Day convolution for algebraic patterns

Day showed that presheaf categories admit a symmetric monoidal structure, known as the convolution product. This structure satisfies a universal property in the ambient category of operads, where it arises as an exponential object. In this talk, we study the exponentiable objects in $\infty$-categories of operad-like structures, including $\infty$-operads, equivariant $\infty$-operads, and virtual double $\infty$-categories. To do so, we work within the framework of algebraic patterns of Chu–Haugseng, which encode these operad-like structures in terms of so-called weak Segal fibrations.
I will explain how to characterize the exponentiable weak Segal fibrations. Even in the case of $\infty$-operads, such a characterization was previously unknown. The proof relies on a (partly new) description of weak Segal fibrations in terms of generalized Segal spaces on tree categories.
This is joint work with Thomas Blom and Félix Loubaton.