Day convolution for algebraic patterns

TL;DR

This paper extends Day convolution to a broad class of algebraic structures in $$-categorical algebra, providing a criterion for exponentiable objects in these contexts.

Contribution

It introduces a necessary and sufficient criterion for exponentiable objects in structures like $$-operads and virtual double $$-categories, generalizing existing frameworks.

Findings

Provides a new description of weak Segal fibrations via generalized Segal spaces.

Defines the 'underlying graph' of weak Segal fibrations, extending the concept of underlying $$-categories.

Explicitly characterizes the underlying graph of exponential objects in weak Segal fibrations.

Abstract

We characterize the exponentiable objects for a wide range of structures prevalent in $\infty$-categorical algebra, extending the construction of Day convolution to more general structures than $\infty$-operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) $\infty$-operads and virtual double $\infty$-categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying $\infty$-category for $\infty$-operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations.