# Homotopy-coherent algebra via Segal conditions

@article{Chu2021HomotopycoherentAV,
title={Homotopy-coherent algebra via Segal conditions},
author={Hongyi Chu and Rune Haugseng},
year={2021}
}
• Published 9 July 2019
• Mathematics

### $$\infty$$-Operads via symmetric sequences

We construct a generalization of the Day convolution tensor product of presheaves that works for certain double $$\infty$$ ∞ -categories. Using this construction, we obtain an $$\infty$$ ∞

### $\infty$-operads as symmetric monoidal $\infty$-categories

• Mathematics
• 2021
. We use Lurie’s symmetric monoidal envelope functor to give two new descriptions of ∞ -operads: as certain symmetric monoidal ∞ -categories whose underlying symmetric monoidal ∞ -groupoids are free,

### Free algebras through Day convolution

• Mathematics
• 2020
Building on the foundations in our previous paper, we study Segal conditions that are given by finite products, determined by structures we call cartesian patterns. We set up Day convolution on

### On rectification and enrichment of infinity properads

• Mathematics
Journal of the London Mathematical Society
• 2022
We develop a theory of infinity properads enriched in a general symmetric monoidal infinity category. These are defined as presheaves, satisfying a Segal condition and a Rezk completeness condition,

### Envelopes for Algebraic Patterns

• Mathematics
• 2022
We generalize Lurie’s construction of the symmetric monoidal envelope of an ∞-operad to the setting of algebraic patterns. This envelope becomes fully faithful when sliced over the envelope of the

A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an

### Categories of graphs for operadic structures

We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalized operads (e.g. cyclic operads, modular operads, properads, and so on), and give new,

### Segal conditions for generalized operads

. This note is an introduction to several generalizations of the dendroidal sets of Moerdijk–Weiss. Dendroidal sets are presheaves on a category of rooted trees, and here we consider indexing

### Twisted arrow categories, operads and Segal conditions

We introduce twisted arrow categories of operads and of algebras over operads. Up to equivalence of categories, the simplex category ∆ , Segal’s category Γ , Connes cyclic category Λ , Moerdijk–Weiss

• Mathematics
• 2021
We extend Bourke and Garner’s idempotent adjunction between monads and pretheories to the framework of ∞-categories and we use this to prove many classical results about monads in the∞-categorical

## References

SHOWING 1-10 OF 40 REFERENCES

### From operator categories to higher operads

In this paper we introduce the notion of an operator category and two different models for homotopy theory of $\infty$-operads over an operator category -- one of which extends Lurie's theory of

### On the equivalence between Θ_{}-spaces and iterated Segal spaces

We give a new proof of the equivalence between two of the main models for $(\infty,n)$-categories, namely the $n$-fold Segal spaces of Barwick and the $\Theta_{n}$-spaces of Rezk, by proving that

• Mathematics
Algebraic & Geometric Topology
• 2019
We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $\Xi$ of trees, which carries a tight

### Stratified categories, geometric fixed points and a generalized Arone-Ching theorem

We develop a theory of Mackey functors on epiorbital categories which simultaneously generalizes the theory of genuine $G$-spectra for a finite group $G$ and the theory of $n$-excisive functors on

### A Cellular Nerve for Higher Categories

Abstract We realise Joyal' cell category Θ as a dense subcategory of the category of ω-categories. The associated cellular nerve of an ω-category extends the well-known simplicial nerve of a small

### Dendroidal Segal spaces and ∞‐operads

• Mathematics
• 2013
We introduce the dendroidal analogues of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the

### Familial 2-functors and parametric right adjoints

We define and study familial 2-functors primarily with a view to the devel- opment of the 2-categorical approach to operads of (Weber, 2005). Also included in this paper is a result in which the