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Integration of a categorical operad

Dominik Trnka

TL;DR

This paper develops an operadic Grothendieck construction for non-symmetric operads valued in Cat, producing a 2-category called IntP that is fibered over $Delta_s$. It proves an equivalence between constant-free non-symmetric categorical operads and split-fibered operadic 2-categories over $Delta_s$, with an inverse reconstruction from a split-fibered 2-category. The results unify operadic and order-theoretic structures (e.g., grafting and edge contractions on planar rooted trees) and connect to the classical presheaf/fibration correspondence in the discrete case. The framework sets the stage for generalizations to broader operadic categories and provides tools to combine operadic and poset structures in a coherent 2-categorical setting.

Abstract

We describe a Grothendieck construction for non-symmetric operads with values in categories, and hence in groupoids and posets. The construction produces a 2-category which is operadically fibered over the category D of finite non-empty ordinals and surjections. We describe an inverse for the construction, yielding an equivalence of constant-free non-symmetric categorical operads and operadic 2-categories (split-)fibered over D, which resembles the correspondence of categorical presheaves and fibered categories. The result provides a new characterization of non-symmetric categorical operads and tools to study them.

Integration of a categorical operad

TL;DR

This paper develops an operadic Grothendieck construction for non-symmetric operads valued in Cat, producing a 2-category called IntP that is fibered over . It proves an equivalence between constant-free non-symmetric categorical operads and split-fibered operadic 2-categories over , with an inverse reconstruction from a split-fibered 2-category. The results unify operadic and order-theoretic structures (e.g., grafting and edge contractions on planar rooted trees) and connect to the classical presheaf/fibration correspondence in the discrete case. The framework sets the stage for generalizations to broader operadic categories and provides tools to combine operadic and poset structures in a coherent 2-categorical setting.

Abstract

We describe a Grothendieck construction for non-symmetric operads with values in categories, and hence in groupoids and posets. The construction produces a 2-category which is operadically fibered over the category D of finite non-empty ordinals and surjections. We describe an inverse for the construction, yielding an equivalence of constant-free non-symmetric categorical operads and operadic 2-categories (split-)fibered over D, which resembles the correspondence of categorical presheaves and fibered categories. The result provides a new characterization of non-symmetric categorical operads and tools to study them.

Paper Structure

This paper contains 4 sections, 8 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and Examples
  3. The Equivalence
  4. Closing Remarks

Key Result

proposition 1

Let $\EuScript{P}$ be a non-symmetric categorical operad. There is a strict factorization system on $\int\EuScript{P}$ given by The subcategory $\mathtt{E}$ consists of morphisms $[f;a_1,\ldots,a_k;\alpha]$ with $f=\mathbb{1}$ and all $a_i$'s are the operad unit $e$, and $\mathtt{M}$ consists of morphisms with $\alpha=\mathbb{1}$.

Figures (1)

  • Figure 1: Three isomorphic leveled trees.

Theorems & Definitions (27)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • proposition 1
  • definition 5
  • definition 6
  • definition 7
  • remark 1
  • definition 8
  • ...and 17 more