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Algebras of higher operads as enriched categories II
 In preparation
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the ..."
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of nglobular sets from any normalised (n + 1)operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)category is something like a category enriched in weak ncategories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.
Multitensors and monads on categories of enriched graphs
"... Abstract. In this paper we unify the developments of [Batanin, 1998], [BataninWeber, 2011] and [Cheng, 2011] into a single framework in which the interplay between multitensors on a category V, and monads on the category GV of graphs enriched in V, is taken as fundamental. The material presented he ..."
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Abstract. In this paper we unify the developments of [Batanin, 1998], [BataninWeber, 2011] and [Cheng, 2011] into a single framework in which the interplay between multitensors on a category V, and monads on the category GV of graphs enriched in V, is taken as fundamental. The material presented here is the conceptual background for subsequent work: in [BataninCisinskiWeber, 2012] the Gray tensor product of 2categories and the Crans tensor product [Crans, 1999] of Gray categories are exhibited as existing within our framework, and in [Weber, 2013] the explicit construction of the funny tensor product of categories is generalised to a large class of Batanin operads. 1.
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
Multitensor lifting and strictly unital higher category theory
"... Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories ..."
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Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories and the Crans tensor product of Gray categories as part of this framework. We define weak ncategories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak ncategories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)categories with strict units. 1.
DISTRIBUTIVE LAWS IN PROGRAMMING STRUCTURES
, 2009
"... Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approac ..."
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Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approaches. These applications have always meant leaps in understanding the nature of the subject. However, distributive laws have not yet been given the attention they deserve. One of the reasons for this omission is certainly the lack of a formal notion of distributive laws in their full generality. This hinders the discovery and formal description of occurrences of distributive laws, which is the precursor of any formal manipulation. In this thesis, an approach to formalisation of distributive laws is presented based on the functorial approach to formal Category Theory pioneered by Lawvere and others, notably Gray. The proposed formalism discloses a rather simple nature of distributive laws of the kind found in programming structures based on lax 2naturality and Gray’s tensor product of 2categories. It generalises the existing more specific notions of distributive
ON THE ITERATION OF WEAK WREATH PRODUCTS
"... Based on a study of the 2category of weak distributive laws, we describe a method of iterating Street’s weak wreath product construction. That is, for any 2category K and for any nonnegative integer n, we introduce 2categories Wdl (n) (K), of (n + 1)tuples of monads in K pairwise related by wea ..."
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Based on a study of the 2category of weak distributive laws, we describe a method of iterating Street’s weak wreath product construction. That is, for any 2category K and for any nonnegative integer n, we introduce 2categories Wdl (n) (K), of (n + 1)tuples of monads in K pairwise related by weak distributive laws obeying the YangBaxter equation. The first instance Wdl (0) (K) coincides with Mnd(K), the usual 2category of monads in K, and for other values of n, Wdl (n) (K) contains Mnd n+1 (K) as a full 2subcategory. For the local idempotent closure K of K, extending the multiplication of the 2monad Mnd, we equip these 2categories with n possible ‘weak wreath product ’ 2functors Wdl (n) (K) → Wdl (n−1) (K), such that all of their possible nfold composites Wdl (n) (K) → Wdl (0) (K) are equal; that is, such that the weak wreath product is ‘associative’. Whenever idempotent 2cells in K split, this leads to pseudofunctors Wdl (n) (K) → Wdl (n−1) (K) obeying the associativity property upto isomorphism. We present a practically important occurrence of an iterated weak wreath product: the algebra of observable quantities in an Ising type quantum spin chain where the spins take their values in a dual pair of finite weak Hopf algebras. We also construct a fully faithful embedding of Wdl (n) (K) into the 2category of commutative n + 1 dimensional cubes in Mnd(K) (hence into the 2category of commutative n + 1 dimensional cubes in K whenever K has EilenbergMoore objects and its idempotent 2cells split). Finally we give a sufficient and necessary condition on a monad in K to be isomorphic to an nary weak wreath product.