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A prop structure on partitions

Coline Emprin, Dana Hunter, Muriel Livernet, Christine Vespa, Inna Zakharevich

Abstract

Motivated by its link with functor homology, we study the prop freely generated by the operadic suspension of the operad Com. We exhibit a particular family of generators, for which the composition and the symmetric group actions admit simple descriptions. We highlight associated subcategories of its Karoubi envelope which allows us to compute extensions groups between simple functors from free groups. We construct a particular prop structure on partitions whose composition corresponds to the Yoneda product of extensions between exterior power functors.

A prop structure on partitions

Abstract

Motivated by its link with functor homology, we study the prop freely generated by the operadic suspension of the operad Com. We exhibit a particular family of generators, for which the composition and the symmetric group actions admit simple descriptions. We highlight associated subcategories of its Karoubi envelope which allows us to compute extensions groups between simple functors from free groups. We construct a particular prop structure on partitions whose composition corresponds to the Yoneda product of extensions between exterior power functors.
Paper Structure (8 sections, 9 theorems, 98 equations)

This paper contains 8 sections, 9 theorems, 98 equations.

Table of Contents

  1. A graded linear prop spanned by surjections
  2. Recollections on props
  3. The prop freely generated by the operadic suspension of $\mathcal{C}om$
  4. On subcategories of the Karoubi envelope of a prop
  5. A graded linear prop spanned by partitions
  6. The composition in the category $\mathcal{E}_{\Lambda}$
  7. A particular prop structure
  8. Relation with functor homology on free groups

Key Result

Theorem 1.7

Let $\mathcal{E}$ be the graded linear prop $\Omega s\mathcal{C}om$.

Theorems & Definitions (25)

  • Definition 1.1: Graded linear prop
  • Definition 1.4: Freely generated prop
  • Theorem 1.7
  • proof
  • Definition 2.1: Karoubi envelope
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 15 more