2-Groups

For a brief review of Group Object internal to a category \(C\) with binary products and terminal object \(1\in C\) consists of an object \(G\in C\) equipped with

• a unit \(e\colon 1\to G\)
• an inverse mapping \((-)^{-1}\colon G\to G\)
• a multiplication map \(m\colon G\times G\to G\)

such that:

• multiplication is associative

\begin{equation} \require{AMScd} \begin{CD} G\times G\times G @>e\times m>> G\times G\\ @Vm\times eVV @VVmV \\ G\times G @>m>> G \end{CD} \end{equation}

• the unit map picks out an element which is a left and right identity:

\begin{equation} \begin{CD} G\times 1 @<{!}<< G @>{!}>> 1\times G\\ @V{\mathrm{id}\times e}VV @V\mathrm{id}VV @VV{e\times\mathrm{id}}V \\ G\times G @>{m}>> G @<{m}<< G\times G \end{CD} \end{equation}

• the inverse map is an honest-to-goodness inverse

\begin{equation} \begin{CD} G\times G @<{\Delta}<< G @>{\Delta}>> 1\times G\\ @V{\mathrm{id}\times(-)^{-1}}VV @V{e\circ!}VV @VV{(-)^{-1}\times\mathrm{id}}V \\ G\times G @>{m}>> G @<{m}<< G\times G \end{CD} \end{equation}

where we recall \({e\circ!}\colon G\to G\) is the composition of \(!\colon G\to 1\) with \(e\colon 1\to G\).