On Tautological Globular Operads

TL;DR

This paper compares classical nonsymmetric operads with globular operads, aiming to clarify their relationships and constructions, thereby making the understanding of globular operads more intuitive for researchers.

Contribution

It provides a detailed comparison and construction of tautological operads in both classical and globular contexts, enhancing conceptual understanding.

Findings

Globular operads are a strict generalization of classical operads.

Construction of tautological globular operad uses internal hom functor in monoidal categories.

Comparison clarifies the relationship between classical and globular operads.

Abstract

The purpose of this exposition is to compare the constructions of classical nonsymmetric operads (and their algebras) to that of the globular operads of Leinster and Batanin. It is hoped that, through this comparison, understanding algebras for globular operads can be made more intuitive and approachable. We begin by giving a description of the construction of the classical tautological, or endomorphism, operad $taut(X)$ on a set $X$. We then describe how globular operads are a strict generalization of classical operads. From this perspective a description is given of the construction for the tautological globular operad $Taut(\mathcal{X})$ on a globular set $\mathcal{X}$ by way of describing the internal hom functor for the monoidal category $\boldsymbol{Col}$, of collections and collection homomorphisms, with respect to the monoidal composition tensor product used to define globular operads, all the while emphasizing comparisons to the analogous construction in the category of graded sets.