Results 1  10
of
22
Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
Abstract

Cited by 74 (6 self)
 Add to MetaCart
We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.
Parallel transport and functors
"... Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. Th ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to substitute categories of functors for categories of smooth fibre bundles with connection. We indicate that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport. Table of Contents
From Coherent Structures to Universal Properties
 J. Pure Appl. Algebra
, 1999
"... Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coh ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coherent structures (pseudoTalgebras) are transformed into universally characterised ones (adjointpseudoSalgebras). The 2category L consists of lax algebras for the pseudomonad induced by T on the bicategory of bimodules of K. We give an intrinsic characterisation of pseudoSalgebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudoalgebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudofunctors into Cat.
Specifying Interaction Categories
, 1997
"... We analyse two complementary methods for obtaining models of typed process calculi, in the form of interaction categories. These methods allow adding new features to previously captured notions of process and of type, respectively. By combining them, all familiar examples of interaction categories, ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
We analyse two complementary methods for obtaining models of typed process calculi, in the form of interaction categories. These methods allow adding new features to previously captured notions of process and of type, respectively. By combining them, all familiar examples of interaction categories, as well as some new ones, can be built starting from some simple familiar categories. Using the presented constructions, interaction categories can be analysed without fixing a set of axioms, merely in terms of the way in which they are specified  just like algebras are analysed in terms of equations and relations, independently on abstract characterisations of their varieties.
Higher gauge theory I: 2Bundles
 University of California Riverside
"... Stevenson for helpful discussion about covers. And of course I thank John Baez for all of the above, as well as inspiration, guidance, and encouragement. iv Abstract of the Dissertation ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Stevenson for helpful discussion about covers. And of course I thank John Baez for all of the above, as well as inspiration, guidance, and encouragement. iv Abstract of the Dissertation
Higher gauge theory
"... I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to ce ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to certain 2categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2category of 2bundles over a given 2space under a given 2group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2space is the 2space corresponding to a given space and the 2group is the automorphism 2group of a given group, then this 2category is equivalent to the 2category of gerbes over that space under that group (being described by the same cohomological data).
Homotopy algebras for operads
"... We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy Palgebra in M is, provided only that some of the morphisms in M have been marked out as ‘homotopy equivalences’. The bulk of the paper consists of examples of homotopy algebras. We show that any loop space is a homotopy monoid, and, in fact, that any nfold loop space is an nfold homotopy monoid in an appropriate sense. We try to compare weakened algebraic structures such as A∞spaces, A∞algebras and nonstrict monoidal categories to our homotopy algebras, with varying degrees of success. We also prove results on ‘change of base’, e.g. that the classifying space of a homotopy monoidal category is a homotopy topological monoid. Finally, we
First Order Logic with Dependent Sorts, with Applications to Category Theory. Book manuscript
, 1995
"... §1. Logic with dependent sorts p. 14 §2. Formal systems p. 32 §3. Quantificational fibrations p. 39 ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
§1. Logic with dependent sorts p. 14 §2. Formal systems p. 32 §3. Quantificational fibrations p. 39
Twisted differential nonabelian cohomology Twisted (n−1)brane nbundles and their ChernSimons (n+1)bundles with characteristic (n + 2)classes
, 2008
"... We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shif ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian ngroup B n−1 U(1). Notable examples are String 2bundles [9] and Fivebrane 6bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spinstructures to Stringstructures [13] and further to Fivebranestructures [133, 52], are abelian ChernSimons 3 and 7bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞Lieintegrating the L∞algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2 and twisted Fivebrane 6bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted Ktheory. We explain the GreenSchwarz mechanism in heterotic string theory in terms of twisted String 2bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6bundles. We close by transgressing differential cocycles to mapping
LOCALLY CARTESIAN CLOSED CATEGORIES WITHOUT CHOSEN CONSTRUCTIONS
"... Abstract. We show how to formulate the notion of locally cartesian closed category without chosen pullbacks, by the use of Makkai’s theory of anafunctors. 1. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We show how to formulate the notion of locally cartesian closed category without chosen pullbacks, by the use of Makkai’s theory of anafunctors. 1.