Free Operad Research Articles

Möbius Functions of Directed Restriction Species and Free Operads, via the Generalised Rota Formula

We present some tools for providing situations where the generalised Rota formula of Carlier (Algebr Geometr Topol 20:169–213, 2020) applies. As an example of this, we compute the Möbius function of the incidence algebra of any directed restriction species, free operad, or more generally free monad on a finitary polynomial monad.

The incidence comodule bialgebra of the Baez–Dolan construction

Starting from any operad P, one can consider on one hand the free operad on P, and on the other hand the Baez–Dolan construction on P. These two new operads have the same space of operations, but with very different notions of arity and substitution. The main result of this paper is that the incidence bialgebras of the two-sided bar constructions of the two operads constitute together a comodule bialgebra. The result is objective: it concerns comodule-bialgebra structures on groupoid slices, and the proof is given in terms of equivalences of groupoids and homotopy pullbacks. Comodule bialgebras in the usual sense are obtained by taking homotopy cardinality. The simplest instances of the construction cover several comodule bialgebras of current interest in analysis. If P is the identity monad, then the result is the Faà di Bruno comodule bialgebra (dual to multiplication and substitution of power series). If P is any monoid Ω (considered as a one-coloured operad with only unary operations), the resulting comodule bialgebra is the dual of the near-semiring of Ω-moulds under product and composition, as employed in Écalle's theory of resurgent functions in local dynamical systems. If P is the terminal operad, then the result is essentially the Calaque–Ebrahimi-Fard–Manchon comodule bialgebra of rooted trees, dual to composition and substitution of B-series in numerical analysis (Chartier–Hairer–Vilmart). The full generality is of interest in category theory. As it holds for any operad, the result is actually about the Baez–Dolan construction itself, providing it with a new algebraic perspective.

Tree series and pattern avoidance in syntax trees

A syntax tree is a planar rooted tree where internal nodes are labeled on a graded set of generators. There is a natural notion of occurrence of contiguous pattern in such trees. We describe a way, given a set of generators G and a set of patterns P, to enumerate the trees constructed on G and avoiding P. The method is built around inclusion-exclusion formulas forming a system of equations on formal power series of trees, and composition operations of trees. This does not require particular conditions on the set of patterns to avoid. We connect this result to the theory of nonsymmetric operads. Syntax trees are the elements of such free structures, so that any operad can be seen as a quotient of a free operad. Moreover, in some cases, the elements of an operad can be seen as trees avoiding some patterns. Relying on this, we use operads as devices for enumeration: given a set of combinatorial objects we want enumerate, we endow it with the structure of an operad, understand it in term of trees and pattern avoidance, and use our method to count them. Several examples are provided.

Faà di Bruno for operads and internal algebras

For any coloured operad R, we prove a Faà di Bruno formula for the ‘connected Green function’ in the incidence bialgebra of R. This generalises on one hand the classical Faà di Bruno formula (dual to composition of power series), corresponding to the case where R is the terminal reduced operad, and on the other hand the Faà di Bruno formula for P-trees of Gálvez–Kock–Tonks ( P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following Gálvez–Kock–Tonks, we work at the objective level of groupoid slices, hence all proofs are ‘bijective’: the formula is established as the homotopy cardinality of an explicit equivalence of groupoids, in turn derived from a certain two-sided bar construction. In fact we establish the formula more generally in a relative situation, for algebras of one polynomial monad internal to another. This covers in particular nonsymmetric operads (for which the terminal reduced case yields the noncommutative Faà di Bruno formula of Brouder–Frabetti–Krattenthaler).

A proof of the existence of Batanin's initial operad

In this paper we prove the existence of the n-globular operad used in Batanin's definition of weak n-category. This operad is initial in the category of n-globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n-globular operad used in his definition of weak n-category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n-globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions.

Free cumulants, Schröder trees, and operads

The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faà di Bruno algebra, and then to the group of a free operad over Schröder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.

The operad of finite labeled tournaments

We study the operad of finite labeled tournaments. We describe the structure of suboperads of this operad generated by simple tournaments. We prove that a suboperad generated by a tournament with two vertices (i.e., the operad of finite linearly ordered sets) is isomorphic to the operad of symmetric groups, and a suboperad generated by a simple tournament with more that two vertices is isomorphic to the quotient operad of the free operad with respect to a certain congruence. We obtain this congruence explicitly.

Free Monoid in Monoidal Abelian Categories

We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. We apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad, free properad and free prop.

A canonical semi-classical star product

We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controlling the deformation theory of associative algebras. We prove that there is a canonical solution (choice independent) within the class of graphs without circuits, i.e. at the level of the free operad, without imposing the Jacobi identity. The proof is a consequence of the unique factorization property of the pre-Lie algebra of graphs (tree operad), where composition is the insertion of graphs. The restriction to graphs without circuits, i.e. at “tree level”, accounts for the interpretation as a semi-classical solution. The fact that this solution is canonical should not be surprising, in view of the Hausdorff series, which lies at the core of almost all quantization prescriptions.

Weak n-categories: comparing opetopic foundations

We construct for each tidy symmetric multicategory Q a cartesian monad ( E Q,T Q) , and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the ‘free operad’ monad construction on suitable monads. We use this to give an explicit description of the relationship between Baez–Dolan and Leinster opetopes.

681 The FDG standardized uptake value in predicting the outcome in head and neck cancer patients

In this paper we prove the existence of the n-globular operad used in Batanin's definition of weak n-category. This operad is initial in the category of n-globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n-globular operad used in his definition of weak n-category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n-globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions.

Operads as abstract algebras, and the Koszul property

The purpose of this work is to clarify some aspects of the theory of operads – especially of Koszul operads. We describe the free operad and give the categorical bar construction in terms of it. After giving a description of operads as abstract “algebras”, we give a self-contained proof of Theorem 4.1.13 of Ginzburg and Kapranov (1994). Our approach uses two spectral sequences associated to the categorical cobar complex of a Koszul operad.