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Lax formal theory of monads, monoidal approach to bicategorical structures and generalized operads
Abstract
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 bicategorical structures.
... We use this method to show how several 2-monads on the 2-category Cat of small categories and functors can be extended to pseudomonads on the bicategory Prof of small categories and profunctors (also known as bimodules or distributors) [8,45,60]. This result has applications in the theory of variable binding [22,24,55,61], concurrency [13], species of structures [23], models of the differential λ-calculus [21], and operads and multicategories [15,16,17,25,27]. ...
We introduce the notion of a relative pseudomonad, which generalises the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonas on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way, thus providing a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.
... The study of the dichotomy between the theory of enriched categories and internal categories, including unification theories, is still of much interest. For instance, within the more general setting of generalized multicategories, we have the introduction of (T,V )-categories [22] (which generalizes enriched categories), T -categories [14,45] (which generalizes internal categories) and possible unification theories [16,24]. ...
This thesis consists of one introductory chapter and four
single-authored papers written during my PhD studies, with minor adaptations.
The original contributions of the papers are mainly within the study of
pseudomonads and descent objects, including applications
to descent theory, commutativity of weighted bilimits,
coherence and (presentations of) categorical structures.
... The study of the dichotomy between the theory of enriched categories and internal categories, including unification theories, is still of much interest. For instance, within the more general setting of generalized multicategories, we have the introduction of (T,V )-categories [22] (which generalizes enriched categories), T -categories [14,45] (which generalizes internal categories) and possible unification theories [16,24]. ...
... The study of the dichotomy between the theory of enriched categories and internal categories, including unification theories, is still of much interest. For instance, within the more general setting of generalized multicategories, we have the introduction of (T,V )-categories [8] (which generalizes enriched categories), T -categories [4,21] (which generalizes internal categories) and possible unification theories [5,10]. ...
This is the introductory chapter of my PhD Thesis. This thesis consists of one introductory chapter and four single-authored papers written during my PhD studies at the University of Coimbra under supervision of Maria Manuel Clementino. In this first chapter, we give a glance of the scope of our work and briefly describe elements of the original contributions of each paper, including some connections between them. We also give a brief exposition of our main setting, which is 2-dimensional category theory. In this direction: (1) we give an exposition on the doctrinal adjunction, focusing on the Beck-Chevalley condition as used in Chapter "Pseudo-Kan Extensions and Descent" (arXiv: 1606.04999), (2) we apply the results of "On lifting of biadjoints and lax algebras" (arXiv: 1607.03087) in a generalized setting of the formal theory of monads and (3) we apply the biadjoint triangle theorem of "On biadjoint triangles" (aXiv: 1606.05009) to study (pseudo)exponentiable pseudocoalgebras.
... We use this method to show how several 2-monads on the 2-category Cat of small categories and functors can be extended to pseudomonads on the bicategory Prof of small categories and profunctors (also known as bimodules or distributors) [8,42,60]. This result has applications in the theory of variable binding [22,24,55,61], concurrency [13], species of structures [23], models of the differential λ-calculus [21], and operads and multicategories [15][16][17]25,27]. ...
... We use this method to show how several 2-monads on the 2-category Cat of small categories and functors can be extended to pseudomonads on the bicategory Prof of small categories and profunctors (also known as bimodules or distributors) [8,42,60]. This result has applications in the theory of variable binding [22,24,55,61], concurrency [13], species of structures [23], models of the differential λ-calculus [21], and operads and multicategories [15][16][17]25,27]. ...
We introduce the notion of a relative pseudomonad, which generalises the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonas on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way, thus providing a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.
... Example 2.4. Consider the tricategory of category-matrices Mat(Cat) which is a higher dimensional analogue of the bicategory Span described in [Ch15]. The identity morphism 1 X is the collection of categories indexed by the elements of X × X which has the one-object category on the diagonal and the empty category everywhere else. ...
In \cite{LS14} the analogy between the Kleisli construction and the
construction of "warping a skew monoidale category" in the sense of \cite{LS12}
was outlined. In this note we present the same work in a slightly more formal
way.
Multimodal normal incestual systems are investigated in terms of multiple
categories. The different sorted composition of operators are exhibited as
2-cells in multiple categories built up from 2-categories giving rise to
different axioms. Subsequently, coherence results are proved pointing the
connections with (usual and mixed) Distributive Laws. This is given as a
geometrical description of certain axioms inside various systems with a number
of necessity and possibility operators.
Working in the framework of -categories, for a symmetric monoidal closed category V and a (not necessarily cartesian) monad , we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for -categories.
The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic
functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart
to the composition of generalised analytic functors, is also put forward. These definitions encompass most notions of combinatorial
species considered in the literature — including of course Joyal's original notion — together with their associated substitution
operation. Our first main result exhibits the substitution calculus of generalised species as arising from a Kleisli bicategory
for a pseudo-comonad on profunctors. Our second main result establishes that the bicategory of generalised species of structures
is cartesian closed.
The definition of a category of (T, V)-algebras, where V is a unital com-mutative quantale and T is a Set-monad, requires the existence of a certain lax extension of T. In this article, we present a general construction of such an extension. This leads to the formation of two categories of (T, V)-algebras: the category Alg(T, V) of canonical (T, V)-algebras, and the category Alg(T ′ , V) of op-canonical (T, V)-algebras. The usual topological-like examples of categories of (T, V)-algebras (preordered sets, topological, metric and approach spaces) are obtained in this way, and the category of closure spaces appears as a category of canonical (P, V)-algebras, where P is the powerset monad. This unified presentation allows us to study how these categories are related, and it is shown that under suitable hypotheses both Alg(T, V) and Alg(T ′ , V) embed coreflectively into Alg(P, V).
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), one can define
generalized multicategories in a way that unifies all previous examples, while
at the same time simplifying and clarifying much of the theory.
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 denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O + whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I 0+ = I, I (i+1)+ = (I i+ ) + , we call the operations of I (nGamma1)+ the `n-dimensional opetopes'. Opetopes form a category, and presheaves on this category are called `opetopic sets'. A weak n-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street's simplicial approach to weak !-categories. Similarly, starting from an arbitrary operad O instead of I, we define `n-coherent O-algebras', which are n times categorified analogs of algebras of O. Examples include `monoidal n-categories', `stable n-categories', `virtual n-functors' and `representabl...
A general notion of operad is given, which includes as instances, the operads originally conceived to study loop spaces, as well as the higher operads that arise in the globular approach to higher dimensional algebra. In the framework of this paper, one can also describe symmetric and braided analogues of higher operads, likely to be important to the study of weakly symmetric, higher dimensional monoidal structures.
Category theory and related topics of mathematics have been increasingly applied to computer science in recent years. This book contains selected papers from the London Mathematical Society Symposium on the subject which was held at the University of Durham. Participants at the conference were leading computer scientists and mathematicians working in the area and this volume reflects the excitement and importance of the meeting. All the papers have been refereed and represent some of the most important and current ideas. Hence this book will be essential to mathematicians and computer scientists working in the applications of category theory.
Working in the framework of -categories, for a symmetric monoidal closed category V and a (not necessarily cartesian) monad , we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for -categories.
Working in the framework of -categories, for a symmetric monoidal
closed category V and a (not necessarily cartesian) monad T, we present a
common account to the study of ordered compact Hausdorff spaces and stably
compact spaces on one side and monoidal categories and representable
multicategories on the other one. In this setting we introduce the notion of
dual for -categories.
We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as \rel\K, \spn\K, \par\K, and \pro\K for a suitable category \K, along with related constructs such as the \V-\pro arising from a suitable monoidal category \V . We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors F:\eL -> \K induce equipment arrows \rel F:\rel\eL ->\rel\K, \spn F:\spn\eL -> \spn\K, and so on, and similarly for arbitrary monoidal functors \V -> \W. The article I with the title above dealt with those equipments \M having each \M(A,B) only an ordered set, and contained a detailed analysis of the case \M =\rel\K; in the present article we allow the \M(A,B) to be general categories, and illustrate our results by a detailed study of the case \M=\spn\K. We show in particular that \spn is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those \spn G which arise from a geometric morphism G.
We give an explicit description of the free completion EM ( K ) of a 2-category K under the Eilenberg–Moore construction, and show that this has the same underlying category as the 2-category Mnd ( K ) of monads in K . We then demonstrate that much of the formal theory of monads can be deduced using only the universal property of this completion, provided that one is willing to work with EM ( K ) as the 2-category of monads rather than Mnd ( K ) . We also introduce the wreaths in K ; these are the objects of EM ( EM ( K )) , and are to be thought of as generalized distributive laws. We study these wreaths, and give examples to show how they arise in a variety of contexts.
This paper has two objectives. The first is to develop the theory of
bicategories enriched in a monoidal bicategory -- categorifying the classical
theory of categories enriched in a monoidal category -- up to a description of
the free cocompletion of an enriched bicategory under a class of weighted
bicolimits. The second objective is to describe a universal property of the
process assigning to a monoidal category V the equipment of V-enriched
categories, functors, transformations, and modules; we do so by considering,
more generally, the assignation sending an equipment C to the equipment of
C-enriched categories, functors, transformations, and modules, and exhibiting
this as the free cocompletion of a certain kind of enriched bicategory under a
certain class of weighted bicolimits.
The formal theory of monads can be developed in any 2-category, but when it comes to pseudomonads, one is forced to move from 2-categories to Gray-categories (semistrict 3-categories). The first steps in developing a formal theory of pseudomonads have been taken by F. Marmolejo, and here we continue that program. We exhibit a Gray-category Psm such that a Gray-functor from Psm to a Gray-category is precisely a pseudomonad in ; this may be viewed as a complete coherence result for pseudomonads. We then describe the pseudoalgebras for a pseudomonad, the morphisms of pseudoalgebras, and so on, in terms of a weighted limit in the sense of Gray-enriched category theory. We also exhibit a Gray-category Psa such that a Gray-functor from Psa to is precisely a pseudoadjunction in , show that every pseudoadjunction induces a pseudomonad, and that every pseudomonad is induced by a pseudoadjunction, provided that admits the limits referred to above. Finally we define a Gray-category PSM() of pseudomonads in and show that it contains as a full reflective subcategory, which is coreflective if and only if admits these same limits.
We introduce the notion of representable multicategory, which stands in the same relation to that of monoidal category as fibration does to contravariant pseudofunctor (into at). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe represent ability in elementary terms via universal arrows. We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2-category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a several object version of representable multicategories.
We construct the 2-category obtained from a category by freely adjoining a right adjoint for each morphism and isolate its universal property. Some others basic properties are also studied. Some examples in which the category is freely generated by a graph are discussed in detail. For these categories, the 2-cells are given a geometric interpretation and shown to be similar to certain diagrams which have appeared in the literature on C∗-algebras.
In this paper, we give a novel abstract description of Szabo's polycategories. We use the theory of double clubs - a generalisation of Kelly's theory of clubs to 'pseudo' (or 'weak') double categories - to construct a pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the 'two-sided Kleisli bicategory' of this pseudo-distributive law are precisely symmetric polycategories. (c) 2008 Elsevier Inc. All rights reserved.
We generalise the concepts introduced by Baez and Dolan to define opetopes constructed from symmetric operads with a category, rather than a set, of objects. We describe the category of 1-level generalised multicategories, a special case of the concept introduced by Hermida, Makkai and Power, and exhibit a full embedding of this category in the category of symmetric operads with a category of objects. As an analogy to the Baez–Dolan slice construction, we exhibit a certain multicategory of function replacement as a slice construction in the multitopic setting, and use it to construct multitopes. We give an explicit description of the relationship between opetopes and multitopes.
We give a definition of weakn-categories based on the theory of operads. We work with operads having an arbitrary setSof types, or “S-operads,” and given such an operadO, we denote its set of operations by elt(O). Then for anyS-operadOthere is an elt(O)-operadO+whose algebras areS-operads overO. LettingIbe the initial operad with a one-element set of types, and definingI0+=I,I(i+1)+=(Ii+)+, we call the operations ofI(n−1)+the “n-dimensional opetopes.” Opetopes form a category, and presheaves on this category are called “opetopic sets.” A weakn-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street's simplicial approach to weakω-categories. Similarly, starting from an arbitrary operadOinstead ofI, we define “n-coherentO-algebras,” which arentimes categorified analogs of algebras ofO. Examples include “monoidaln-categories,” “stablen-categories,” “virtualn-functors” and “representablen-prestacks.” We also describe hown-coherentO-algebra objects may be defined in any (n+1)-coherentO-algebra.
TABLA DE CONTENIDO
TABLA DE CONTENIDO
Operads and Spaces. The little Cubes Operands. The approximation Theorem. Cofibrations and Quasi-Fibrations. A categorical Constructions.
For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T , V)-algebra and show that various old and new structures are instances of such algebras. Lawvere's presentation of a metric space as a V-category is included in our setting, via the Betti-Carboni-Street-Walters interpretation of a V-category as a monad in the bicategory of V-matrices, and so are Barr's presentation of topological spaces as lax algebras, Lowen's approach spaces, and Lambek's multicategories, which enjoy renewed interest in the study of n-categories. As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category.
Given a 2-category K admitting a calculus of bimodules, and a 2-monad T on it compatible with such calculus, we construct a 2-category L with a 2-monad S on it such that: S has the adjoint-pseudo-algebra property. The 2-categories of pseudo-algebras of S and T are equivalent. Thus, coherent structures (pseudo-T-algebras) are transformed into universally characterised ones (adjoint-pseudo-S-algebras). The 2-category L consists of lax algebras for the pseudo-monad induced by T on the bicategory of bimodules of K. We give an intrinsic characterisation of pseudo- S-algebras in terms of representability . Two major consequences of the above transformation are the classications of lax and strong morphisms, with the attendant coherence result for pseudo-algebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classiers) as well as pseudo-functors into Cat. Contents 1
. We define distributive laws between pseudomonads in a Gray-category A, as the classical two triangles and the two pentagons but commuting only up to isomorphism. These isomorphisms must satisfy nine coherence conditions. We also define the Gray-category PSM(A) of pseudomonads in A, and define a lifting to be a pseudomonad in PSM(A). We define what is a pseudomonad with compatible structure with respect to two given pseudomonads. We show how to obtain a pseudomonad with compatible structure from a distributive law, how to get a lifting from a pseudomonad with compatible structure, and how to obtain a distributive law from a lifting. We show that one triangle suffices to define a distributive law in case that one of the pseudomonads is a (co-)KZ-doctrine and the other a KZ-doctrine. 1. Introduction Distributive laws for monads were introduced by J. Beck in [2]. As pointed out by G. M. Kelly in [7], strict distributive laws for higher dimensional monads are rare. We need then a study ...
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