Hinich's model for Day convolution revisited

TL;DR

This paper revisits Hinich's construction of the Day convolution operad, proving it forms an exponential in the category of ∞-operads and providing an explicit description of algebra formation as a bivariant functor.

Contribution

It offers a new proof that Hinich's Day convolution is an exponential and explicitly describes algebra formation in this context.

Findings

Hinich's construction is an exponential in the ∞-category of ∞-operads.

Provides an explicit description of algebra formation as a bivariant functor.

Enhances understanding of Day convolution in ∞-categories.

Abstract

We prove that Hinich's construction of the Day convolution operad of two $\mathcal{O}$-monoidal $\infty$-categories is an exponential in the $\infty$-category of $\infty$-operads over $\mathcal{O}$, and use this to give an explicit description of the formation of algebras in the Day convolution operad as a bivariant functor.