English

Free rigid commutative algebras

Category Theory 2026-04-03 v1 Algebraic Topology K-Theory and Homology

Abstract

We describe free rigid commutative algebras in 22-presentably symmetric monoidal (,2)(\infty,2)-categories as oplax colimits over the 11-dimensional framed cobordism category. The special case of the (,2)(\infty,2)-category PrL\mathrm{Pr}^\mathrm{L} itself provides a description of the free symmetric monoidal (,1)(\infty,1)-category with duals on a given (,1)(\infty,1)-category, while the case of ModV(PrL)\mathrm{Mod}_{\mathcal{V}}(\mathrm{Pr}^\mathrm{L}) provides a description of a similar object in the V\mathcal{V}-enriched context, for V\mathcal{V} a presentably symmetric monoidal (,1)(\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 Sp\mathrm{Sp} embeds into a compactly-rigidly generated one.

Keywords

Cite

@article{arxiv.2604.01854,
  title  = {Free rigid commutative algebras},
  author = {Maxime Ramzi},
  journal= {arXiv preprint arXiv:2604.01854},
  year   = {2026}
}

Comments

36 pages, comments welcome!

R2 v1 2026-07-01T11:50:43.337Z