Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Day convolution for algebraic patterns

Thomas Blom, Félix Loubaton, Jaco Ruit

Abstract

We characterize the exponentiable objects for a wide range of structures prevalent in $\infty$-categorical algebra, extending the construction of Day convolution to more general structures than $\infty$-operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) $\infty$-operads and virtual double $\infty$-categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying $\infty$-category for $\infty$-operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations.

Day convolution for algebraic patterns

Abstract

We characterize the exponentiable objects for a wide range of structures prevalent in -categorical algebra, extending the construction of Day convolution to more general structures than -operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) -operads and virtual double -categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying -category for -operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations.

Paper Structure

This paper contains 43 sections, 79 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. A Conduché criterion for weak Segal fibrations
  3. The underlying graph of an exponential object
  4. Algebrads as complete Segal presheaves on the tree category
  5. Relation to other results
  6. Outline of the paper
  7. Future work
  8. Algebrads
  9. Recollections on algebraic patterns
  10. Algebraic patterns as double categories
  11. Algebrads in the double categorical context
  12. The forest and tree categories of an algebraic pattern
  13. Examples
  14. The slice categories of the forest category
  15. Comparing presheaves on the forest and tree categories
  16. ...and 28 more sections

Key Result

Theorem A

Let $\mathscr{O}$ be an algebraic pattern and $\mathscr{P} \to \mathscr{O}$ an algebrad. Then $\mathscr{P}$ is exponentiable, as an object in $\mathrm{Algad}(\mathscr{O})$, if the following condition is satisfied:

Theorems & Definitions (238)

  • Theorem A
  • Theorem B
  • Remark 1
  • Theorem C
  • Theorem D
  • Definition 1
  • Definition 2: Chu--Haugseng
  • Remark 2
  • Definition 3
  • Example 1
  • ...and 228 more