Algebraic structures such as monoids, groups and categories can be defined inside any category with finite products by replacing element-wise axioms with commutative diagrams over morphisms. Learners see how these internal definitions specialise — group objects in Set are ordinary groups, in Top topological groups, in the category of smooth manifolds Lie groups, and in Grp exactly abelian groups (forced by the interchange law) — and how the construction dualises to cogroup objects. The internal-category construction over pullbacks then yields the Brown-Spencer equivalence: internal categories in Grp correspond to group objects in Cat and to crossed modules of groups.
An expository essay by Magnus Forrester-Barker covering how algebraic structures (monoids, groups, categories) can be formulated within an arbitrary category using commutative diagrams, rather than v…