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Koszul duality for operadic categories

Michael Batanin, Martin Markl

Abstract

The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of operads over a large class of operadic categories. In particular, for these operadic categories we will study concrete examples of binary quadratic operads, describe their Koszul duals and prove that they are Koszul. This includes operads whose algebras are the most important operad- and PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1/2PROPs, and still more exotic objects such as permutads and pre-permutads.

Koszul duality for operadic categories

Abstract

The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of operads over a large class of operadic categories. In particular, for these operadic categories we will study concrete examples of binary quadratic operads, describe their Koszul duals and prove that they are Koszul. This includes operads whose algebras are the most important operad- and PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1/2PROPs, and still more exotic objects such as permutads and pre-permutads.

Paper Structure

This paper contains 20 sections, 29 theorems, 178 equations.

Table of Contents

  1. Recollections
  2. The Axioms
  3. Markl operads
  4. Markl operads and virtual isomorphisms
  5. Free Markl operads
  6. Chains of elementary morphisms
  7. Free operads
  8. Quadratic Markl operads and duality
  9. Modular and odd modular operads
  10. Other operad-like structures
  11. Cyclic operads
  12. Ordinary operads
  13. Pre-permutads.
  14. PROP-like structures and permutads
  15. Wheeled properads
  16. ...and 5 more sections

Key Result

Lemma 1.1

Consider the commutative diagram in an operadic category \xymatrix@C=4em{S' \ar[d]_{f'}\ar[dr]^{f} \ar[r]^\pi & S'' \ar[d]^{f''} \\ T' \ar[r]^\sigma &\ T''\,. }Let $j\in |T"|$ and $|\sigma|^{-1}(j) = \{i\}$ for some $i\in |T'|$. Diagram (s1) determines: If ${\sigma}^{-1}(j)$ is trivial, in particular if $\sigma$ is a quasi-bijection, then $\pi$ induces a map which is a quasi-bijection if $\pi$

Theorems & Definitions (86)

  • Lemma 1.1: Lemma 2.4 of part1
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5: Lemma 5.5 of part1
  • Definition 1.6
  • Corollary 1.7: Corollary 5.8 of part1
  • Definition 1.8
  • Definition 1.9
  • Lemma 2.1
  • proof
  • ...and 76 more