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Monad interleaving: a construction of the operad for Leinster’s weak ωcategories, Preprint, 2003, available at http://arxiv.org/abs/math/0309336
"... We show how to “interleave ” the monad for operads and the monad for contractions on the category Coll of collections, to construct the monad for the operadswithcontraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions ea ..."
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We show how to “interleave ” the monad for operads and the monad for contractions on the category Coll of collections, to construct the monad for the operadswithcontraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions each of which acts on only one dimension of the underlying globular sets at a time. We then exhibit mutual stability conditions that enable us to alternate the dimensionbydimension free functors. Hence we give an explicit construction of a left adjoint
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
Customizing an XML–Haskell data binding with type isomorphism inference in Generic Haskell
"... Customizing an XML–Haskell data binding ..."
Duality For Simple omegaCategories And Disks
"... A. Joyal [J] has introduced the category D of the socalled finite disks, and used it to define the concept of #category, a notion of weak #category. We introduce the notion of an #graph being composable (meaning roughly that 'it has a unique composite'), and call an #category simple if it is fr ..."
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A. Joyal [J] has introduced the category D of the socalled finite disks, and used it to define the concept of #category, a notion of weak #category. We introduce the notion of an #graph being composable (meaning roughly that 'it has a unique composite'), and call an #category simple if it is freely generated by a composable #graph. The category S of simple #categories is a full subcategory of the category, with strict #functors as morphisms, of all #categories. The category S is a key ingredient in another concept of weak #category, called protocategory [MM1], [MZ]. We prove that D and S are contravariantly equivalent, by a duality induced by a suitable schizophrenic object living in both categories. In [MZ], this result is one of the tools used to show that the concept of #category and that of protocategory are equivalent in a suitable sense. We also prove that composable #graphs coincide with the #graphs of the form T # considered by M.Batanin [B], which were characterized by R. Street (as announced in [S]) and called `globular cardinals'. Batanin's construction, using globular cardinals, of the free #category on a globular set plays an important role in our paper. We give a selfcontained presentation of Batanin's construction that suits our purposes.
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.
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
Combinatorial nfold monoidal categories and nfold operads
, 2008
"... Operads were originally defined as Voperads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad ..."
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Operads were originally defined as Voperads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. A sequence of categorical types that filter the category of monoidal categories and monoidal functors is given by Balteanu, Fiedorowicz, Schwänzl and Vogt in [2]. These subcategories of MonCat have objects that are called nfold monoidal categories. A k– fold monoidal category is nfold monoidal for all n ≤ k, and a symmetric monoidal category is nfold monoidal for all n. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce the lower branches of an extended family tree of simple examples. Then we generalize the definition of operad by defining nfold operads and their algebras in an iterated monoidal category. It is seen that the interchanges in an iterated monoidal category are the natural requirement for expressing operad associativity. The definition is developed from the starting point of iterated monoids in a category of collections. Since monoids are special cases of enriched categories this allows us to describe the iterated monoidal and higher dimensional categorical structure of iterated operads. We show that for V kfold monoidal the structure of a (k − n)fold monoidal strict ncategory is possessed by the category of nfold operads in V. We discuss examples of these operads that live in the previously described categories. Finally we describe the algebras of nfold Voperads and their products.