Group object - Knowlpedia

Definition

Let $\mathcal C$ be a category with finite products and terminal object $1$ . A group object in $\mathcal C$ is an object $G$ with morphisms

m:G\times G\to G,\qquad e:1\to G,\qquad i:G\to G

that satisfy the associativity, identity, and inverse diagrams obtained by writing the ordinary group axioms using products, the diagonal $G\to G\times G$ , and the unique map $G\to1$ .

Generalized elements

For every object $T$ , composition with $m,e,i$ makes

\operatorname{Hom}_{\mathcal C}(T,G)

a group, naturally in $T$ . This is a reliable way to read the internal axioms, although ordinary elements alone may not detect all morphisms in every category.

Examples

Group objects in sets are ordinary groups.
Group objects in topological spaces are topological groups.
Group objects in smooth manifolds are Lie groups.
Group objects in schemes are group schemes.

References

Francis Borceux, Handbook of Categorical Algebra 2: Categories and Structures, Cambridge University Press, 1994. DOI record. Relevant: internal algebraic structures.