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 n-category. Included is a full explanation of why the proposed definition of n-category is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to n-categories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.

A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem.

Abstract. In some bicategories, the 1-cells are ‘morphisms ’ between the 0-cells, such as functors between categories, but in others they are ‘objects ’ over the 0-cells, such

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This article shows that the distributive laws of Beck in the bicategory of sets and matrices determine strict factorization systems on their composite monads (=categories). Conversely, it is shown that strict factorization systems on categories give rise to distributive laws. Moreover, these processes are shown to be mutually inverse in a precise sense. Further, an extension of the distributive law concept provides a correspondence with the classical orthogonal factorization systems.

We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. This becomes the object part of a 2-functor N: NHom → [ ∆ op,Cat], where NHom is a 2-category whose objects are bicategories and whose 1-cells are normal homomorphisms of bicategories. The 2-functor N is fully faithful and has a left biadjoint, and we characterize its image. The 2-nerve of a bicategory is always a weak 2-category in the sense of Tamsamani, and we show that NHom is biequivalent to a certain 2-category whose objects are Tamsamani weak 2-categories. This paper concerns a notion of “2-nerve”, or Cat-valued nerve, of bicategories. To every category, one can associate its nerve; this is the simplicial set whose 0-simplices are the objects, whose 1-simplices are the morphisms, and whose n-simplices are the composable n-tuples of morphisms. The face maps encode the domains and codomains of morphisms, the composition law, and the associativity property, while the degeneracies record information about the identities. This construction is the object part of a functor N: Cat1 → [ ∆ op,Set] from the category of categories and functors, to the category of simplicial sets. This functor is fully faithful and has a left adjoint. It arises in a natural way, as the “singular functor ” (see Section 1 below) of the inclusion

Abstract. In some bicategories, the 1-cells are ‘morphisms ’ between the 0-cells, such as functors between categories, but in others they are ‘objects’ over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the ‘morphisms between 0-cells’, such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1-cells. We avoid complicated coherence problems by describing base change ‘nonalgebraically’, using categorical fibrations. The resulting ‘framed bicategories ’ assemble into 2-categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a ‘monoidal fibration’, meaning a parametrized family of monoidal categories, and produces an analogue of the framed bicategory of spans. Combining the two, we obtain a construction which includes both enriched and internal categories as special cases.

Abstract. In some bicategories, the 1-cells are ‘morphisms ’ between the 0-cells, such as functors between categories, but in others they are ‘objects’ over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the ‘morphisms between 0-cells’, such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1-cells. We avoid complicated coherence problems by describing base change ‘nonalgebraically’, using categorical fibrations. The resulting ‘framed bicategories ’ assemble into 2-categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a ‘monoidal fibration’, meaning a parametrized family of monoidal categories, and produces an analogue of