Koszul Symmetric Set-Operads

• Koszul symmetric set-operads are algebraic structures that encode symmetric multilinear operations using intrinsic symmetry and combinatorial models.
• They leverage Koszul duality, compact resolutions, and bar constructions to derive minimal models and support efficient homological computations.
• Their classification, particularly among quadratic operads, reveals diverse structures and explicit Hilbert series that guide both theoretical and applied research.

Koszul symmetric set-operads are algebraic structures encoding multilinear operations with intrinsic symmetry, whose foundational homological properties are illuminated via Koszul duality, compact resolutions, and bar constructions. The theory synthesizes categorical, combinatorial, and homotopical perspectives to produce explicit models, classification results, and computational criteria for Koszulity in the context of symmetric set-operads. Recent advances have clarified the minimal models, functional identities, and Hilbert series governing such operads, particularly in the case of quadratic operads generated by one binary operation (Livernet, 2011, &&&1&&&, &&&2&&&).

1. Koszul Categories and Operadic Resolutions

The Koszul property for categories, as established for the category of reduced trees $\mathrm{TI}$, underpins compact resolutions of operads through the explicit construction of the Koszul complex. The objects of $\mathrm{TI}$ are reduced trees with leaves labelled by a finite set; morphisms correspond to the contraction of internal edges, formalized as $t \to s = t/E$ for $E$ a subset of internal edges. The Koszul complex is given by the top exterior power on the set of contracted edges:

$$K(b_s, b_t) = \begin{cases} \Lambda^{|E|}([E]) & \text{if } s = t/E, \ 0 & \text{otherwise.} \end{cases}$$

where $[E]$ denotes the span of $E$.

The normalized bar complex $N(b_s, b_t)$ is shown to be quasi-isomorphic to $K(b_s, b_t)$, i.e., homology is concentrated in top degree and isomorphic to the Koszul complex. This provides a canonical resolution supporting derived functor computations. The minimal model (cobar construction $\Omega(K)$) yields resolutions for operads up to homotopy.

2. Bar Constructions and Duality of Operadic Models

Several approaches to the bar construction for operads are unified via Koszul duality:

• The reduced Ginzburg–Kapranov bar complex is realized as a two-sided bar construction via the category $\mathrm{TI}$, given for a left $\mathrm{TI}$-module operad $P$ by $B^{GK}_n(P)(I) = K_{n-1}(b_I, P)$.
• Viewing operads as modules over the free operad monad $F$ provides $B(F, F, P)$.
• Considering operads as monoids in the category of connected species yields $B^\circ(R, P, L)$.

The paper shows these bar constructions are quasi-isomorphic—there exists a chain of quasi-isomorphisms, often mediated by the levelization morphism of Fresse, such that all encode the same derived data. This equivalence leverages the Koszulity of $\mathrm{TI}$ to provide minimal and combinatorially efficient resolutions. The essential fact is:

$B(F, F, P)(I) \cong B(b_I, P) \cong B^{GK}(P)(I),$

with levelization morphisms interpolating between them.

3. Operads Up to Homotopy and Minimal Models

The Koszul property gives a minimal resolution $R(\mathrm{TI}) = \Omega(K)$, and operads up to homotopy are precisely functors from $R(\mathrm{TI})$ to chain complexes with an explicit "up to homotopy" differential. For a vector species $M$, the functor is defined on a tree $t$ by

$\underline{M}(t) = \bigotimes_{v \in V_t} M((v)),$

with $(v)$ the incoming edges to vertex $v$.

The differential for compositions $\circ_E: M(t) \to M(s)$, where $E = e_1 \wedge ... \wedge e_n \in K$, is given by

$$\partial(\circ_E) = \sum_{F \sqcup G = E, F,G \neq \emptyset} \varepsilon(F,G) \circ_F \circ_G,$$

reproducing the standard coherence identities for homotopy operads. This generalizes the classical operad structure to one encoded by higher homotopies, making explicit the connection between Koszul minimal models and homotopy theory for operads.

4. Constructing Koszul Symmetric Set-Operads: Combinatorial Methods and Presentations

Functorial constructions from monoids to set-operads enable broad combinatorial modeling (Giraudo, 2012, Giraudo, 2013). For a monoid $(M, \cdot, 1)$, define the operad $T M$ by words of length $n$ over $M$ with composition:

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

Symmetric group actions and quotients produce symmetric set-operads encoding parking functions, packed words, and various rooted tree structures. Suboperads and presentations often yield classical examples (magmatic, associative commutative, diassociative), many of which are Koszul or have Koszul duals. Presentations by generators and quadratic relations admit combinatorial interpretations; for instance, operads on planar trees correspond to normal forms arising from rewriting systems, establishing PBW bases and facilitating Koszul duality.

5. Classification and Hilbert Series of Binary Koszul Symmetric Set-Operads

A comprehensive classification of Koszul symmetric set-operads generated by a single binary operation is now available (Laubie, 17 Sep 2025). Every such operad is a quotient of the magmatic operad $\mathrm{Mag}/\langle R \rangle$ via an equivariant relation $R$ among arity-3 monomials, splitting into orbits under $\mathbb{S}_3$ symmetry. After reduction by automorphism, exactly 11 distinct Koszul operads remain, including magmatic, associative, permutative, commutative, Lie-admissible, commutative magmatic, NAP, and four new examples constructed via connected sums or additional relations.

For each, the Hilbert series $f_P(t)$—enumerating n-ary operations in $P$—is explicit and solves a first-order differential algebraic equation over $\mathbb{Z}[t]$:

• Mag: $f_{\mathrm{Mag}}(t) = \frac{1}{2}(1 - \sqrt{1-4t})$.
• Ass: $f_{\mathrm{Ass}}(t) = \frac{t}{1-t}$.
• Perm: $f_{\mathrm{Perm}}(t) = t\,\exp(t)$.
• Com: $f_{\mathrm{Com}}(t) = \exp(t) - 1$.
• $P_{10}$: $f_{P_{10}}(t) = 1 - \sqrt{1 - 2t - t^2}$.
• $P_{2;2}$: $f_{P_{2;2}}(t) = 2 - t - \sqrt{1-2t}$.

Every such Hilbert series satisfies a first-order differential equation:

$F_P(t, f_P(t), f'_P(t)) = 0,$

for some polynomial $F_P$ over $\mathbb{Z}$. This supports the conjecture that Hilbert series for Koszul symmetric operads generated by one binary operation are always differential algebraic of order 1.

6. Rewriting Systems, PBW Bases, and Koszul Criteria

Modern criteria for Koszulity exploit rewriting systems on tree monomials (Malbos et al., 2020, Hivert et al., 2019). In the shuffle operad framework, quadratic convergent rewriting systems (i.e., ones that terminate and yield unique normal forms) provide explicit PBW bases and guarantee Koszulity for the presented operad. For example, in $k$-signaletic operads, right-comb trees give a PBW basis; for operads from combinatorial monoids, trees or words encoded by normal forms yield dimensions matching Hilbert series predictions.

Polygraphic methods, extending Gröbner bases to categorical rewriting systems, remove the monomial order constraint and allow higher-dimensional resolutions. The homology of the associated chain complex is concentrated on the diagonal if and only if the operad is Koszul. These techniques underpin the systematic verification of Koszulity for symmetric set-operads arising from combinatorial or algebraic data.

7. Implications and Applications

The synthesis of Koszul duality, explicit resolutions, classification, and computational tools has several ramifications:

• Efficient computations in homology and minimal models for operads, enabling derived functor analyses and deformation theory applications.
• The ability to treat operads up to homotopy via extensions of functors from minimal resolutions, providing a homotopical language compatible with differential graded structures.
• Comprehensive understanding of the generating functions and combinatorial content of operads constructed from monoids or posets, clarifying the enumeration and structure of multilinear operations.

These advances not only unify disparate approaches to symmetric set-operads and their Koszul properties but also yield practical criteria and algebraic models for further development in operad theory, homotopical algebra, and higher category theory.