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From the outset, the theories of ordinary categories and of additive categories were developed in parallel. Indeed additive category theory was dominant in the early days. By additivity for a category I mean that each set of morphisms between two objects (each " hom ") is equipped with the structure of abelian group and composition on either side, with any(More)
This survey of categorical structures, occurring naturally in mathematics, physics and computer science, deals with monoidal categories; various structures in monoidal categories; free monoidal structures; Penrose string notation; 2-dimensional categorical structures; the simplex equations of field theory and statistical mechanics; higher-order categories(More)
Soon after the appearance of enriched category theory in the sense of Eilenberg-Kelly 1 , I wondered whether V-categories could be the same as W-categories for non-equivalent monoidal categories V and W. It was not until my four-month sabbatical in Milan at the end of 1981 that I made a serious attempt to properly formulate this question and try to solve(More)
There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict n-categories and outlined for weak n-categories. The categorical aspects concern the development of descent theory in low dimensions in(More)
This is an expanded, revised and corrected version of the rst author's preprint [1]. The discussion of one-dimensional cohomology H 1 in a fairly general category E involves passing to the (2-)category Cat(E) of categories in E. In particular, the coecient object is a category B in E and the torsors that H 1 classies are particular functors in E. We only(More)
Distributive laws between monads (triples) were dened by Jon Beck in the 1960s; see [1]. They were generalized to monads in 2-categories and noticed to be monads in a 2-category of monads; see [2]. Mixed distributive laws are comonads in the 2-category of monads [3]; if the comonad has a right adjoint monad, the mate of a mixed distributive law is an(More)