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Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n < ..."
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Cited by 38 (2 self)
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
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.
SYMMETRY AND CAUCHY COMPLETION OF QUANTALOIDENRICHED CATEGORIES
"... Abstract. We formulate an elementary condition on an involutive quantaloid Q under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Qenriched categories. For such quantaloids, which we call Cauchybilateral quantaloids, it follows ..."
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Abstract. We formulate an elementary condition on an involutive quantaloid Q under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Qenriched categories. For such quantaloids, which we call Cauchybilateral quantaloids, it follows that the Cauchy completion of any symmetric Qenriched category is again symmetric. Examples include Lawvere’s quantale of nonnegative real numbers and Walters ’ small quantaloids of closed cribles.
MODULES
 THEORY AND APPLICATIONS OF CATEGORIES, VOL. 11, NO. 17, 2003, PP. 375396.
, 2003
"... This paper studies lax higher dimensional structure over bicategories. The general notion of a module between two morphisms of bicategories is described. These modules together with their (multi)2cells, which we call modulations, organize themselves into a multibicategory. The usual notion of a m ..."
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This paper studies lax higher dimensional structure over bicategories. The general notion of a module between two morphisms of bicategories is described. These modules together with their (multi)2cells, which we call modulations, organize themselves into a multibicategory. The usual notion of a module can be recovered from this general notion by simply choosing the domain bicategory to be the terminal or final bicategory. The composite of two such modules need not exist. However, when the domain bicategory is small and the codomain bicategory is locally cocomplete then the composite of any two modules does exist and has a simple construction using the local colimits. These modules and their modulations then give rise to a bicategory. Recall that neither transformations nor optransformations (respectively lax natural transformations and oplax natural transformations) between morphisms of bicategories give rise to a smooth 3dimensional structure. However, there is a smooth 3dimensional structure for modules, and both transformations and optransformations give rise to associated modules. Furthermore, the modulations between two modules associated with transformations can then be described directly as a new sort of modification between the transformations. This provides a locally full and faithful homomorphism from transformations and modifications into the bicategory of modules. Finally, if each 1cell component of a transformation is a leftadjoint then the rightadjoints provide an optransformation. In the module bicategory the module associated with this optransformation is rightadjoint to the module associated with the transformation. Therefore the inclusion of transformations whose 1cells have left adjoints into the (multi)bicategory of modules provides a sou...
fcmulticategories
, 1999
"... What fcmulticategories are, and two uses for them. fcmulticategories are a very general kind of twodimensional structure, encompassing bicategories, monoidal categories, double categories and ordinary multicategories. ..."
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What fcmulticategories are, and two uses for them. fcmulticategories are a very general kind of twodimensional structure, encompassing bicategories, monoidal categories, double categories and ordinary multicategories.