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Disquisitions on Monoidal Categories and Operads
Disquisitions on Monoidal Categories and Operads
Operadic Perspectives on Gerstenhaber-Schack Theory
Operadic Perspectives on Gerstenhaber-Schack Theory
Organizers
Speaker
Alexander Voronov
Time
Tuesday, April 14, 2026 12:00 PM - 2:30 PM
Venue
A3-2-201
Online
Zoom 537 192 5549
(BIMSA)
Abstract
In the 1980s, Murray Gerstenhaber and Samuel Schack studied the deformation theory of the incidence algebra $I(P)=I(P,k)$ of a poset $P$. They identified the relative Hochschild cochain complex of $I(P)$ with the simplicial cochain complex of the nerve $N(P)$, matching cup products and relating the composition product to Steenrod’s cup-1 product. A further result described the moduli space of formal deformations of $I(P)$ as the cohomology group $H^2(N(P);1+tk[[t]])$.
In the 1990s, Gerstenhaber and the speaker introduced operads with multiplication and constructed that structure on both Hochschild and simplicial cochains, yielding homotopy $G$-algebra structures that encode the cup product, cup-1 product, and Gerstenhaber bracket. In the 2000s, Frédéric Patras showed that the Gerstenhaber-Schack isomorphism respects these homotopy $G$-algebra structures.
In joint work with Andy Yu, we revisit these results from an operadic and topological viewpoint, studying the deformation theory governed by the dg-Lie algebra $C^*(X;k)$ of a simplicial set $X$. We prove that the Gerstenhaber–Schack isomorphism is in fact an isomorphism of operads with multiplication. We also observe that the cohomology Lie algebra $H^*(X;k)$ is abelian, and identify the Maurer-Cartan moduli space of $C^*(X;k[[t]])$ with $H^2(X;1+tk[[t]])$. We conjecture that $C^*(X;k)$ is homotopy abelian and provide supporting evidence.
In the 1990s, Gerstenhaber and the speaker introduced operads with multiplication and constructed that structure on both Hochschild and simplicial cochains, yielding homotopy $G$-algebra structures that encode the cup product, cup-1 product, and Gerstenhaber bracket. In the 2000s, Frédéric Patras showed that the Gerstenhaber-Schack isomorphism respects these homotopy $G$-algebra structures.
In joint work with Andy Yu, we revisit these results from an operadic and topological viewpoint, studying the deformation theory governed by the dg-Lie algebra $C^*(X;k)$ of a simplicial set $X$. We prove that the Gerstenhaber–Schack isomorphism is in fact an isomorphism of operads with multiplication. We also observe that the cohomology Lie algebra $H^*(X;k)$ is abelian, and identify the Maurer-Cartan moduli space of $C^*(X;k[[t]])$ with $H^2(X;1+tk[[t]])$. We conjecture that $C^*(X;k)$ is homotopy abelian and provide supporting evidence.