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Generalized enrichment of categories
 Also Journal of Pure and Applied Algebra
, 1999
"... We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmultica ..."
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We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmulticategory extends the (more or less wellknown) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fcmulticategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper is elementary and selfcontained, we also explain why, from one point of view, fcmulticategories are the natural structures in which to enrich categories.
Customizing an XML–Haskell data binding with type isomorphism inference in Generic Haskell
"... Customizing an XML–Haskell data binding ..."
A PREHISTORY OF nCATEGORICAL PHYSICS
, 2008
"... We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, me ..."
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We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, membranes and spin foams.
Comparing operadic theories of ncategory
, 2008
"... We give a framework for comparing on the one hand theories of ncategories that are weakly enriched operadically, and on the other hand ncategories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (ChengGurski) and examples of ..."
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We give a framework for comparing on the one hand theories of ncategories that are weakly enriched operadically, and on the other hand ncategories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (ChengGurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory of ncategories of the former kind and produce a globular operad whose algebras are the ncategories we started with. We first provide a generalisation of Trimble’s original theory that allows for the use of other parametrising operads in a very general way, via the notion of categories weakly enriched in V where the weakness is parametrised by an operad P in the category V. We define weak ncategories by iterated weak enrichment using a series of parametrising operads Pi. We then show how to construct from such a theory an ndimensional globular operad for each n ≥ 0 whose algebras
Comparing definitions of weak higher categories, I
, 2009
"... The theory of operads, defined through categories of labeled graphs, is generalized to suit definitions of higher categories with arbitrary basic shapes. Constructions of cubical, globular and opetopic weak higher categories are obtained as examples. 1 ..."
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The theory of operads, defined through categories of labeled graphs, is generalized to suit definitions of higher categories with arbitrary basic shapes. Constructions of cubical, globular and opetopic weak higher categories are obtained as examples. 1
2DENDROIDAL SETS
"... that the objects of Ω are trees and a morphism t → t ′ is an operad map from Ω(t) to Ω(t ′), that is, from the operad generated by the vertices of t to that generated by the vertices of t ′. [Globular pasting diagrams are pasting diagrams which correspond to trees with height. Note that the two sort ..."
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that the objects of Ω are trees and a morphism t → t ′ is an operad map from Ω(t) to Ω(t ′), that is, from the operad generated by the vertices of t to that generated by the vertices of t ′. [Globular pasting diagrams are pasting diagrams which correspond to trees with height. Note that the two sorts of trees just mentioned are not directly related. For one thing there are two sorts of tree composition around, one the ordinary grafting, and the other the special composition that reflects composition of pasting diagrams. We won’t draw the trees with height unless they make definitions or proofs more efficient. Sources are Batanin [4] and Leinster [13]. For examples of pasting diagrams together with their trees see page 8 of [6] at
Operads of cellular automata and little ncubes
, 2005
"... We demonstrate how to organize 1dimensional cellular automata into an operad of spaces. The nth term C(k) is the space of radius r = k − 1 automata. The operad composition operation involves both automata composition and shifting of domain. Pointwise operations such as addition of automata become i ..."
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We demonstrate how to organize 1dimensional cellular automata into an operad of spaces. The nth term C(k) is the space of radius r = k − 1 automata. The operad composition operation involves both automata composition and shifting of domain. Pointwise operations such as addition of automata become important when we look at the structure of the individual terms in the operad, the spaces of automata with a given radius. Having adopted the discrete topology on such a space, we demonstrate an action of the little ncubes operad on (n − 1)dimensional radius r automata. There are clear applications of this action to parallel programming issues. Finally we discuss ways of generalizing both the idea of an operad of automata and the ncubes action to higher dimensional automata, using higher dimensional operads.
A Syntactical Approach to Weak ωGroupoids
"... Abstract—When moving to a Type Theory without proof irrelevance the notion of a setoid has to be generalized to the notion of a weak ωgroupoid. As a first step in this direction we study the formalisation of weak ωgroupoids in Type Theory. This is motivated by Voevodsky’s proposal of univalent typ ..."
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Abstract—When moving to a Type Theory without proof irrelevance the notion of a setoid has to be generalized to the notion of a weak ωgroupoid. As a first step in this direction we study the formalisation of weak ωgroupoids in Type Theory. This is motivated by Voevodsky’s proposal of univalent type theory which is incompatible with proofirrelevance and the results by Lumsdaine and Garner/van de Berg showing that the standard eliminator for equality gives rise to a weak ωgroupoid.