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Dioperads, Frobenius monoidal functors and duality

Valerio Melani, Hugo Pourcelot

Abstract

Motivated by duality phenomena for derived global sections on derived local systems on compact oriented manifolds, we introduce the notion of a $d$-duality context between symmetric monoidal enriched categories. In this setting, the right adjoint of a symmetric monoidal functor carries compatible lax and colax structures twisted by an invertible object $d$. For any enriched dioperad $\mathcal{P}$, we define a $d$-twist $\mathcal{P}\{d\}$ and prove that, in a $d$-duality context, the right adjoint sends $\mathcal{P}$-algebras to $\mathcal{P}\{-d\}$-algebras. To achieve this, the key conceptual result is that Frobenius monoidal functors between symmetric monoidal categories are precisely those functors inducing morphisms between the underlying dioperads. We also develop a dioperadic Day convolution, yielding an alternative proof of the main theorem and suggesting an $\infty$-categorical extension of the theory.

Dioperads, Frobenius monoidal functors and duality

Abstract

Motivated by duality phenomena for derived global sections on derived local systems on compact oriented manifolds, we introduce the notion of a -duality context between symmetric monoidal enriched categories. In this setting, the right adjoint of a symmetric monoidal functor carries compatible lax and colax structures twisted by an invertible object . For any enriched dioperad , we define a -twist and prove that, in a -duality context, the right adjoint sends -algebras to -algebras. To achieve this, the key conceptual result is that Frobenius monoidal functors between symmetric monoidal categories are precisely those functors inducing morphisms between the underlying dioperads. We also develop a dioperadic Day convolution, yielding an alternative proof of the main theorem and suggesting an -categorical extension of the theory.

Paper Structure

This paper contains 37 sections, 15 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Recollections on properads and dioperads
  4. Graphs
  5. Definition of properads and dioperads
  6. Polynatural transformations
  7. Underlying properad and dioperad of a symmetric monoidal category
  8. Envelopes
  9. Properadic envelope of a dioperad
  10. Symmetric monoidal envelope of a properad
  11. Symmetric monoidal envelope of a dioperad
  12. Frobenius monoidal functors as morphisms of dioperads
  13. Frobenius monoidal structures
  14. Characterization of Frobenius monoidal functors
  15. Proof of the properadic statement.
  16. ...and 22 more sections

Key Result

Theorem A

Let $\zP$ be a dioperad. Given a $d$-duality context $F \colon \zD \rightleftharpoons\zC \colon G$, the right adjoint $G$ sends $\zP$-algebras in $\zC$ to $\zP\{-d\}$-algebras in $\zD$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Two morphisms $\sigma$ and $\tau$ in $\zE^p\zP$, represented as union of colored corollas, and their composition obtained by grafting and contraction of inner edges. Here the operation $\rho$ is obtained by the appropriate properadic composition of $\sigma_a$, $\sigma_b$, $\tau_e$ and $\tau_f$ in $\zP$.

Theorems & Definitions (55)

  • Theorem A: Cor. \ref{['cor:main']}
  • Theorem B: Thm. \ref{['thm:frob_as_diop_map']}
  • Theorem C: Thm. \ref{['thm:day']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 45 more