Koszul duality for algebras over infinity-operads

In this paper, we introduce a new notion of algebra over a linear $\infty$-operad and a corresponding notion of coalgebra over an $\infty$-cooperad. We next extend the Koszul duality between linear $\infty$-operads and linear $\infty$-cooperads from our previous paper (arXiv:2105.11943) to their categories of algebras and coalgebras. This duality theorem specialises to the known duality in the case of algebras over classical (non-infinity) operads, but our proof is different. In fact, it is based on a much more general duality between presheaves and copresheaves on a category of trees. The latter duality is a priori independent of the (co)algebra structures, but we show that it can be lifted to (co)presheaves supporting such a structure. Based on this duality, we define the homology of an algebra over an $\infty$-operad, and prove that it can be described in terms of the homology of the same category of trees with coefficients in a presheaf.

💡 Research Summary

The paper develops a comprehensive Koszul duality theory for algebras over linear ∞‑operads, extending the authors’ previous work on operads and cooperads to the level of algebras and coalgebras. The authors begin by recalling that a linear ∞‑operad can be modeled as a presheaf X on a category A of finite rooted trees, equipped with structure maps X(S ∘ₑ T) → X(S) ⊗ X(T) for each grafting of a tree T onto a leaf e of a tree S. These maps are required to be quasi‑isomorphisms; when they are actual isomorphisms the object is a strict operad. Dually, a linear ∞‑cooperad is a copresheaf with analogous structure maps.

To treat algebras, the authors enlarge the tree category to R, which contains A but also allows external face maps (prunings). An X‑pre‑algebra is a presheaf M on R together with structure maps τ_α : M(T) → X(S) ⊗ M(T/S) for each morphism α : S → T in R. When X is an ∞‑operad and each τ_α is a quasi‑isomorphism for pruning maps, M is called an ∞‑algebra over X. The dual notion of a conilpotent ∞‑coalgebra over an ∞‑cooperad Y is defined analogously using conilpotent copresheaves on R.

The central technical achievement is a bar–cobar adjunction at the level of presheaves and copresheaves on R. The authors construct functors B ∨ : CoPsh_c(R) → Psh(R) and B : Psh(R) → CoPsh_c(R) which are mutually inverse up to quasi‑isomorphism. The unit and counit of this adjunction are themselves quasi‑isomorphisms, establishing a homotopy equivalence between the homotopy categories of (conilpotent) copresheaves and presheaves. This result is first proved for the underlying categories of trees, independent of any operadic structure.

Next, the authors lift this adjunction to the operadic setting. For an ∞‑operad X, the bar construction B X is a conilpotent ∞‑cooperad, and the cobar construction B ∨ Y of a conilpotent ∞‑cooperad Y is an ∞‑operad. Moreover, the adjunction between X‑∞‑algebras and B X‑∞‑coalgebras is established: the bar functor sends an X‑algebra M to a B X‑coalgebra B(M), while the cobar functor sends a B X‑coalgebra N to an X‑algebra B ∨ (N). When X arises from a classical differential graded operad P, the construction recovers the usual bar complex of a P‑algebra A via the presheaf N(P,A) on R, and evaluation at specific trees reproduces the classical bar‑cobar duality.

A significant application is the definition of “dendroidal homology” for both ∞‑operads and ∞‑algebras. For an ∞‑operad X, the homology groups DH_ℓ⁎(X) are defined as the homology of the chain complex B(X)(C_ℓ), where C_ℓ is the corolla with ℓ leaves. The authors show that DH_ℓ⁎(X) can be computed as the homology of a pair of categories (C_ℓ/A, C_ℓ//A) with coefficients in X, linking the construction to classical combinatorial models such as partition complexes and the Lie operad. For an ∞‑algebra M over X, the homology DH⁎(M) is defined as H⁎(B(M)(η)), where η is the trivial tree with a single leaf. This homology inherits a conilpotent coalgebra structure over the cooperad DH⁎(X). When M comes from an ordinary P‑algebra, DH⁎(M) coincides with the André‑Quillen homology defined in the literature.

The paper also discusses functoriality: a morphism of (∞‑)operads f : X → X′ induces a push‑forward functor f_! on (pre‑)algebras, and when f is a quasi‑isomorphism this functor is an equivalence up to homotopy. The dual statements for coalgebras are treated similarly.

Overall, the work provides a new conceptual framework for Koszul duality that starts from a purely categorical bar‑cobar equivalence on tree‑indexed (co)presheaves and then lifts it to the level of algebras and coalgebras over ∞‑operads. This approach unifies and extends classical results, offers a clean homotopical proof, and yields concrete computational tools for the homology of ∞‑algebras via tree‑category homology.