Category of groups

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This article describes a category (in the mathematical sense) where the notion of "object" is groupand the notion of morphism is homomorphism of groups. In other words, it gives a category structure to the collection of all groups.
View other category structures on groups

This article describes a way of viewing the collection of groups as a structure in its own right

Definition

The category of groups is defined as follows:

Aspect Name Definition/description
objects groups A group is a set with associative binary operation admitting an identity element and inverse map.
morphisms homomorphisms of groups A homomorphism between groups $G$ and $H$ is a set map $\varphi:G \to H$ such that $\varphi(xy) = \varphi(x)\varphi(y)$ for all $x,y \in G$. Note that this also forces that it preserves the identity element and the inverse map, and some definitions include these additional (redundant) conditions.
composition of morphisms compose the homomorphisms as set maps Given homomorphisms $\alpha:G \to H$ and $\beta:H \to K$, the composite is the set composition $\beta \circ \alpha$, a homomorphism $G \to K$.
identity morphism identity map from a group to itself. For a group $G$, the identity map is a map $\operatorname{id}_G: G \to G$ given by $\operatorname{id}_G(x) = x$ for all $x \in G$.

Categorical constants and constructs

Construct Name in this category Definition/description
isomorphism isomorphism of groups A bijective homomorphism; equivalently, a homomorphism whose inverse map is also a homomorphism.
monomorphism injective homomorphism The kernel of the mapping is trivial. Alternatively, it can be identified with a subgroup inclusion mapping. This is relatively straightforward to prove; see monomorphism iff injective in the category of groups
epimorphism surjective homomorphism The mapping is surjective. This is not immediately obvious, but relies on the amalgamated free product construction. For more, see epimorphism iff surjective in the category of groups.
zero object trivial group The group with one element, namely its identity element.
categorical product the usual external direct product We take the external direct product with the coordinate-wise projection maps.
categorical coproduct the usual external free product We take the external free product with the natural inclusions.

Important functors

Target category Description of functor Important facts about this functor
category of sets send each group to its underlying set and send each homomorphism to its underlying set map. This functor is often called the forgetful functor to Sets because it involves forgetting the group structure. This is a faithful functor but not a full functor, and it gives the category of groups the structure of a concrete category.
category of monoids send each group to its underlying monoid (i.e., forget the inverse operation) and send each homomorphism to its underlying monoid homomorphism. This functor is also often called a forgetful functor because it involves forgetting part of the group structure. This functor is faithful and full (which means that the monoidal homomorphisms between groups are precisely the same as the group homomorphisms between them). For more, see groups form a full subcategory of monoids.
category of semigroups send each group to its underlying semigroup (i.e., retain only the binary operation; forget the identity element and inverse operation) and send each homomorphism to its underlying semigroup homomorphism. This functor is also often called a forgetful functor because it involves forgetting part of the group structure. This functor is faithful and full (which means that the semigroup homomorphisms between groups are precisely the same as the group homomorphisms between them). For more, see groups form a full subcategory of semigroups.