Rigid Algebras and Cospans

We introduce rigid algebras, a generalization of rigid categories to arbitrary symmetric monoidal $(\mathrm{\infty },2)$-categories. We develop their general theory, showing in particular that the a priori $(\mathrm{\infty },2)$-category of rigid algebras is in fact an $(\mathrm{\infty },1)$-category. For the $(\mathrm{\infty },2)$-category of cospans in an $(\mathrm{\infty },1)$-category $\mathcal{C}$, we show that the $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },2)$-category its $(\mathrm{\infty },1)$-category of rigid commutative algebras.