Nonsymmetric Operad with Multiplication

• Nonsymmetric operads with multiplication are algebraic structures defined by planar operads combined with an associative product, generalizing classic algebraic systems.
• They employ combinatorial frameworks based on planar trees and grafting operations, enabling precise modeling in homotopy theory and algebraic topology.
• Their categorical framework supports robust model structures and polynomial realizations, bridging applications in combinatorics, deformation theory, and mathematical physics.

A nonsymmetric operad with multiplication is an algebraic structure in which one has operations with multiple inputs and one output, composed via planar (non-permutative) substitution together with a distinguished associative “multiplication” operation. These structures generalize associative algebras and encode richer compositions, appearing fundamentally in areas such as homotopy theory, algebraic combinatorics, higher category theory, and algebraic topology. The multiplication—formally a map $\mu: \mathcal{O} \circ \mathcal{O} \to \mathcal{O}$—endows the operad with additional algebraic structure, supporting the transfer of homotopical and algebraic properties to wide classes of objects (such as categories enriched over monoidal model categories) (Muro, 2011).

1. Formal Structure and Definitions

A nonsymmetric operad $\mathcal{O}$ in a symmetric monoidal category $V$ is a monoid in the category of sequences $V^{\mathbb{N}}$ with respect to the composition product $\circ$, given by

$$(U \circ V)(m) = \coprod_{n \ge 0} \coprod_{p_1+\cdots+p_n = m} U(n) \otimes V(p_1) \otimes \cdots \otimes V(p_n).$$

The operad comes equipped with a multiplication

$$\mu: \mathcal{O} \circ \mathcal{O} \to \mathcal{O},$$

which unravels to partial composition structure maps

$$\circ_i: \mathcal{O}(m) \otimes \mathcal{O}(n) \to \mathcal{O}(m+n-1), \quad 1 \le i \le m,$$

satisfying associativity, unit, and compatibility constraints. For example, the associativity relation takes the form: $$(\mathcal{O}(l) \circ_i \mathcal{O}(m)) \circ_j \mathcal{O}(n) = \mathcal{O}(l) \circ_i (\mathcal{O}(m) \circ_{j-i+1} \mathcal{O}(n)).$$ Alternatively, the operadic structure may be described via planar rooted trees, where grafting trees at specified vertices models the operad multiplication.

A (reduced) free nonsymmetric operad is generated by planar trees (e.g., over a set $\Sigma$), with the multiplication determined by tree grafting. Quotients and suboperads of free operads accommodate additional relations, leading to a vast spectrum of multiplicative nonsymmetric operads.

2. Model Structures and Homotopy Theory

Under standard hypotheses (e.g., $V$ cofibrantly generated, closed symmetric monoidal, and satisfying the monoid axiom), the category of nonsymmetric operads $\mathrm{Op}(V)$ acquires a cofibrantly generated model structure in which:

• Weak equivalences and fibrations are defined levelwise (i.e., $f: \mathcal{O} \to \mathcal{P}$ is a weak equivalence iff all $f(n): \mathcal{O}(n) \to \mathcal{P}(n)$ are weak equivalences in $V$).
• Cofibrations are created via the free operad functor $\mathcal{F}: V^{\mathbb{N}} \to \mathrm{Op}(V)$.
• Pushouts and transfinite compositions are described inductively using the combinatorics of planar trees and explicit colimit expressions (Muro, 2011).

This categorical framework enables lifting model structures to the category of algebras over any nonsymmetric operad. In particular, for a monoidal model category $C$ enriched over $V$ and an operad $\mathcal{O}$ in $V$, the category $\mathrm{Alg}_\mathcal{O}(C)$ has a transferred model structure defined via the free/forgetful adjunction: $$\mathcal{F}_\mathcal{O}: C \rightleftarrows \mathrm{Alg}_\mathcal{O}(C): U,$$ with explicit formulas for free algebra objects

$$\mathcal{F}_\mathcal{O}(Y) = \coprod_{p \ge 0} z(\mathcal{O}(p)) \otimes Y^{\otimes p},$$

where $z: V \to C$ is a strong braided monoidal functor.

Cofibrancy conditions on the operad $\mathcal{O}$ (e.g., $\mathcal{O}(n)$ cofibrant in $V$) propagate to ensure cofibrancy of free algebras and rectification (i.e., Quillen equivalence) between categories of algebras over weakly equivalent operads.

3. Multiplicative Structures: Combinatorics and Examples

The multiplicative structure is rooted in the operadic composition. In combinatorial realizations, the free nonsymmetric operad on a set $\Sigma$ corresponds to planar trees whose internal vertices are decorated by $\Sigma$. The multiplication corresponds to the grafting of trees—substituting one operation into an “input slot” of another.

For instance, a magmatic operad is the free nonsymmetric operad on one binary generator; the associative operad is its quotient by the classical associativity relation. These, and a range of others (e.g., diassociative/triassociative, dendriform, and their deformations) are constructed as suboperads or quotients, sometimes arising functorially from monoids $(M,\cdot,1)$ by the construction $T(M)$ (Giraudo, 2013), which defines partial composition via

$$(x_1, ..., x_n) \circ_i (y_1, ..., y_m) = (x_1, ..., x_{i-1}, x_i \cdot y_1, ..., x_i \cdot y_m, x_{i+1}, ..., x_n).$$

Such operads organize classical combinatorial families—trees, parking functions, Motzkin words, and more—under a unified composition law linked to the monoid's multiplication.

Further, in the context of modular and cyclic extensions, the gluing maps and contraction maps provide an internal multiplication in the absence of symmetric group actions, as in non-$\Sigma$ modular operads that appear in models of open strings and related moduli spaces (Markl, 2014).

4. Algebraic and Homological Observables

Nonsymmetric operads with multiplication enable the construction of algebraic and homological invariants. For an operad $\mathcal{O}$ with multiplication $\mu\in \mathcal{O}(2)$ and unit $e\in \mathcal{O}(0)$, one defines chain-level cap products, Lie derivatives, Gerstenhaber brackets, and even Batalin–Vilkovisky structures on associated modules. For modules $M$ over an operad with multiplication, the following structures arise (Kowalzig, 2013):

• Cap product: $\iota_y(x) = (\mu \circ_2 y) \circ^0 x$.
• Lie derivative: $L_y(x) = \sum_{i} (-1)^{\cdots} t^{i-1}(x)$ (where $t$ is a cyclic operator), which together with the boundary operators relate through Cartan–Rinehart-type formulas.

In homology, these pairings endow objects with Gerstenhaber and BV module structures, central in noncommutative geometry and deformation theory.

Examining the operadic chain complexes can lead to explicit DGLAs governing Nijenhuis and Rota–Baxter structures (Baishya et al., 4 May 2025), in which certain bracket operations and differentials encode the failure of an element to be a Nijenhuis or Rota–Baxter operator, offering a Maurer–Cartan framework for rich deformation and compatibility theory among multiplicative operations.

5. Polynomial Realizations and Hopf Algebra Connections

Polynomial realizations provide a combinatorial and algebraic bridge between nonsymmetric operads and Hopf algebras (Giraudo, 2023, Giraudo, 18 Jun 2024). For any sufficiently structured operad (especially free operads on trees or forests), one can define a polynomial realization by mapping basis elements (forests, i.e., reduced operad-words) into sums of monomials in an explicitly constructed noncommutative alphabet $A$ with specified unary (root, decoration) and binary (edge) relations. The encoding is governed by:

• The length of the word matching the degree of the forest.
• Compatibility of letters with root positions, decorations, and parent–child edges (using relations $R^A$, $D_s^A$, and $\{j\}^A$).

Crucially, the concatenation product in the Hopf algebra corresponds exactly to polynomial multiplication. By varying the alphabet and its relations, the polynomial images project onto well-known combinatorial and algebraic Hopf algebras, including:

• Hivert's Hopf algebra of word quasi-symmetric functions (WQSym),
• Decorated versions of the noncommutative Connes–Kreimer Hopf algebra,
• Noncommutative Faà di Bruno Hopf algebras,
• Multi-symmetric function algebras,
• The double tensor Hopf algebra of Ebrahimi-Fard and Patras.

Binary operations at the combinatorial (tree or forest) level, such as “over/under” products and shuffle operations, are reflected directly in these polynomial realizations (e.g., “over” and “under” products match the concatenation and reassembly of forests) (Giraudo, 2 Jul 2025).

These connections yield a triple of explicit combinatorial bases (elementary, fundamental, homogeneous) for the Hopf algebra, related via Möbius inversion in explicit poset structures (e.g., the “easterly wind” lattices on forests or trees) (Giraudo, 2023, Giraudo, 2 Jul 2025). Such basis changes bring the algebraic theory in alignment with standard structures featured in classical combinatorial Hopf algebras (e.g., NCSF, Loday–Ronco, Malvenuto–Reutenauer).

6. Applications, Enriched Examples, and Homotopical Implications

Nonsymmetric operads with multiplication, together with the Hopf and poset-theoretic machinery above, underpin a variety of concrete applications:

• Homotopy theory: Model structures on operads and their categories of algebras enable the transfer of homotopical invariants, strictification, and rectification results (e.g., Quillen equivalences between “strict” and A$_\infty$-algebraic structures) (Muro, 2011).
• Combinatorial and algebraic encoding: The use of forest-like alphabets and poset lattices provides explicit encodings of combinatorial families, such as trees, compositions, parking functions, and related objects, with their multiplication reflecting algebraic graph gluing and substitutions (Giraudo, 2013, Chenavier et al., 2018).
• Deformation, compatibility, and averaging: Maurer–Cartan theory for graded Lie algebras constructed from nonsymmetric operads with multiplication enables the systematic paper of deformations, Nijenhuis-Rota–Baxter hierarchies, averaging operators, and the rich algebraic structures appearing in Loday-type algebras (Baishya et al., 4 May 2025).
• Modular and field theoretic models: Non-$\Sigma$ modular operads supply algebraic models for moduli spaces of surfaces with boundary, relevant in open string field theory, where the gluings and contractions reflect both operadic multiplication and surface geometry (Markl, 2014).
• Polynomial and Hopf algebra frameworks: Polynomial realizations enable the transfer of operadic multiplication into a well-understood algebraic context (noncommutative polynomial rings), immediately yielding associativity and compatibility with known Hopf algebraic structures (Giraudo, 2023, Giraudo, 18 Jun 2024).

7. Classification, Quotients, and Future Directions

The classification of nonsymmetric operads with multiplication leverages presentations by generators and relations, rewriting systems on trees, combinatorial lattices, and functorial constructions arising from monoids, with explicit criteria for freeness, nilpotence, and combinatorial Hilbert series (Giraudo, 2013, Bremner et al., 2015, Chenavier et al., 2018). Lattice structures on quotients of magmatic operads and their deeper connections (e.g., with Tamari, Fuss–Catalan, and other posets) frame the possible multiplicative operad types.

Recent advances in the fibration theory of operads (e.g., via filtrations of lattice path operads) suggest a homotopy-theoretic hierarchy: operads of complexity $m$ generalize monoids ($m=1$), nonsymmetric operads ($m=2$), and higher-arity (planar) gluing gadgets ($m>2$) (Leger, 19 Mar 2025). This suggests further categorical and homotopical paper of nonsymmetric operads with multiplication as a foundational tool in the broader algebraic landscape.

The accumulation of these structures, constructions, and equivalences positions the theory of nonsymmetric operads with multiplication as a cornerstone for modern algebraic and combinatorial research, with ongoing developments in polynomial realization, homotopical algebra, combinatorics, and mathematical physics.