nLab relative operad

Idea

Relative operads are the operadic generalization of relative categories. In generalization of how the latter model $(\infty,1)$-categories, so relative operads model $(\infty,1)$-operads.

Definition

A relative operad is a pair $(O,W)$, where $O$ is a colored operad and $W$ is a replete subcategory of the category of unary operations of $O$.

Properties

Relation to $(\infty,1)$-operads

Given a relative operad $O$, one can formally invert the maps in $W$ to obtain an $(\infty,1)$-operad $O[W^{-1}]$, called its localization. This process defines a functor

$$L \;\colon\; \mathsf{RelOp} \longrightarrow \mathsf{Op}_{(\infty,1)}$$

from the category of relative operads to the (∞,1)-category of $(\infty,1)$-operads. The main theorem of ACP25 asserts that $L$ is in fact a localization of (∞,1)-categories. Therefore, relative operads model $(\infty,1)$-operads.

References

• Kensuke Arakawa, Victor Carmona, Francesca Pratali: Relative operads model $\infty$-operads [arXiv:2512.16374]