Rigid algebras and cospans

Abstract We introduce rigid algebras, a generalization of rigid categories to arbitrary symmetric monoidal $(\infty ,2)$-categories. We develop their general theory, showing in particular that the a priori $(\infty ,2)$-category of rigid algebras is in fact an $(\infty ,1)$-category. For the $(\infty ,2)$-category of cospans in an $(\infty ,1)$-category $\mathcal{C}$, we show that the $(\infty ,1)$-category of rigid commutative algebras is canonically identified with $\mathcal{C}$. This identification is used to construct an adjunction between the cospan construction and the functor assigning to a symmetric monoidal $(\infty ,2)$-category its $(\infty ,1)$-category of rigid commutative algebras.