Abstract. J.-L. Loday introduced the concept of coherent unit actions on a regular operad and showed that such actions give Hopf algebra structures on the free algebras. Hopf algebras obtained this way include the Hopf algebras of shuffles, quasi-shuffles and planar rooted trees. We characterize coherent unit actions on binary quadratic regular operads in terms of linear equations of the genera...

This is an exposition of some of the constructions which have arisen in higher-dimensional category theory. We start with a review of the general theory of operads and multicategories (cf. Bur], Her], Lei1]). Using this we give a new deenition of n-category (a variation on Batanin's); we also give an informal dee-nition in pictures. Next we discuss Gray-categories and their place in coherence p...

The purpose of these notes is to define an equivalence between the natural homology theories associated to operads and the homology of functors over certain categories of operators (PROPs) related to operads.

We review definitions and basic properties of operads, PROPs and algebras over these structures. Dedicated to the memory of Jakub Jan Ryba (1765–1815) Operads involve an abstraction of the family {Map(X, X)}n≥0 of composable functions of several variables together with an action of permutations of variables. As such, they were originally studied as a tool in homotopy theory, specifically for it...

Working in the context of symmetric spectra, we prove higher homotopy excision and higher Blakers–Massey theorems, and their duals, for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra).

Shuffle operads were introduced to forget the symmetric group actions on while preserving all possible operadic compositions. Rewriting methods then applied via shuffle operads: in particular, a notion of Gröbner basis was for with respect total order tree monomials. In this article, we introduce structure polygraphs as categorical model rewriting operads, which generalizes bases approach by re...

Some binary quadratic operads are endowed with anticyclic structures and their characteristic functions as anticyclic operads are determined, or conjectured in one case. AMS Classification 18D50; 05E05

Homotopy theories of algebras over operads, including operads over “little ncubes,” are defined. Spectral sequences are constructed and the corresponding homotopy groups are calculated.

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to this task. The resulting theory includes the Gray tensor product of 2-categories and the Crans tensor product [12] of Gray categories. Moreover much of the pre...

We prove that the homotopy theory of $N_\infty$ operads is equivalent to a discrete operads, and we construct free associative operadic realizations every indexing system. This resolves conjecture Blumberg Hill in affirmative.