• Corpus ID: 7542271

A unified framework for generalized multicategories

@article{Cruttwell2009AUF,
  title={A unified framework for generalized multicategories},
  author={Geoff S. H. Cruttwell and Michael Shulman},
  journal={arXiv: Category Theory},
  year={2009}
}
Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the "lax algebras" or "Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified… 
View PDF on arXiv
59 Citations
Highly Influential Citations
13
Background Citations
31
Methods Citations
11
Results Citations
3

Tables from this paper

59 Citations

LAX FORMAL THEORY OF MONADS, MONOIDAL APPROACH TO BICATEGORICAL STRUCTURES AND GENERALIZED OPERADS

Generalized operads, also called generalized multicategories and T -monoids, are dened as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads…

Lax monads, equipments and generalized multicategory theory

Generalized multicategories, also called $T$-monoids, are well known class of mathematical structures, which include diverse set of examples. In this paper we construct a generalization of the…

Left Adjoints for Generalized Multicategories

We construct generalized multicategories associated to an arbitrary operad in Cat that is $\Sigma$-free. The construction generalizes the passage to symmetric multicategories from permutative…

Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures

We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient…

Fibered Multicategory Theory

Given a fibration in groupoids d ∶ D → I, we define a fibered multicategory as a particular functor p ∶ M → I, whereM has the same objects as D, and its arrows a ∶X → Y should be thought of as…

Kleisli enriched

Foundations of Algebraic Theories and Higher Dimensional Categories.

Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields…

Regular patterns, substitudes, Feynman categories and operads

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…

Multivariable adjunctions and mates

We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2-variable adjunction is the…

Globular Multicategories with Homomorphism Types

We introduce a notion of globular multicategory with homomorphism types. These structures arise when organizing collections of "higher category-like" objects such as type theories with identity…
...

References

SHOWING 1-10 OF 92 REFERENCES

Metric, topology and multicategory—a common approach

Generalized enrichment of categories

Operads in Higher-Dimensional Category Theory

The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak n-category. Included is a full…

Higher Operads, Higher Categories

Part I. Background: 1. Classical categorical structures 2. Classical operads and multicategories 3. Notions of monoidal category Part II. Operads. 4. Generalized operads and multicategories: basics…

Weak n -categories: opetopic and multitopic foundations

Framed bicategories and monoidal fibrations

In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors,…

One Setting for All: Metric, Topology, Uniformity, Approach Structure

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T, (T,V)-algebras are considered and a network of functors at the general level which describe the basic connections between the structures are established.

Rings, modules, and algebras in infinite loop space theory

From coherent structures to universal properties

Extensions in the theory of lax algebras

Recent investigations of lax algebras—in generalization of Barr’s relational algebras—make an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and V-relations…
...