E_n-monoidal extension of the oplax-colimit construction for free rigid algebras

Establish an E_n-monoidal analogue of Construction cons:cocone by constructing an appropriate oplax cocone in the E_n-monoidal (∞,2)-categorical setting, so that the paper’s oplax-colimit method extends from symmetric monoidal to E_n-monoidal contexts; in particular, determine whether, in the universal E_n-monoidal case, there exists an oplax cocone sufficient to carry through the proof (for example via a generalization of Neuhauser’s framework).

Background

The paper constructs free rigid commutative algebras in weakly 2-presentably symmetric monoidal (∞,2)-categories via oplax colimits over the 1-dimensional framed cobordism category, recovering and extending enriched and classical cases.

The author notes that a parallel set of questions arises for rigid E_n-algebras in E_n-monoidal (∞,2)-categories, where one would expect a description in terms of the free E_n-monoidal category on a dualizable object (as suggested by tangle/cobordism frameworks).

However, a key step used in the symmetric monoidal case—Construction cons:cocone—does not readily generalize to the E_n setting, and the author indicates not knowing how to adapt the proof. A suggested approach is to generalize Neuhauser’s results to the E_n-monoidal case and test for the existence of a suitable oplax cocone in the universal situation.

References

A lot of our methods work, but as far as we can tell, not all of them directly generalize, and in particular at this point the author does not know how to make the proof go through in that generality. It would certainly be an interesting question to investigate.

For the interested reader, the particular point I cannot replicate in the E_n-story is \Cref{cons:cocone}. One possible approach would be to generalize the work of Neuhauser in the E_n-monoidal case and see whether in the universal case there is such an oplax cocone.

— Free rigid commutative algebras  (2604.01854 - Ramzi, 2 Apr 2026) in Remark (rmk:En), Introduction