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 [2].

References

 [1] Yu.I. Manin, "The theory of commutative formal groups over fields of finite characteristic" Russian Math. Surveys , 18 (1963) pp. 1–80 Uspekhi Mat. Nauk , 18 : 6 (1963) pp. 3–90 [2] M. Demazure, A. Grothendieck, "Schémas en groupes I" , Lect. notes in math. , 151–153 , Springer (1970)

Group objects, in particular categories, are often objects of interest in their own right. For example, topological groups (cf. Topological group) are group objects in the category of topological spaces and continuous mappings; Lie groups (cf. Lie group) are group objects in the category of smooth manifolds; and sheaves of groups on a given space $X$ are group objects in the category of sheaves of sets on $X$. A group object in a category of the form $\mathop{\rm Gr} - C$ is an object of $C$ equipped with two commuting group structures; it is easily seen that in this case the two structures must coincide and be Abelian, and conversely an Abelian group structure commutes with itself, so that $\mathop{\rm Gr} - \mathop{\rm Gr} - C$ is isomorphic to the category $\mathop{\rm Ab} - C$ of Abelian group objects in $C$. A functor which preserves finite products (including the final object) preserves group objects; using this and the above identification, one obtains an easy proof of the result that the fundamental group of a topological group is Abelian.