Corpus ID: 119618908

# Regular patterns, substitudes, Feynman categories and operads

@article{Batanin2015RegularPS,
title={Regular patterns, substitudes, Feynman categories and operads},
author={M. Batanin and Joachim Kock and Mark Weber},
journal={arXiv: Category Theory},
year={2015}
}
• Published 2015
• Mathematics, Physics
• arXiv: Category Theory
We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the 2-category of coloured operads (with invertible 2-cells). These biequivalences induce equivalences between the corresponding categories of algebras. There are three main ingredients in establishing these biequivalences. The first is a strictification theorem… Expand
15 Citations
Homotopy theory of algebras of substitudes and their localisation.
• Mathematics
• 2020
We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler) equipped with a (semi)model structure lifted from the model structure on theExpand
• Mathematics
• 2018
Our aim is to set up the cornerstones of Koszul duality in general operadic categories. In particular, we will prove that operads (in our generalized sense) governing the most important operad-Expand
Feynman Categories
• Mathematics, Physics
• 2017
In this paper we give a new foundational categorical formulation for operations and relations and objects parameterizing them. This generalizes operads and all their cousins including but not limitedExpand
A criterion for existence of right-induced model structures
• Mathematics
• 2017
Suppose that $F: \mathcal{N} \to \mathcal{M}$ is a functor whose target is a Quillen model category. We give a succinct sufficient condition for the existence of the right-induced model categoryExpand
$\infty$-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, andExpand
Algebraic Kan extensions along morphisms of internal algebra classifiers
Abstract An \algebraic left Kan extension" is a left Kan extension which interacts well with the alge- braic structure present in the given situation, and these appear in various subjects such as theExpand
C T ] 1 1 M ay 2 02 1 KOSZUL DUALITY FOR OPERADIC CATEGORIES
• 2021
The aim of this sequel to [10] is to set up the cornerstones of Koszul duality and Koszulity in the context of a large class of operadic categories. In particular, we will prove that operads, in theExpand
The equivalence between Feynman transform and Verdier duality
The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality toExpand
Deformation theory of Cohomological Field Theories
• Mathematics, Physics
• 2020
We develop the deformation theory of cohomological field theories (CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduceExpand
Comprehensive factorisation systems
• Mathematics
• 2017
Abstract We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categoriesExpand

#### References

SHOWING 1-10 OF 37 REFERENCES
ALGEBRAIC KAN EXTENSIONS IN DOUBLE CATEGORIES
We study Kan extensions in three weakenings of the Eilenberg-Moore dou- ble category associated to a double monad, that was introduced by Grandis and Par e. To be precise, given a normal oplax doubleExpand
• Mathematics
• 2003
in which moving down along a side of gradient 1 imposes invertibility on constraints, while moving down along a side of gradient Ð1 imposes representability on the multihoms. A strong form ofExpand
The Eckmann–Hilton argument and higher operads
Abstract The classical Eckmann–Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative.Expand
Feynman Categories
• Mathematics, Physics
• 2017
In this paper we give a new foundational categorical formulation for operations and relations and objects parameterizing them. This generalizes operads and all their cousins including but not limitedExpand
On a bicomma object condition for KZ-doctrines
• Mathematics
• 1999
Abstract We study Kock–Zoberlein doctrines that satisfy a certain bicomma object condition. Such KZ-doctrines we call admissible. Our investigation is mainly motivated by the example of the symmetricExpand
Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes
• Mathematics
• 1997
Abstract We give a definition of weak n -categories based on the theory of operads. We work with operads having an arbitrary set S of types, or “ S -operads,” and given such an operad O , we denoteExpand
The cartesian closed bicategory of generalised species of structures
• Mathematics
• 2008
AbstractThe concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operationExpand
Homotopy theory for algebras over polynomial monads
• Mathematics
• 2013
We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as forExpand
Operadic categories and Duoidal Deligne's conjecture
• Mathematics
• 2014
The purpose of this paper is two-fold. In Part 1 we introduce a new theory of operadic categories and their operads. This theory is, in our opinion, of an independent value. In Part 2 we use thisExpand
Algebraic Kan extensions along morphisms of internal algebra classifiers
Abstract An \algebraic left Kan extension" is a left Kan extension which interacts well with the alge- braic structure present in the given situation, and these appear in various subjects such as theExpand