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\begin{document}

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\section*{module over a monoid}

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

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{higher_linear_algebra}{}\paragraph*{{Higher linear algebra}}\label{higher_linear_algebra}

[[!include homotopy - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{modules_over_monoids_in_abelian_groups}{Modules over monoids in abelian groups}\dotfill \pageref*{modules_over_monoids_in_abelian_groups} \linebreak
\noindent\hyperlink{GSets}{$G$-sets}\dotfill \pageref*{GSets} \linebreak
\noindent\hyperlink{AbelianGroupsWithGAction}{Abelian groups with $G$-action as modules over the group ring}\dotfill \pageref*{AbelianGroupsWithGAction} \linebreak
\noindent\hyperlink{more_examples}{more examples}\dotfill \pageref*{more_examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In a [[monoidal category]], there is a notion of [[modules]] over [[monoid objects]] which generalizes the classical notion of [[modules]] over [[rings]]. This is a special case of [[module over a monad]] where the monad is taken to be $A \otimes -$, with $A$ some [[monoid object]].

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

Let $(\mathcal{V}, \otimes, I)$ be a [[monoidal category]] and $A$ a [[monoid object]] in $\mathcal{V}$, hence an object $A \in \mathcal{V}$ equipped with a multiplication morphism

\begin{displaymath}
\cdot : A \otimes A \to A
\end{displaymath}
and a unit element

\begin{displaymath}
e : I \to A
\end{displaymath}
satisfying the [[associativity law]] and the [[unit law]].

\begin{defn}
\label{ModuleInMonoidalCategory}\hypertarget{ModuleInMonoidalCategory}{}
A (left) \textbf{module} over $A$ in $(\mathcal{V}, \otimes, I)$ is

\begin{itemize}%
\item an [[object]] $N \in \mathcal{V}$


\item equipped with a morphism

\begin{displaymath}
\lambda : A \otimes N \to N
\end{displaymath}
in $\mathcal{V}$



\end{itemize}
such that this satisfies the axioms of an [[action]], in that the following are [[commuting diagrams]] in $\mathcal{V}$:

\begin{displaymath}
\itexarray{
     A \otimes A \otimes N &\stackrel{id_A \otimes \lambda}{\to}& A \otimes N
     \\
     \downarrow^{\mathrlap{\cdot \otimes id_n}} && \downarrow^{\mathrlap{\lambda}}
     \\
     A \otimes N &\stackrel{\lambda}{\to}& N
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{   
     I \otimes N &&\stackrel{e \otimes id_N}{\to}&& A \otimes N
     \\
     & \searrow && \swarrow_{\mathrlap{\lambda}}
     \\
     && N
  }
  \,.
\end{displaymath}
\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{modules_over_monoids_in_abelian_groups}{}\subsubsection*{{Modules over monoids in abelian groups}}\label{modules_over_monoids_in_abelian_groups}

Recall that a [[ring]], in the classical sense, is a [[monoid object]] in the category [[Ab]] of [[abelian groups]] with [[monoidal structure]] given by the [[tensor product of abelian groups]] $\otimes$. Accordingly a \emph{module over $R$} is a module in $(Ab,\otimes)$ according to def. \ref{ModuleInMonoidalCategory}.

We unwind what this means in terms of [[abelian groups]] regarded as [[sets]] with extra [[structure]]:

\begin{defn}
\label{ModuleOverARing}\hypertarget{ModuleOverARing}{}
A \textbf{module} $N$ over a ring $R$ is

\begin{enumerate}%
\item an [[object]] $N \in$ [[Ab]], hence an [[abelian group]];


\item equipped with a [[morphism]]

\begin{displaymath}
\alpha : R \otimes N \to N
\end{displaymath}
in [[Ab]]; hence a [[function]] of the underlying [[sets]] that sends elements

\begin{displaymath}
(r,n) \mapsto  r n  \coloneqq \alpha(r,n)
\end{displaymath}
and which is a [[bilinear function]] in that it satisfies

\begin{displaymath}
(r, n_1 + n_2) \mapsto r n_1 + r n_2
\end{displaymath}
and

\begin{displaymath}
(r_1 + r_2, n) \mapsto r_1 n + r_2 n
\end{displaymath}
for all $r, r_1, r_2 \in R$ and $n,n_1, n_2 \in N$;


\item such that the [[diagram]]

\begin{displaymath}
\itexarray{
     R \otimes R \otimes N 
     &\stackrel{\cdot_R \otimes Id_N}{\to}& R \otimes N
     \\
     {}^{\mathllap{Id_R \otimes \alpha}}\downarrow 
     && 
     \downarrow^{\mathrlap{\alpha}}
     \\
     R \otimes N &\to& N 
  }
\end{displaymath}
[[commuting diagram|commutes]] in [[Ab]], which means that for all elements as before we have

\begin{displaymath}
(r_1 \cdot r_2) n = r_1 (r_2 n)
   \,.
\end{displaymath}

\item such that the diagram

\begin{displaymath}
\itexarray{
     1 \otimes N &&\stackrel{1 \otimes id_N}{\to}&& R \otimes N
     \\
     & \searrow && \swarrow_{\mathrlap{\alpha}}
     \\
     && N
  }
\end{displaymath}
commutes, which means that on elements as above

\begin{displaymath}
1 \cdot n = n
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The category of all modules over all commutative rings is [[Mod]]. It is a [[bifibration]]

\begin{displaymath}
Mod \to CRing
\end{displaymath}
over [[CRing]].

This fibration may be characterized intrinsically, which gives yet another way of defining $R$-modules. This we turn to \hyperlink{ModulesOverARingInTermsOfStabilizedSlices}{below}.

\end{remark}
\hypertarget{GSets}{}\subsubsection*{{$G$-sets}}\label{GSets}

Simpler than the traditionally default notion of a module in $(Ab,\otimes)$, as \hyperlink{Rings}{above} is that of a module in [[Set]], equipped with its [[cartesian monoidal category|cartesian monoidal structure]]. (These days one may want to think of this as a notion of modules over [[F1]].)

A [[monoid object]] in $(Set,\times)$ is just a [[monoid]], for instance a [[discrete group]] $G$. A $G$-module in $(Set,\times)$ is simpy an \emph{[[action]]}, say a [[group action]].

\begin{defn}
\label{ActionOfDiscreteGroupOnSet}\hypertarget{ActionOfDiscreteGroupOnSet}{}
For $S \in$ [[Set]] and $G$ a [[discrete group]], a \textbf{$G$-[[action]]} of $G$ on $S$ is a [[function]]

\begin{displaymath}
\lambda \colon G \times S \to S
\end{displaymath}
such that

\begin{enumerate}%
\item the neutral element acts trivially

\begin{displaymath}
\itexarray{
     * \times S &&\stackrel{\simeq}{\to}&& S
     \\  
     & {}_{(e,id_S)}\searrow && \nearrow_{\mathrlap{\lambda}}
     \\
     && G \times S
  }
\end{displaymath}

\item the action property holds: for all $g_1, g_2 \in G$ and $s \in S$ we have $\lambda(g_1,\lambda(g_2, s)) = \lambda(g_1 \cdot g_2, s)$.



\end{enumerate}
\end{defn}
\hypertarget{AbelianGroupsWithGAction}{}\subsubsection*{{Abelian groups with $G$-action as modules over the group ring}}\label{AbelianGroupsWithGAction}

If a [[discrete group]] acts, as in def. \ref{ActionOfDiscreteGroupOnSet}, on the set underlying an [[abelian group]] and acts by [[linear maps]] (abelian group [[homomorphisms]]), then this action is equivalently a module over the [[group ring]] $\mathbb{Z}[G]$ as in def. \ref{ModuleOverARing}.

\begin{defn}
\label{}\hypertarget{}{}
For $G$ a [[discrete group]], write $\mathbb{Z}[G] \in$ [[Ring]] for the [[ring]]

\begin{enumerate}%
\item whose underlying [[abelian group]] is the [[free abelian group]] on the set underlying $G$;


\item whose multiplication is given on [[basis]] elements by the group operation in $G$.



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For $G$ a [[finite group]] an element $r$r of $\mathbb{Z}[G]$ is for the form

\begin{displaymath}
r = \sum_{g \in G} r_g g
\end{displaymath}
with $r_g \in \mathbb{Z}$. Addition is given by addition of the [[coefficients]] $r_g$ and multiplication is given by the formula

\begin{displaymath}
\begin{aligned}
    r \cdot \tilde r 
    & = 
    \sum_{g \in G} \sum_{\tilde g \in G} (r_g \tilde r_{\tilde g}) g \cdot \tilde g
    \\
    & = 
    \sum_{q \in G}
    \left(
      \sum_{g \tilde g = q} r_g \tilde r_{\tilde g}
    \right)
    q
  \end{aligned}
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
For $A \in$ [[Ab]] an [[abelian group]] with underlying set $U(A)$, $G$-[[actions]] $\lambda \colon G \times U(A) \to U(A)$ such that for each element $g \in G$ the function $\lambda(g,-) \colon U(A) \to U(A)$ is an abelian group homomorphism are equivalently $\mathbb{Z}[G]$-[[module]] structures on $A$.

\end{prop}
\begin{proof}
Since the underlying abelian group of $\mathbb{Z}[G]$ is a [[free abelian group|free]] by definition, a [[bilinear map]] $\mathbb{Z}[G] \times A \to A$ is equivalently for each [[basis]] element $g \in G$ a [[linear map]] $A \to A$. Similarly the module property is determined on basis elements, where it reduces manifestly to the action property of $G$ on $U(A)$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
This reformulation of linear $G$-[[actions]] in terms of [[modules]] allows to treat $G$-actions in terms of [[homological algebra]]. See at \emph{\href{Ext#RelationToGroupCohomology}{Ext -- Relation to group cohomology}}.

\end{remark}
\hypertarget{more_examples}{}\subsubsection*{{more examples}}\label{more_examples}

\begin{itemize}%
\item In a [[symmetric monoidal category|symmetric monoidal]] [[category of chain complexes]] equipped with the [[tensor product of chain complexes]], then a [[monoid]] is a [[dg-algebra]], and a module is a \emph{[[dg-module]]}. See there for more.

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

The basic properties of categories of modules over [[monoid objects]] in [[symmetric 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}, and the references at [[modules over a monad]].

For the classical case of the [[symmetric monoidal category]] [[Ab]], a standard textbook is

\begin{itemize}%
\item F.W. Anderson, K.R. Fuller, \emph{Rings and Categories of Modules}, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York, (1992).

\end{itemize}
[[!redirects modules over a monoid]] [[!redirects modules over monoids]] [[!redirects modules over a monoid]]



\end{document}