Actads

TL;DR

This paper introduces $n$-actads, a new generalization of operads that incorporates novel permutation structures at multiple levels, leading to new insights in higher category theory and homotopy types.

Contribution

The paper defines $n$-actads, explores their algebraic structures, and demonstrates their applications in constructing new homotopy types and connecting to ordinal notation.

Findings

$n$-actads generalize operads with multi-level permutation structures.

Examples of $2$-actad algebras illustrate their use in homotopy theory.

Connections between actads and ordinal notation are established.

Abstract

In this paper, I introduce a new generalization of the concept of an operad, further generalizing the concept of an opetope introduced by Baez and Dolan, who used this for the definition of their version of non-strict $n$-categories. Opetopes arise from iterating a certain construction on operads called the $+$-construction, starting with monoids. The first step gives rise to plain operads, i.e. operads without symmetries. The permutation axiom in a symmetric operad, however, is an additional structure resulting from permutation of variables, independent of the structure of a monoid. Even though we can apply the $+$-construction to symmetric operads, there is the possibility of introducing a completely different kind of permutations on the higher levels by again permuting variables without regard to the structure on the previous levels. Defining and investigating these structures is the main purpose of this paper. The structures obtained in this way is what I call $n$-actads. In $n$-actads with $n>1$, the permutations on the different levels give rise to a certain special kind of $n$-fold category. I also explore the concept of iterated algebras over an $n$-actad (generalizing an algebra and module over an operad), and various types of iterated units. I give some examples of algebras over $2$-actads, and show how they can be used to construct certain new interesting homotopy types of operads. I also discuss a connection between actads and ordinal notation.