# Category of groups with central homomorphisms

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

The category of groups with central homomorphisms is a category 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 central homomorphisms A homomorphism of groups $\varphi:G \to H$ is termed a central homomorphism if $\varphi(Z(G)) \le Z(H)$.
composition of morphisms compose the homomorphisms as set maps Given central homomorphisms $\alpha:G \to H$ and $\beta: H \to K$, the composite is the homomorphism $\beta \circ \alpha: G \to K$.
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 ? An injective homomorphism where the image is a subgroup whose center is contained in the center of the whole group
epimorphism  ? A surjective homomorphism (the idea is to adapt the proof of epimorphism iff surjective in the category of groups by using a variant of the amalgamated free product that makes the homomorphism central)
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  ? We take the external free product with the natural inclusions, then quotient out by relations saying that the center of each of the factor groups centralizes all the other groups as well.