Category of groups

This article describes a category

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:

• The objects of this category are groups
• The morphisms of this category are group homomorphisms.
• The identity map is designated as the identity morphism.
• Composition of morphisms is defined by function composition.

Categorical constants and constructs

Zero object

The zero object in this category is the: trivial group

The category of groups has a zero object, viz an object that is both an initial object and a terminal object. This is the trivial group. In other words, given any group, there is a unique map from the trivial group to that group, and a unique map from that group to the trivial group.

Product in this category

The notion of product in this category is: external direct product

The category of groups has a well-defined notion of products. The product in the category of groups is the external direct product of groups.

Coproduct in this category

The notion of coproduct in this category is: external free product

The category of groups has a well-defined notion of coproducts: the external free product.

Important functors

For lists of important functors involving the category of groups, refer the following:

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Forgetful functor to sets

This functor takes a group and outputs the underlying set of the group. The forgetful functor is a faithful functor, because any group homomorphism is completely determined by the corresponding map at the set-theoretic level. Thus, the category of groups is a concrete category. This in particular shows that the category of groups is a locally small category.