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\begin{document}

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\section*{cogroup}

\hypertarget{cogroup_objects}{}\section*{{Cogroup objects}}\label{cogroup_objects}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{relationship_to_group_objects}{Relationship To Group Objects}\dotfill \pageref*{relationship_to_group_objects} \linebreak
\noindent\hyperlink{relationship_to_other_objects}{Relationship to Other Objects}\dotfill \pageref*{relationship_to_other_objects} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\textbf{Cogroup objects} are [[group object]] in an [[opposite category]], and often one takes the [[opposite category]] of group object in an opposite category to be the category of cogroup objects.

The defining property of a cogroup object is that morphisms \emph{out} of it form a [[group]]. Specifically, if $C$ is a category, then $G$ is a cogroup object in $C$ if $\operatorname{Hom}(G,X)$ is a group for any object $X$ in $C$ (and the group structure must be natural in $X$).

There are many examples of cogroup objects. Perhaps the most well-known are the [[n-sphere]] in the [[homotopy category]] of [[pointed topological spaces]], $\operatorname{hTop}_*$. Then the fact that $S^n$ is a cogroup object in $\operatorname{hTop}$ is precisely the statement that the [[homotopy group]] $\pi_n(X)$ for $n \geq 1$ is indeed a [[group]], naturally in $X$, for all topological spaces $X$.

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

The basic definition is as follows.

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be a category. To give an object $G$ of $C$ a \textbf{cogroup structure} in $C$ is to give the functor $\operatorname{Hom}(G,-)$ a [[lift]] from $\operatorname{Set}$ to $\operatorname{Grp}$.

A \textbf{cogroup object} in $C$ is an object $G$ together with a choice of cogroup structure.

A \textbf{morphism of cogroup objects} $G_1 \to G_2$ is a morphism in $C$ between the underlying objects of the $G_i$ such that the [[natural transformation]] $\operatorname{Hom}(G_2,-) \to \operatorname{Hom}(G_1,-)$ lifts to a natural transformation of functors into $\operatorname{Grp}$.

\end{defn}
Thus cogroup objects and their morphisms can be thought of as the category of [[representable functors]] from $C$ to $\operatorname{Grp}$.

Providing $C$ has enough coproducts of $G$ (the $0,1,2,3$th copowers to be precise), the concept of a cogroup structure on $G$ can be internalised.

\begin{theorem}
\label{}\hypertarget{}{}
To give an object $G$ of $C$ a cogroup structure is equivalent to choosing morphisms $\mu \colon G \to G \amalg G$, $\eta \colon G \to 0_C$, and $\iota \colon G \to G$ satisfying the diagrams for associativity, unit, and inverse but \emph{the other way around}.

\end{theorem}
Here, the phrase ``the other way around'' means: take the normal diagrams for a [[group object]] that express the properties of associativity, unit, and inverses, invert all the arrows, and replace products by coproducts.

\hypertarget{relationship_to_group_objects}{}\subsection*{{Relationship To Group Objects}}\label{relationship_to_group_objects}

A cogroup object in a category, say $C$, is nothing more than a [[group object]] in the [[opposite category]]: $C^{op}$. However, morphisms in the cogroup category go the other way around. That is to say, with the obvious notation:

\begin{displaymath}
C\operatorname{coGrp} = (C^{op}\operatorname{Grp})^{op}
\end{displaymath}
\hypertarget{relationship_to_other_objects}{}\subsection*{{Relationship to Other Objects}}\label{relationship_to_other_objects}

Of course, there is nothing special about groups here. The same style of definition works for any [[variety of algebras]] in the sense of universal algebra, where $C coAlg_T \coloneqq (C^{op}Alg_T)^{op}$.

Some terminological care should be taken in the case of [[comonoid]], which makes sense in any [[monoidal category]], not just cocartesian monoidal categories which is the general default environment for discussing co-$T$-algebras. Thus, check with the author to see which monoidal product is meant; in the case of comonoid it's likely that it's \emph{not} the cocartesian notion that is intended. Whereas in the case of cogroups, confusion is not so likely: one needs the cocartesian structure (codiagonals, etc.) essentially because the axioms of a group involve duplication of variables, whereas this is not the case for axioms of a monoid. (Cf. the distinction between [[operad]] and [[Lawvere theory]], where the latter can be viewed as a kind of ``cartesian operad''.)

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

\begin{enumerate}%
\item All [[n-spheres]] for all $n$ are cogroup objects in the [[homotopy category]] of [[based topological spaces]], $\operatorname{hTop}_*$.

This is the origin of the [[group]] structure on [[homotopy groups]]. it is also crucial in the structure of the [[Brown representability theorem]].

The higher spheres are actually \emph{abelian} cogroup objects, as demonstrated by the fact that $\pi_n(X)$ is abelian for $n \ge 2$.


\item More generally, any [[suspension]] is a cogroup object with the ``pinch'' map as the comultiplication. See at \emph{[[suspensions are H-cogroup objects]]}. (Since the $0$-sphere is not a suspension in $\operatorname{hTop}_*$, but only in $\operatorname{hTop}$, it need not be a cogroup and in fact is not.) This is dual to, and equivalent to, the statement that (based) [[loop spaces]] are group objects in $\operatorname{hTop}_*$ since there is an [[adjunction]], internal to $\operatorname{hTop}_*$:

\begin{displaymath}
\operatorname{Hom}(\Sigma X,Y) \cong \operatorname{Hom}(X,\Omega Y)
\end{displaymath}

\item There are examples of spaces that are cogroups in $\operatorname{hTop}_*$ that are \textbf{not} suspensions, see Bernstein \& Harper \emph{Cogroups which are not suspensions}. Note that cogroups in $\operatorname{hTop}_*$ are the same as [[co-H-spaces]] which are additionally (co-)associative and have (co-)inverses.


\item Cogroup objects in the [[Grp|category of groups]] are [[free groups]], and to give a free group the structure of a cogroup object is the same a choosing a generating set. This is an old result of D.M. Kan's.


\item On the other hand, every abelian group is again an abelian cogroup since $\operatorname{Ab}$ is self-enriched. Indeed, in an [[abelian category]] every object is simultaneously an abelian group object and an abelian cogroup object. In $\operatorname{Ab}$, the abelian cogroup object structure is unique, with comultiplication given by the [[diagonal morphism]].


\item In [[Set]], the only cogroup object (abelian or otherwise) is the [[empty set]]. This is because the counit map must be a morphism from $X$ to the terminal object \emph{of the opposite category}. In the case of $\operatorname{Set}$, this is the empty set.


\item This extends further: any category with a [[faithful functor]] to $\operatorname{Set}$ which preserves an [[initial object]] will have no non-trivial cogroup objects. In particular, the category [[Top]] of \emph{unbased} topological spaces has only the [[empty space]] as a cogroup object.


\item The case of cogroups, and some other co-things, in certain other varieties of algebras has been extensively studied by Bergman and Hausknecht in \emph{Co-groups and co-rings in categories of associative rings}, (MR1387111)

In particular, a co-group in the category of (unital) commutative rings is a commutative [[Hopf ring]] and a cogroup in the category of (unital) commutative $k$-algebras is a commutative [[Hopf algebra|Hopf]] $k$-algebra; a fact highlighted in homotopy theory by [[Haynes Miller]] (in view of his generalization to [[commutative Hopf algebroid]]s as cogroupoids in commutative algebra) in the context of discussion of [[dual Steenrod algebras]], see (\hyperlink{Ravenel86}{Ravenel 86, appendix A}) for review.



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

Discussion of commutative [[Hopf algebras]] as cogroups is in

\begin{itemize}%
\item [[Doug Ravenel]], appendix A1 of \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, Academic Press 1986

\end{itemize}
Cogroups in the category of (graded or not) associative algebras are very rare (unlike Hopf algebras) -- in fact the underlying algebras are free; this has been clear since

\begin{itemize}%
\item Israel Berstein, On cogroups in the category of graded algebras. Trans. Amer. Math. Soc. 115 (1965), 257--269 \href{http://www.jstor.org/stable/1994268}{jstor}

\end{itemize}
This fact is later observed in bigger generality in

\begin{itemize}%
\item [[Benoit Fresse]], \emph{Cogroups in algebras over an operad are free algebras}, Commen. Math. Helv. \textbf{73}:4, 1998, 637--676 \href{http://dx.doi.org/10.1007/s000140050072}{doi}

\end{itemize}
[[!redirects cogroups]]

[[!redirects co-H-object]]

[[!redirects cogroup object]] [[!redirects cogroup objects]]

[[!redirects co-group object]] [[!redirects co-group]]



\end{document}