The category of groups has groups as objects and group homomorphisms as morphisms. Its product is the direct product of groups, and its coproduct is the free product, as expressed by the universal property of a free product.
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In mathematics, particularly in the field of abstract algebra and category theory, a **category of groups** is a concept that arises from the framework of category theory, which is a branch of mathematics that deals with objects and morphisms (arrows) between them. ### Basic Definitions 1. **Category**: A category consists of: - A collection of objects. - A collection of morphisms (arrows) between those objects, which can be thought of as structure-preserving functions.