Free rigid commutative algebras

TL;DR

This paper characterizes free rigid commutative algebras in advanced categorical contexts using oplax colimits, providing new insights and proofs in higher category theory and rigidity concepts.

Contribution

It offers a novel description of free rigid commutative algebras in $( olinebreak ext{infinity,}2)$-categories via oplax colimits, extending understanding of rigidity and embeddings.

Findings

Describes free rigid commutative algebras as oplax colimits over the framed cobordism category.

Provides new proofs of rigidity results and embedding theorems for categories over spectra.

Connects the theory to enriched and symmetric monoidal $( ext{infinity,}1)$-categories.

Abstract

We describe free rigid commutative algebras in $2$-presentably symmetric monoidal $(\infty,2)$-categories as oplax colimits over the $1$-dimensional framed cobordism category. The special case of the $(\infty,2)$-category $\mathrm{Pr}^\mathrm{L}$ itself provides a description of the free symmetric monoidal $(\infty,1)$-category with duals on a given $(\infty,1)$-category, while the case of $\mathrm{Mod}_{\mathcal{V}}(\mathrm{Pr}^\mathrm{L})$ provides a description of a similar object in the $\mathcal{V}$-enriched context, for $\mathcal{V}$ a presentably symmetric monoidal $(\infty,1)$-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 $\mathrm{Sp}$ embeds into a compactly-rigidly generated one.