Your favorite group is secretly equivalent to the space of loops in some topological space.

Both groups and loops are example of $\mathbb A_\infty\!$-groups: a floppier kind of group in which the group axioms hold only up to (coherent) homotopy. Adding in commutativity up to (coherent) homotopy leads to the notion of an $\mathbb E_\infty\!$-group, which plays a central role in modern algebraic topology. I'll explain how important invariants like cohomology groups secretly arise from $\mathbb E_\infty\!$-groups.