# Group object

of a category

An object $X$ of a category $C$ such that for any $Y \in \mathop{\rm Ob} ( C)$ the set of morphisms $\mathop{\rm Hom} _ {C} ( Y, X)$ is a group, while the correspondence $Y \rightarrow \mathop{\rm Hom} _ {C} ( Y, X)$ is a functor from $C$ into the category of groups Gr. A homomorphism of a group object $X$ into a group object $X _ {1}$ is a morphism $f: X \rightarrow X _ {1}$ of $C$ such that for any $Y \in \mathop{\rm Ob} ( C)$ the corresponding mapping $\mathop{\rm Hom} _ {C} ( Y, X) \rightarrow \mathop{\rm Hom} _ {C} ( Y, X _ {1} )$ is a homomorphism of groups. The group objects of a category $C$ and homomorphisms between them form the category $\mathop{\rm Gr} - C$. The functor $X \rightarrow h _ {X} = \mathop{\rm Hom} _ {C} ( \cdot , X)$ establishes an equivalence between the category $\mathop{\rm Gr} - C$ and the category of representable pre-sheaves of groups on $C$. If the values of the functor $h _ {X} : Y \rightarrow \mathop{\rm Hom} _ {C} ( Y, X)$ belong to the subcategory Ab of Abelian groups, then the group object $X$ is said to be commutative or Abelian. If $C$ has finite products and a final object $e$, a group object $X$ of $C$ is defined by the following properties.

There exist morphisms $m: X \times X \rightarrow X$( multiplication), $r: X \rightarrow X$( inversion) and $\beta : e \rightarrow X$( a unit) satisfying the following axioms.

Associativity. The diagram

$$\begin{array}{rcr} X \times X \times X & \mathop \rightarrow \limits ^ { {m \times id }} X \times X \rightarrow ^ { m } & X \\ {} _ {id \times m } \searrow &{} &\nearrow _ {id \times m } \\ {} &X \times X &{} \\ \end{array}$$

is commutative.

Existence of a unit element. The diagram

$$\begin{array}{rcr} X \times X & \rightarrow ^ { {p _ x} \times id } \ e \times X \mathop \rightarrow \limits ^ { {\beta \times id }} &X \times X \\ {} _ \Delta \nwarrow &{} &\swarrow _ {m} \\ {} &X \ \rightarrow _ { id } X &{} \\ \end{array}$$

is commutative.

Existence of an inverse element. The diagram

$$\begin{array}{rcr} X \times X & \mathop \rightarrow \limits ^ { {r \times id }} X \times X \rightarrow ^ { m } & X \\ {} _ \Delta \nwarrow &{} &\nearrow _ \beta \\ {} &X \mathop \rightarrow \limits _ { { p _ {x} }} X &{} \\ \end{array}$$

is commutative. Here $p _ {X} : X \rightarrow e$ is the canonical morphism of $X$ into the final object $e$, while $\Delta : X \rightarrow X \times X$ is the diagonal morphism.

If $C$ is the category of sets Ens, group objects are precisely groups. The final object of the category Ens is the set $\{ e \}$ consisting of the single element $e$. Axiom a) denotes the associativity of the binary operation given by the morphism $m: X \times X \rightarrow X$. The morphism $r: X \rightarrow X$ is the mapping of inversion, while the morphism $\beta : \{ e \} \rightarrow X$ is the mapping of the set $\{ e \}$ into $X$, whose image is equal to the unit element in $X$.

In a similar manner it is possible to define a ring object of a category and, generally, to specify an algebraic structure on an object of a category .

How to Cite This Entry:
Group object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_object&oldid=47144
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article