref*{preservation_by_lax_monoidal_functors} \linebreak
\noindent\hyperlink{category_of_monoids}{Category of monoids}\dotfill \pageref*{category_of_monoids} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

Generalizing the classical notion of [[monoid]], one can define a \emph{monoid} (or \emph{monoid object}) in any [[monoidal category]] $(C,\otimes,I)$. Classical monoids are of course just monoids in [[Set]] with the [[cartesian product]].

By the [[microcosm principle]], in order to define monoid objects in $C$, $C$ itself must be a ``categorified monoid'' in some way. The natural requirement is that it be a [[monoidal category]]. In fact, it suffices if $C$ is a [[multicategory]]. Contrast this with a [[group object]], which can only be defined in a [[cartesian monoidal category]] (or a [[cartesian multicategory]]).

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

Namely, a \textbf{monoid in $C$} is an object $M$ equipped with a multiplication $\mu: M \otimes M \to M$ and a unit $\eta: I \to M$ satisfying the \textbf{associative law}:

\begin{displaymath}
\itexarray{
  & (M \otimes M) \otimes M
    & \stackrel{\alpha}{\longrightarrow}
  & M \otimes (M \otimes M)
    & \stackrel{1 \otimes \mu}{\longrightarrow}
  & M \otimes M
  \\
  & {}_{\mu \otimes 1}\searrow
    &&
    && \swarrow_{\mu}
  &
  \\
  &&
  M \otimes M
    & \stackrel{\mu}{\longrightarrow}
  M
  &&
}
\end{displaymath}
and the \textbf{left and right unit laws}:

\begin{displaymath}
\itexarray{
  & I \otimes M
  & \stackrel{\eta \otimes 1}{\longrightarrow}
  & M \otimes M
  & \stackrel{1 \otimes \eta}{\longleftarrow}
  & M \otimes I
  \\
  &
  & {}_{\lambda}\searrow
  & {}_{\mu}\downarrow
  & \swarrow_{\rho}
  &
  \\
  &
  &
  & M
  &
  &
}
\end{displaymath}
Here $\alpha$ is the [[associator]] in $C$, while $\lambda$ and $\rho$ are the left and right [[unitor|unitors]].

\hypertarget{morphism_of_monoids}{}\subsection*{{Morphism of monoids}}\label{morphism_of_monoids}

The analogue of a monoid homomorphism, called a morphism of monoids, is a morphism, $\f: M \to M'$ between two monoid objects, satisfying the equations;

$f \circ \mu = \mu' \circ (f \otimes f)$

$f \circ \eta = \eta'$

corresponding to the commutative diagrams;

\begin{displaymath}
\itexarray{
  & M \otimes M
  & \stackrel{f \otimes f}{\longrightarrow}
  & M' \otimes M'
  \\
  & {}_{\mu}\downarrow
  &
  & \downarrow_{\mu'}
  \\
  & M
  & \stackrel{f}{\longrightarrow}
  & M'
}
\end{displaymath}
\begin{displaymath}
\itexarray{
  & I
  & \stackrel{\eta}{\longrightarrow}
  & M
  \\
  &
  & {}_{\eta'}\searrow
  & \downarrow_{f}
  \\
  &
  &
  & M'
}
\end{displaymath}
\hypertarget{as_categories_with_one_object}{}\subsection*{{As categories with one object}}\label{as_categories_with_one_object}

Just as the category of regular [[monoid|monoids]] is equivalent to the category of locally small (i.e. Set-enriched) categories with one object, the category of monoids in $C$ (with the obvious morphisms) is equivalent to the category of $C$-[[enriched category|enriched categories]] with one object.

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

\hypertarget{preservation_by_lax_monoidal_functors}{}\subsubsection*{{Preservation by lax monoidal functors}}\label{preservation_by_lax_monoidal_functors}

Monoid structure is preserved by [[lax monoidal functor]]s. Comonoid structure by [[oplax monoidal functor]]s. See there for more.

\hypertarget{category_of_monoids}{}\subsubsection*{{Category of monoids}}\label{category_of_monoids}

For special properties of categories of monoids, see [[category of monoids]].

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

\begin{itemize}%
\item A monoid object in [[Ab]] (with the usual tensor product of $\mathbb{Z}$-modules as the tensor product) is a [[ring]]. A monoid object in the category of vector spaces over a field $k$ (with the usual tensor product of vector spaces) is an [[algebra]] over $k$.
\item A monoid in a [[category of modules]] is an [[associative unital algebra]].
\item A monoid object in [[Top]] (with cartesian product as the tensor product) is a [[topological monoid]].
\item A monoid object in [[Ho(Top)]] is an [[H-monoid]].
\item A monoid object in the category of monoids (with cartesian product as the tensor product) is a [[commutative monoid]]. This is a version of the [[Eckmann-Hilton argument]].
\item A monoid object in the category of complete join-[[semilattice]]s (with its tensor product that represents maps preserving joins in each variable separately) is a unital [[quantale]].
\item Given any monoidal category $C$, a monoid in the monoidal category $C^{op}$ is called a [[comonoid]] in $C$.
\item In a [[cocartesian monoidal category]], every object is a monoid object in a unique way.
\item For any category $C$, the [[endofunctor]] category $C^C$ has a monoidal structure induced by composition of endofunctors, and a monoid object in $C^C$ is a [[monad]] on $C$.

\end{itemize}
These are examples of monoids internal to monoidal categories. More generally, given any [[bicategory]] $B$ and a chosen object $a$, the [[hom-category]] $B(a,a)$ has the structure of a monoidal category. So, the concept of monoid makes sense in any [[bicategory]] $B$: we define a \textbf{monoid in $B$} to be a monoid in $B(a,a)$ for some object $a \in B$. This often called a [[monad]] in $B$. The reason is that a monad in [[Cat]] is the same as monad on a category.

A monoid in a bicategory $B$ may also be described as the [[hom-object]] of a $B$-[[enriched category]] with a single object.

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

\begin{itemize}%
\item [[monoid]]
\item [[commutative monoid in a symmetric monoidal category]]
\item [[module over a monoid]]
\item [[category of monoids]]
\item [[monoid in a monoidal (infinity,1)-category]]

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

Categorical properties of [[monoid objects]] in [[monoidal categories]] are spelled out in sections 1.2 and 1.3 of

\begin{itemize}%
\item Florian Marty, \emph{Des Ouverts Zariski et des Morphismes Lisses en G\'e{}om\'e{}trie Relative}, Ph.D. Thesis, 2009, \href{http://thesesups.ups-tlse.fr/540/}{web}

\end{itemize}
A summary is in section 4.1 of

\begin{itemize}%
\item [[Martin Brandenburg]], \emph{Tensor categorical foundations of algebraic geometry}, \href{http://arxiv.org/abs/1410.1716}{arXiv:1410.1716}.

\end{itemize}
See also \href{http://mathoverflow.net/questions/180673/category-of-modules-over-commutative-monoid-in-symmetric-monoidal-category}{MO/180673}.

[[!redirects monoids in a monoidal category]] [[!redirects monoids in monoidal categories]]

[[!redirects monoid object in a monoidal category]] [[!redirects monoid objects in a monoidal category]] [[!redirects monoid objects in monoidal categories]]

[[!redirects monoid object]] [[!redirects monoid objects]] [[!redirects internal monoid]] [[!redirects internal monoids]]

[[!redirects algebra object]] [[!redirects algebra objects]]



\end{document}