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\begin{document}

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\section*{group object}

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

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

[[!include category theory - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_internal_group_objects}{In terms of internal group objects}\dotfill \pageref*{in_terms_of_internal_group_objects} \linebreak
\noindent\hyperlink{InTermsOfPresheavesOfGroups}{In terms of presheaves of groups}\dotfill \pageref*{InTermsOfPresheavesOfGroups} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{theory}{Theory}\dotfill \pageref*{theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


A \emph{group object} in a [[category]] $C$ is a [[group]] [[internalization|internal]] to $C$.

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

\hypertarget{in_terms_of_internal_group_objects}{}\subsubsection*{{In terms of internal group objects}}\label{in_terms_of_internal_group_objects}

A \textbf{group object} or \textbf{internal group} in a category $C$ with binary [[product]]s and a [[terminal object]] $*$ is an object $G$ in $C$ and arrows

\begin{displaymath}
1:* \to G
\end{displaymath}
(the unit map)

\begin{displaymath}
(-)^{-1}:G\to G
\end{displaymath}
(the inverse map) and

\begin{displaymath}
m:G\times G \to G
\end{displaymath}
(the multiplication map), such that the following diagrams commute:

\begin{displaymath}
\itexarray{
G\times G\times G & \stackrel{id\times m}{\to} & G\times G\\
 m\times id\downarrow && \downarrow m \\
 G\times G & \stackrel{m}{\to} &G
}
\end{displaymath}
(expressing the fact multiplication is associative),

\begin{displaymath}
\itexarray{
G & \stackrel{(1,id)}{\to} & G\times G\\
 (\id,1)\downarrow &\underset{=}{\searrow}& \downarrow m \\
 G\times G & \stackrel{m}{\to} &G
}
\end{displaymath}
(telling us that the unit map picks out an element that is a left and right identity), and

\begin{displaymath}
\itexarray{
G & \stackrel{(id,(-)^{-1})\circ\Delta}{\to} & G\times G\\
 ((-)^{-1},id)\circ\Delta\downarrow & \underset{1}{\searrow}& \downarrow m \\
 G\times G & \stackrel{m}{\to} &G
}
\end{displaymath}
(telling us that the inverse map really does take an inverse), where we have let $1: G \to G$ denote the composite $G \to * \stackrel{1}{\to} G$ and $\Delta$ is a [[diagonal morphism]].

Even if $C$ doesn't have \emph{all} binary products, as long as products with $G$ (and the terminal object $*$) exist, then one can still speak of a group object $G$ in $C$.

\hypertarget{InTermsOfPresheavesOfGroups}{}\subsubsection*{{In terms of presheaves of groups}}\label{InTermsOfPresheavesOfGroups}

\begin{prop}
\label{}\hypertarget{}{}
Given a [[cartesian monoidal category]] $C$, the category of internal groups in $C$ is equivalent to the [[full subcategory]] of the category of [[presheaves]] of [[groups]] $Grp^{C^{op}}$ on $C$, spanned by those presheaves whose underlying set part in $Set^{C^{op}}$ is [[representable functor|representable]].

\end{prop}
This is a special case of the general theory of \emph{[[structures in presheaf toposes]]}.

In other words, the [[forgetful functor]] from $Grp^{C^{op}}$ to $Set^{C^{op}}$ (obtained by composing with the [[forgetful functor]] [[Grp]] $\to$ [[Set]]) creates representable group objects from representable objects.

An object $G$ in $C$ with an internal group structure is a diagram

\begin{displaymath}
\itexarray{
    && Grp
    \\
    & {}^{(G,\cdot)}\nearrow & \downarrow
    \\
    C^{op} &\stackrel{Y(G)}{\to}& Set
  }
  \,.
\end{displaymath}
This equips each object $S \in C$ with an ordinary group $(G(S), \cdot)$ structure, so in particular a product operation

\begin{displaymath}
\cdot_S : G(S) \times G(S) \to G(S)
  \,.
\end{displaymath}
Moreover, since morphisms in $Grp$ are group homomorphisms, it follows that for every morphism $f : S \to T$ we get a commuting diagram

\begin{displaymath}
\itexarray{
    G(S) \times G(S) &\stackrel{\cdot_S}{\to}& G(S)
    \\
    \uparrow^{G(f)\times G(f)}  && \uparrow^{G(f)}
    \\
    G(T) \times G(T) &\stackrel{\cdot_T}{\to}& G(T)     
  }
\end{displaymath}
Taken together this means that there is a morphism

\begin{displaymath}
Y(G \times G)  \to Y(G)
\end{displaymath}
of representable presheaves. By the [[Yoneda lemma]], this uniquely comes from a morphism $\cdot : G \times G \to G$, which is the product of the group structure on the object $G$ that we are after.

etc.

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

\begin{itemize}%
\item A group object in [[Set]] is a [[group]].
\item A group object in [[Top]] is a [[topological group]].
\item A group object in [[Ho(Top)]] is an [[H-group]].
\item A group object in [[Diff]] is a [[Lie group]].
\item A group object in [[SDiff]] is a [[supergroup|super Lie group]].
\item A group object in [[Grp]] is an [[abelian group]] (using the [[Eckmann-Hilton argument]]).
\item A group object in [[Ab]] is an abelian group again.
\item A group object in [[Cat]] is a strict [[2-group]].
\item A group object in [[Grpd]] is a strict $2$-group again.
\item A group object in [[CRing]]$^{op}$ is a commutative [[Hopf algebra]].
\item A group object in a [[functor category]] is a [[group functor]].
\item A group object in [[schemes]] is a [[group scheme]].
\item A group object in an [[opposite category]] is a [[cogroup object]].
\item A group object in [[stacks]] is a [[group stack]].

\end{itemize}
\hypertarget{theory}{}\subsection*{{Theory}}\label{theory}

The basic results of elementary group theory apply to group objects in any category with finite products. (Arguably, it is precisely the \emph{elementary} results that apply in any such category.)

The theory of group objects is an example of a [[Lawvere theory]].

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

\begin{itemize}%
\item [[monoid]], [[monoid object]], [[monoid object in an (∞,1)-category]]


\item [[group]], \textbf{group object}, [[group object in an (∞,1)-category]]


\item [[groupoid]], [[groupoid object]], [[groupoid object in an (∞,1)-category]]


\item [[infinity-groupoid]], [[infinity-groupoid object]], [[groupoid object in an (∞,1)-category]]


\item [[ring]], [[ring object]]



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

\begin{itemize}%
\item [[Saunders MacLane]], chapter III, section 6 in \emph{[[Categories for the Working Mathematician]]}, Springer (1971)

\end{itemize}
[[!redirects group objects]]

[[!redirects internal group]] [[!redirects internal groups]] [[!redirects inner group]] [[!redirects inner groups]] [[!redirects group object]] [[!redirects group objects]]

[[!redirects abelian group object]] [[!redirects abelian group objects]]



\end{document}