Free rigid commutative algebras

We describe free rigid commutative algebras in $2$-presentably symmetric monoidal $(\mathrm{\infty },2)$-categories as oplax colimits over the $1$-dimensional framed cobordism category. The special case of the $(\mathrm{\infty },2)$-category ${\mathrm{P}\mathrm{r}}^{\mathrm{L}}$ itself provides a description of the free symmetric monoidal $(\mathrm{\infty },1)$-category with duals on a given $(\mathrm{\infty },1)$-category, while the case of ${\mathrm{M}\mathrm{o}\mathrm{d}}_{\mathcal{V}}({\mathrm{P}\mathrm{r}}^{\mathrm{L}})$ provides a description of a similar object in the $\mathcal{V}$-enriched context, for $\mathcal{V}$ a presentably symmetric monoidal $(\mathrm{\infty },1)$-category. As a byproduct, we obtain new proofs of some results about rigidification of locally rigid categories, as well as a proof that any rigid category over $\mathrm{S}\mathrm{p}$ embeds into a compactly-rigidly generated one.