# Category of groups

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.