Notions of `operad' and `multicategory' abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term `(S,*)-multicategory', subject to certain conditions on S and *. Different choices of S and * give some of the existing notions of operad and multicategory. We then describe the `algebras' for an (S,*)-multicategory and, finally, present a tentative selection of further developments. Our approach makes possible concise descriptions of Baez and Dolan's opetopes and Batanin's operads; both of these are included.
In this paper we answer the question: `what kind of a structure can a general multicategory be enriched in?' The answer is, in a sense to be made precise, that a multicategory of one type can be enriched in a multicategory of the type one level up. In the case of ordinary categories this reduces to something surprising: a category may be enriched in an `fc-multicategory', a very general kind of 2-dimensional structure encompassing monoidal categories, plain multicategories, bicategories and double categories. (It turns out that fc-multicategories also provide a natural setting for the bimodule…
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 adjunction between strict monoidal categories and multicategories, where the latter are replaced by $T$-monoids. To do this we introduce lax monads in a 3-category, and establish their relationship with equipments, which are bicategory like structures appropriate for the generalized multicategory theory.
We define the phrase `category enriched in an fc-multicategory' and explore some examples. An fc-multicategory is a very general kind of 2-dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fc-multicategory extends the (more or less well-known) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fc-multicategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper …
We construct generalized multicategories associated to an arbitrary operad in Cat that is $Σ$-free. The construction generalizes the passage to symmetric multicategories from permutative categories, which is the case when the operad is the categorical version of the Barratt-Eccles operad. The main theorem is that there is an adjoint pair relating algebras over the operad to this sort of generalized multicategory. The construction is flexible enough to allow for equivariant generalizations.
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 types. We show how these globular multicategories can be used to construct various weak higher categorical structures of types and terms.
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 explanation of why the proposed definition of n-category is a reasonable one, and of what happens when n=2. Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to n-categories expounded by Baez, Dolan and others, and are a natural language in which to discuss enr…
The purpose of this paper is to study the derived category of simplicial multicategories with arbitrary sets of objects (also known as, colored operads in simplicial sets). Our main result is a derived Morita theory for operads-where we describe the derived mapping spaces between two multicategories P and Q in terms of the nerve of a certain category of P-Q-bimodules. As an application, we show that the derived category possesses internal Hom-objects.
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 hom/tensor/cotensor trio of functors; we generalise this situation to n+1 functors of n variables. Furthermore, we generalise the mates correspondence, which enables us to pass between natural transformations involving left adjoints to those involving right adjoints. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of "…
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 framework: by working with monads on double categories and related structures (rather than bicategories),…
Generalized operads, also called generalized multicategories and $T$-monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including symmetric multicategories, topological spaces, globular operads and Lawvere theories. In this paper we study functoriality of the Kleisli construction, and correspondingly that of generalized operads. Motivated by this problem we develop a lax version of the formal theory of monads, and study its connection to …
The subject of this paper is the higher structure of the strictification adjunction, which relates the two fundamental bases of three-dimensional category theory: the $\mathbf{Gray}$-category of $2$-categories and the tricategory of bicategories. We show that -- far from requiring the full weakness provided by the definitions of tricategory theory -- this adjunction can be $\textit{strictly}$ enriched over the symmetric closed multicategory of bicategories defined by Verity. Moreover, we show that this adjunction underlies an adjunction of bicategory-enriched symmetric multicategories. An appe…
We show how non-symmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo-)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day's tensor products, or via analytical functors) fit with the profunctor approach.
We propose a new model for multicategories with symmetries with respect to Zhang's group operads. The fully faithful embedding of the category of group operads into that of crossed interval groups is made use of, and it is shown that every multicategory gives rise to a fibration, in a sense, over a quotient of the total category of group operads. The symmetric structures can be presented as structures of internal presheaves over a category internal to the category of small categories, in other words a double category.
We develop a new definition of opetopic sets. There are two main technical ingredients. The first is the systematic use of fibrations, which are implicit in most of the approaches in the literature. Their explicit use leads to certain clarifications in the construction of opetopic sets and other constructions. The second is the "web monoid", which plays a role analogous to the "operad for operads" of Baez and Dolan, the "multicategory of function replacement" of Hermida, Makkai and Power. We demonstrate that the web monoid is closely related to the "Baez-Dolan slice construction" as defined by…
We give an introduction to the topics of our forthcoming work, in which we introduce and study new mathematical objects which we call "higher theories" of algebras, where inspiration for the term comes from William Lawvere's notion of "algebraic theory". Indeed, our "theories" are `higher order' generalizations of coloured operad or multicategory, where we see an operad as analogous to Lawvere's theory. Higher theories are obtained by iterating a certain process, which we call "theorization", generalizing categorification in the sense of Louis Crane. The hierarchy of all iterated theorizations…
