Rigid Algebras and Cospans

TL;DR

This paper introduces the concept of rigid algebras within symmetric monoidal $( olinebreak ext{infinity,2})$-categories, establishing their theory and linking rigid commutative algebras to cospans in an $( olinebreak ext{infinity,1})$-category.

Contribution

It generalizes rigid categories to higher categories, develops their theory, and connects rigid commutative algebras with cospan constructions in a novel way.

Findings

The $( olinebreak ext{infinity,2})$-category of rigid algebras is an $( olinebreak ext{infinity,1})$-category.

Rigid commutative algebras in cospan categories are identified with the underlying $( olinebreak ext{infinity,1})$-category.

An adjunction between cospan construction and the functor to rigid commutative algebras is established.

Abstract

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