[Paper Review] Curved cooperads and homotopy unital A-infty-algebras

Volodymyr Lyubashenko|arXiv (Cornell University)|Mar 14, 2014

Advanced Topics in Algebra3 citations

TL;DR

This paper introduces curved cooperads as a framework to model homotopy unital A-infinity algebras, extending classical A-infinity theory by incorporating curvature terms. It establishes a duality between curved cooperads and curved algebras via cobar constructions, proving that the cobar construction of a curved cooperad yields a curved A-infinity algebra, thereby generalizing the classical cobar duality to curved settings.

ABSTRACT

We provide bar and cobar constructions as functors acting between various categories of curved operads and curved cooperads. Cobar and bar constructions are adjoint to each other. Given a twisting cochain between a curved augmented cooperad C with an extra grading and a curved operad O we construct a couple of adjoint functors between the category of curved O-modules and the category of curved C-comodules. The important feature is that the curved operad O is not necessarily augmented.

Motivation & Objective

• To extend the classical cobar duality between cooperads and algebras to include curvature terms.
• To define and study curved cooperads as a new algebraic structure suitable for modeling homotopy unital A-infinity algebras.
• To establish a duality between curved cooperads and curved A-infinity algebras via the cobar construction.
• To provide a homotopical framework that incorporates non-trivial curvature in A-infinity structures.
• To generalize the standard cobar construction to handle curved coalgebras and curved algebras.

Proposed method

• The paper defines curved cooperads using a curved version of the cooperad axioms, incorporating a curvature term in the comultiplication.
• It introduces the cobar construction of a curved cooperad, which produces a curved A-infinity algebra.
• The construction uses the bar-cobar duality framework, adapted to include curvature via modified comultiplication and differential structures.
• Key equations involve the action of weight vectors w and the use of projection maps pr to define the differential on the cobar complex.
• The paper verifies that the cobar construction satisfies the curved A-infinity relations by checking that the differential squares to zero under the curved comultiplication.
• It uses diagrammatic reasoning and algebraic identities, such as equation (LABEL:dia-cooperad-4-OOOOO), to verify the curvature compatibility.

Experimental results

Research questions

• RQ1 How can the classical cobar duality between cooperads and algebras be extended to include curvature terms in the A-infinity structure?
• RQ2 What axioms must a cooperad satisfy to support a curved A-infinity algebra structure via the cobar construction?
• RQ3 How does the presence of curvature affect the differential in the cobar complex and its square-zero property?
• RQ4 Can the cobar construction of a curved cooperad yield a well-defined curved A-infinity algebra?
• RQ5 What role do weight parameters w and the projection maps pr play in ensuring the consistency of the curved A-infinity relations?

Key findings

• The cobar construction of a curved cooperad yields a curved A-infinity algebra, extending the classical duality to the curved setting.
• The differential in the cobar complex squares to zero due to the curvature compatibility encoded in the defining equation involving w and ξT.
• The equation involving wξT(0,1,0) and wξT(0,0,1) ensures the curved A-infinity relations are satisfied under the projection maps pr.
• The framework allows for the modeling of homotopy unital A-infinity algebras through curved cooperads, providing a new algebraic tool for homotopical algebra.
• The use of weight vectors and comultiplication structures enables a systematic treatment of curvature in higher algebraic structures.
• The paper establishes that the curved cobar construction is consistent and closed under the required algebraic identities, such as those derived from equation (LABEL:dia-cooperad-4-OOOOO).