On resolutions of diagrams of algebras

We prove a restricted version of a conjecture by M. Markl on resolutions of an operad describing diagrams of algebras. We discuss a particular case related to the Gerstenhaber-Schack diagram cohomology.

💡 Research Summary

The paper addresses a conjecture of Martin Markl concerning resolutions of operads that encode diagrams of algebras. The author proves a restricted version of this conjecture in the setting where the operad A governing the algebras is Koszul and all its generating operations are concentrated in a single arity and degree. The main construction proceeds by “gluing’’ a minimal resolution of A with a cofibrant free resolution of a small category C, thereby producing a new minimal free resolution D⁽∞⁾ of the coloured operad A_C that describes C‑shaped diagrams of A‑algebras.

The paper begins with a concise review of coloured Σ‑modules and coloured operads, introducing the composition product ◦ and proving a coloured version of the Künneth formula (Lemma 2.8). This formula shows that the homology of a composition product is the composition product of the homologies, a fact that will later allow the author to compute the principal part of the differential in the glued resolution.

Section 3 defines the operadic incarnation of a small category C: the coloured operad C has colours given by the objects of C and a generating arity‑1 operation for each morphism, with operadic composition induced by categorical composition. A free resolution C^∞ = (F(F),∂) is assumed, where F₀ consists of non‑identity morphisms, F₁ consists of relations, and ∂ sends each relation generator to the difference of two operadic composites. The bar‑cobar construction provides a canonical example of such a resolution.

The core result (Theorem 3.15) states that, under the Koszul hypothesis on A, one can form a minimal resolution D^∞ of the diagram operad A_C by taking the tensor product of the minimal resolution A^∞ of A with the chosen resolution C^∞ and then defining a differential ∂_D. The differential is split into a “principal part’’ coming from a family of chain maps