\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Beck module}

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

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

[[!include category theory - contents]]

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

[[!include higher algebra - 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{derivations}{Derivations}\dotfill \pageref*{derivations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{OverAssociativeAlgebras}{Beck modules over associative algebras}\dotfill \pageref*{OverAssociativeAlgebras} \linebreak
\noindent\hyperlink{beck_modules_over_groups}{Beck modules over groups}\dotfill \pageref*{beck_modules_over_groups} \linebreak
\noindent\hyperlink{the_tangent_category}{The tangent category}\dotfill \pageref*{the_tangent_category} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

One usually defines [[cohomology]] with respect to some [[coefficient]] objects:

\begin{itemize}%
\item For [[group cohomology]] of a [[group]] $G$, the coefficients come from a (left) $G$-[[module]].
\item For [[Lie algebra cohomology]] of a [[Lie algebra]] $\mathfrak{g}$, the coefficients come from a (left) $\mathfrak{g}$-module.
\item For [[Hochschild cohomology]] of an [[associative algebra]] $A$, the coefficients come from an $A$-[[bimodule]].

\end{itemize}
\textbf{Beck modules} are a simultaneous generalisation of all three types of module.

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

Let $\mathcal{C}$ be a [[category]] with [[pullback]]s and let $A$ be an object in $\mathcal{C}$. A \textbf{Beck module} over $A$ is an [[abelian group object]] in the slice category $\mathcal{C}_{/ A}$. In particular, if $A$ is the terminal object, this reduces to the notion of an abelian group object in $\mathcal{C}$. We write $\mathbf{Ab}(\mathcal{C}_{/ A})$ for the category of Beck modules over $A$.

(\hyperlink{Beck67}{Beck 67, def. 5})

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

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be an [[effective regular category]] (resp. [[locally presentable category]]) and let $A$ be an object in $\mathcal{C}$. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is an [[abelian category]] (resp. locally presentable category).

\end{prop}
\begin{proof}
If $\mathcal{C}$ is a effective regular category (resp. locally presentable category), then so is $\mathcal{C}_{/ A}$. Thus, the claim reduces to the fact that the category of abelian group objects in an effective regular category (resp. locally presentable category) is an abelian category (resp. locally presentable category).

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be an effective regular category with [[filtered colimits]] and let $A$ be an object in $\mathcal{C}$. If filtered colimits in $\mathcal{C}$ preserve finite [[limits]], then $\mathbf{Ab}(\mathcal{C}_{/ A})$ (is an abelian category and) satisfies axiom AB5.

\end{prop}
\begin{proof}
The forgetful functor $\mathcal{C}_{/ A} \to \mathcal{C}$ [[created limit|creates]] pullbacks and filtered colimits, so filtered colimits in $\mathcal{C}_{/ A}$ also preserve finite limits. The forgetful functor $\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ creates limits and filtered colimits, so filtered colimits in $\mathbf{Ab}(\mathcal{C}_{/ A})$ preserve kernels. In view of the earlier proposition, it follows that $\mathbf{Ab}(\mathcal{C}_{/ A})$ satisfies axiom AB5.

\end{proof}
\begin{cor}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a locally finitely presentable effective regular category and let $A$ be an object in $\mathcal{C}$. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is a [[Grothendieck category]].

\end{cor}
\begin{proof}
Combine the two propositions above.

\end{proof}
\hypertarget{derivations}{}\subsection*{{Derivations}}\label{derivations}

Let $\mathcal{C}$ be a category with pullbacks, let $A$ be an object in $\mathcal{C}$, and let $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ be the forgetful functor. Given a Beck module $M$ over $A$, an $M$-valued \textbf{derivation} of $A$ is a morphism $1_A \to U M$ in $\mathcal{C}_{/ A}$, where $1_A$ is the terminal object in $\mathcal{C}_{/ A}$, and we write

\begin{displaymath}
Der (A, M) = \mathcal{C}_{/ A} (1_A, U M)
\end{displaymath}
for the set of $M$-valued derivations of $A$. The \textbf{Beck module of differentials} over $A$ is an object $\Omega_A$ in $\mathbf{Ab}(\mathcal{C}_{/ A})$ [[representable functor|representing]] the functor $Der (A, -) : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathbf{Set}$.

The Beck module $\Omega_A$ is not guaranteed to exist in general. When the functor $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ has a left adjoint, $\Omega_A$ is simply the value of the left adjoint at $1_A$.

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a locally presentable category and let $A$ be an object in $\mathcal{C}$. Then the forgetful functor $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ has a left adjoint.

\end{prop}
\begin{proof}
The forgetful functor $\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ creates limits and $\kappa$-filtered colimits (for some $\kappa$ large enough), so we may apply the accessible [[adjoint functor theorem]].

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

\hypertarget{OverAssociativeAlgebras}{}\subsubsection*{{Beck modules over associative algebras}}\label{OverAssociativeAlgebras}

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be the category of (not necessarily commutative) [[rings]] and let $A$ be a ring. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is equivalent to the category of $A$-bimodules.

\end{prop}
\begin{proof}
Let $\epsilon : B \to A$ be ring homomorphism. To give it the structure of a Beck module over $A$, we must give ring homomorphisms $\eta : A \to B$ and $\mu : B \times_A B \to B$ such that $\epsilon \circ \eta = id_A$, $\epsilon (\mu (b_0, b_1)) = \epsilon (b_0) = \epsilon (b_1)$, as well as various other equations. Given elements $b_0, b_1, b_2, b_3$ of $B$ such that $\epsilon (b_0) = \epsilon (b_2)$ and $\epsilon (b_1) = \epsilon (b_3)$, we have the following interchange law:

\begin{displaymath}
\mu (b_0 + b_1, b_2 + b_3) = \mu (b_0, b_2) + \mu (b_1, b_3)
\end{displaymath}
Hence, if $a = \epsilon (b_1) = \epsilon (b_2)$,

\begin{displaymath}
\mu (b_1, b_2) = \mu (\eta (a) + b_1 - \eta (a), \eta (a) + b_2 - \eta (a)) = \mu (\eta (a), \eta (a)) + \mu (b_1 - \eta(a), b_2 - \eta (a)) = \eta (a) + \mu (b_1 - \eta(a), b_2 - \eta (a))
\end{displaymath}
but $\epsilon (b_1 - \eta (a)) = \epsilon (b_2 - \eta (a)) = 0$, so

\begin{displaymath}
\mu (b_1 - \eta(a), b_2 - \eta(a)) = \mu (0, b_2 - \eta (a)) + \mu (b_1 - \eta (a), 0) = b_2 - \eta (a) + b_1 - \eta (a)
\end{displaymath}
and we conclude that

\begin{displaymath}
\mu (b_1, b_2) = b_1 - \eta (\epsilon (b_1)) + b_2 = b_1 + b_2 - \eta (\epsilon (b_2))
\end{displaymath}
and in particular, $\mu$ is entirely determined by $\eta$ and $\epsilon$. We also have the following interchange law,

\begin{displaymath}
\mu (b_0 b_1, b_2 b_3) = \mu (b_0, b_2) \mu (b_1, b_3)
\end{displaymath}
and in particular,

\begin{displaymath}
\mu (\eta (\epsilon (b_2)) b_1, b_2 \eta (\epsilon (b_1))) = \mu (\eta (\epsilon (b_2)), b_2) \mu (b_1, \eta (\epsilon (b_1))) = b_2 b_1
\end{displaymath}
hence,

\begin{displaymath}
b_2 b_1 = \eta (\epsilon (b_2)) b_1 - \eta (\epsilon (b_2 b_1)) + b_2 \eta (\epsilon (b_1))
\end{displaymath}
so if $\epsilon (b_1) = \epsilon (b_2) = 0$, then $b_2 b_1 = 0$.

Let $M = \ker \epsilon$. The above shows that the internal abelian group structure on $B$ restricts to the pre-existing abelian group structure on $\ker \epsilon$. In addition, the homomorphism $\eta : A \to B$ gives $M$ the structure of an $A$-bimodule, and we see that $B$ is naturally isomorphic to the [[square-0 extension]] ring $A \oplus M$, with componentwise addition and the multiplication given below,

\begin{displaymath}
(a_0, m_0) \cdot (a_1, m_1) = (a_0 a_1, a_0 m_1 + m_0 a_1)
\end{displaymath}
regarded as a Beck module over $A$ by defining $\epsilon : A \oplus M \to A$, $\eta : A \to A \oplus M$, and $\mu : (A \oplus M) \times_A (A \oplus M) \to A \oplus M$ as follows:

\begin{displaymath}
\epsilon (a, m) = a
\end{displaymath}
\begin{displaymath}
\eta (a) = (a, 0)
\end{displaymath}
\begin{displaymath}
\mu ((a, m_0), (a, m_1)) = (a, m_0 + m_1)
\end{displaymath}
Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ A})$ and the category of $A$-bimodules, as claimed.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Let $A$ be a ring. Then the Beck module $\Omega_A$ is isomorphic to the $A$-bimodule of [[Kähler differentials]] (relative to $\mathbb{Z}$).

\end{prop}
\begin{proof}
Let $M$ be an $A$-bimodule, regard $A \oplus M$ as a ring as above, and let $\epsilon : A \oplus M \to A$ be the obvious projection. A ring homomorphism $\phi : A \to A \oplus M$ satisfying $\epsilon \circ \phi = id_A$ is the same thing as an additive homomorphism $\delta : A \to M$ satisfying the following equations,

\begin{displaymath}
\delta (a_0 a_1) = \delta (a_0) a_1 + a_0 \delta (a_1)
\end{displaymath}
i.e. a derivation $A \to M$ (over $\mathbb{Z}$). Thus, the Beck module $\Omega_A$ has the same universal property as the $A$-bimodule of K\"a{}hler differentials.

\end{proof}
\hypertarget{beck_modules_over_groups}{}\subsubsection*{{Beck modules over groups}}\label{beck_modules_over_groups}

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be the category of (not necessarily abelian) [[groups]] and let $G$ be a group. Then $\mathbf{Ab}(\mathcal{C}_{/ G})$ is equivalent to the category of left $G$-modules.

\end{prop}
\begin{proof}
Let $\epsilon : H \to G$ be group homomorphism. To give it the structure of a Beck module over $G$, we must give group homomorphisms $\eta : G \to H$ and $\mu : H \times_G H \to H$ such that $\epsilon \circ \eta = id_A$, $\epsilon (\mu (h_0, h_1)) = \epsilon (h_0) = \epsilon (h_1)$, as well as various other equations. Given elements $h_0, h_1, h_2, h_3$ of $H$ such that $\epsilon (h_0) = \epsilon (h_2)$ and $\epsilon (h_1) = \epsilon (h_3)$, we have the following interchange law:

\begin{displaymath}
\mu (h_0 h_1, h_2 h_3) = \mu (h_0, h_2) \mu (h_1, h_3)
\end{displaymath}
and in particular,

\begin{displaymath}
\mu (\eta (\epsilon (h_2)) h_1, h_2 \eta (\epsilon (h_1))) = \mu (\eta (\epsilon (h_2)), h_2) \mu (h_1, \eta (\epsilon (h_1))) = h_2 h_1
\end{displaymath}
but on the other hand, if $g = \epsilon (h_1) = \epsilon (h_2)$, then

\begin{displaymath}
\mu (h_1, h_2) = \mu (\eta (g) \eta (g)^{-1} h_1, \eta (g) \eta (g)^{-1} h_2) = \mu (\eta (g), \eta (g)) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \eta (g) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2)
\end{displaymath}
and writing $e$ for the unit of $G$ and $H$, we have $\epsilon (\eta (g)^{-1} h_1) = \epsilon (\eta (g)^{-1} h_2) = e$, hence

\begin{displaymath}
\mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \mu (e, \eta (g)^{-1} h_2) \mu (\eta (g)^{-1} h_1, e) = \eta (g)^{-1} h_2 \eta (g)^{-1} h_1
\end{displaymath}
so we conclude that

\begin{displaymath}
\mu (h_1, h_2) = h_2 \eta (g)^{-1} h_1
\end{displaymath}
and in particular, $\mu$ is entirely determined by $\eta$.

Let $M = \ker \epsilon$. The above shows that the internal abelian group structure on $B$ restricts to the pre-existing group structure on $\ker \epsilon$. (In particular, $M$ is an abelian group!) We make $M$ into a left $G$-module as follows:

\begin{displaymath}
g \cdot m = \eta (g) m \eta (g)^{-1}
\end{displaymath}
We can then construct the [[semi-direct product]] $M \rtimes G$, which has the following multplication:

\begin{displaymath}
(m_0, g_0) \cdot (m_1, g_1) = (m_0 \eta (g_0) m_1 \eta (g_0)^{-1}, g_0 g_1)
\end{displaymath}
There is a group homomorphism $M \rtimes G \to H$ defined by $(m, g) \mapsto m \eta (g)$, and it is bijective: surjectivity is clear, and injectivity is a consequence of the fact that $M \cap \operatorname{im} \eta = \{ e \}$. We may regard $M \rtimes G$ as a Beck module over $G$ by defining $\epsilon : M \rtimes G \to G$, $\eta : G \to M \rtimes G$, and $\mu : (M \rtimes G) \times_G (M \rtimes G) \to M \rtimes G$ as follows:

\begin{displaymath}
\epsilon (m, g) = g
\end{displaymath}
\begin{displaymath}
\eta (g) = (0, g)
\end{displaymath}
\begin{displaymath}
\mu ((m_0, g), (m_1, g)) = (m_1 m_0, g)
\end{displaymath}
Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ G})$ and the category of left $G$-modules, as claimed.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Let $G$ be a group and let $M$ be a left $G$-module. Under the above identification of Beck modules over $G$ with left $G$-modules, $M$-valued derivations of $G$ are precisely crossed homomorphisms $G \to M$, i.e. maps $\delta : G \to M$ satisfying the following equation:

\begin{displaymath}
\delta (g_0 g_1) = \delta (g_0) + g_0 \cdot \delta (g_1)
\end{displaymath}
\end{prop}
\begin{proof}
Let $\epsilon : M \rtimes G \to G$ be the evident projection. A group homomorphism $\phi : G \to M \rtimes G$ such that $\epsilon \circ \phi = id_G$ is the same thing as a map $\delta : G \to M$ satisfying the equation below,

\begin{displaymath}
(\delta (g_0), g_0) \cdot (\delta (g_1), g_1) = (\delta (g_0 g_1), g_0 g_1)
\end{displaymath}
which is equivalent to the defining equation for crossed homomorphisms.

\end{proof}
\hypertarget{the_tangent_category}{}\subsection*{{The tangent category}}\label{the_tangent_category}

One may assemble the individual categories of Beck modules over the objects of $\mathcal{C}$ into a category fibred over $\mathcal{C}$, called the [[tangent category]].

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

The concept is due to

\begin{itemize}%
\item [[Jon Beck]], \emph{Triples, algebras and cohomology}, Ph.D. thesis, Columbia University, 1967, Reprints in Theory and Applications of Categories, No. 2 (2003) pp 1-59 (\href{http://www.tac.mta.ca/tac/reprints/articles/2/tr2abs.html}{TAC})

\end{itemize}
and was popularized in

\begin{itemize}%
\item [[Daniel G. Quillen]], \emph{On the (co-)homology of commutative rings}, in Proc. Symp. on Categorical Algebra, 65 -- 87, American Math. Soc., 1970.

\end{itemize}
See also

\begin{itemize}%
\item [[Michael Barr]], \emph{Acyclic models}, Chapter 6, \S{}1.

\end{itemize}
An application to [[knot theory]] is given in

\begin{itemize}%
\item [[Markus Szymik]], \emph{Alexander-Beck modules detect the unknot}, Fund. Math. 246 (2019) 89-108.

\end{itemize}
[[!redirects Beck modules]]



\end{document}