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Deformations of mixed associators in module categories

Matthieu Faitg, Azat M. Gainutdinov, Christoph Schweigert, Jan-Ole Willprecht

Abstract

We set up a cochain complex $C^\bullet_{\mathrm{mix}}(\mathcal{M})$ whose cohomology controls deformations of the mixed associator of a module category $\mathcal{M}$ over a $\Bbbk$-linear monoidal category $\mathcal{C}$. We show that $C^\bullet_{\mathrm{mix}}(\mathcal{M})$ is isomorphic to the Davydov-Yetter (DY) complex of the representation functor $ρ: \mathcal{C} \to \mathrm{End}(\mathcal{M})$. Using our previous results on DY cohomology (arXiv:2411.19111), we prove that if $\mathcal{C}$ and $\mathcal{M}$ are finite then the cohomology $H^\bullet_{\mathrm{mix}}(\mathcal{M})$ is isomorphic to the relative Ext groups $\mathrm{Ext}^\bullet_{\mathcal{Z}(\mathcal{C}),\mathcal{C}}(\boldsymbol{1},\mathcal{A}_{\mathcal{M}})$ for the usual adjunction between the Drinfeld center $\mathcal{Z}(\mathcal{C})$ and $\mathcal{C}$, where $\mathcal{A}_{\mathcal{M}}$ is the so-called adjoint algebra of $\mathcal{M}$. This allows us to give a dimension formula for $H^n_{\mathrm{mix}}(\mathcal{M})$ in terms of certain Hom spaces in $\mathcal{Z}(\mathcal{C})$, and also to prove that $H^{>0}_{\mathrm{mix}}(\mathcal{C}) = 0$. We also show that the algebra $\mathcal{A}_{\mathcal{M}}$ is the ``full center'' of an algebra in $\mathcal{C}$ realizing $\mathcal{M}$. We furthermore establish a generalized version of Ocneanu rigidity for monoidal functors with coefficients, and provide its application to general (non-exact and non-finite) $\mathcal{C}$-module categories over a fusion category $\mathcal{C}$ such that $\dim(\mathcal{C}) \neq 0$. We spell out these results for module categories defined by finite-dimensional comodule algebras over finite-dimensional Hopf algebras. Examples based on comodule algebras over Sweedler's Hopf algebra are worked out in detail and yield new continuous families of inequivalent non-exact module categories.

Deformations of mixed associators in module categories

Abstract

We set up a cochain complex whose cohomology controls deformations of the mixed associator of a module category over a -linear monoidal category . We show that is isomorphic to the Davydov-Yetter (DY) complex of the representation functor . Using our previous results on DY cohomology (arXiv:2411.19111), we prove that if and are finite then the cohomology is isomorphic to the relative Ext groups for the usual adjunction between the Drinfeld center and , where is the so-called adjoint algebra of . This allows us to give a dimension formula for in terms of certain Hom spaces in , and also to prove that . We also show that the algebra is the ``full center'' of an algebra in realizing . We furthermore establish a generalized version of Ocneanu rigidity for monoidal functors with coefficients, and provide its application to general (non-exact and non-finite) -module categories over a fusion category such that . We spell out these results for module categories defined by finite-dimensional comodule algebras over finite-dimensional Hopf algebras. Examples based on comodule algebras over Sweedler's Hopf algebra are worked out in detail and yield new continuous families of inequivalent non-exact module categories.

Paper Structure

This paper contains 39 sections, 45 theorems, 269 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mixed associator cohomology and DY cohomology
  3. Deformation theory of mixed associators
  4. Module categories and module functors
  5. Infinitesimal deformations of mixed associators
  6. Mixed associator cohomology
  7. Relation with DY cohomology
  8. DY cohomology with coefficients
  9. Monoidal structures of the representation functor
  10. Centralizer of the representation functor
  11. DY cohomology of the representation functor
  12. Higher order deformations and obstructions
  13. Deformations of monoidal functors and of half-braidings
  14. Deformations of -module structures
  15. Deformations and rigidity via the adjoint algebra
  16. ...and 24 more sections

Key Result

Proposition 1

(Prop. propMixCohomDY) For all $\mathsf{F},\mathsf{G} \in \mathrm{End}_{\mathcal{C}}(\mathcal{M}) = \mathcal{Z}(\rho)$, the cochain complexes $\mathrm{C}^\bullet_{\mathrm{mix}}(\mathcal{M};\mathsf{F},\mathsf{G})$ and $\mathrm{C}^\bullet_{\mathrm{DY}}(\rho;\mathsf{F},\mathsf{G})$ are isomorphic. $\bl

Figures (1)

  • Figure 1: Diagrammatic proof of eq. \ref{['formulaHalfBrUniv']}. All symbols $\rhd$ and $\otimes$ are omitted thanks to strictness; for morphisms we write for instance $M$ instead of $\mathrm{id}_M$. The first equality is by definition of $b^{\mathsf{Id}}$ in \ref{['halfBrEnd']}, the second is by definition of $u$ in \ref{['defFactU']}, the third uses the fact that $\beta_{X \rhd M} = (\mathrm{id}_X \rhd \beta_M) \circ (h_X \rhd \mathrm{id}_M)$ which easily follows from the definition of $\beta$ in \ref{['defBeta']} and the half-braiding property of $h$, the fourth is by naturality of $J$ (here in the last variable), the fifth is by the formula for $J$ in \ref{['formulaJ']}, the sixth is by naturality of $\underline{\mathsf{coev}}$\ref{['natEvInt']}, the seventh is by naturality of $\underline{\mathsf{ev}}$ applied within the functor $\underline{\mathop{\mathrm{Hom}}\nolimits}(M,-)$ and by naturality of $\underline{\mathsf{coev}}$ applied in the two bottom boxes \ref{['natEvInt']}, the eighth is by the zig-zag property \ref{['zigZagPropAdj']} used within the functor $\underline{\mathop{\mathrm{Hom}}\nolimits}(M,-)$, the ninth is by naturality of $\underline{\mathsf{coev}}$\ref{['natEvInt']} used here two times and the last is by definition of $u$ in \ref{['defFactU']}.

Theorems & Definitions (116)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • ...and 106 more