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Monad-like structures in 2-categories and soft adjunctions Stone, Peter


One of the fundamental concepts of ordinary category theory is that of an adjunction. There are analogous notions in 2-category theory one of which is the "soft adjunction" of this thesis. In order to construct examples which exhibit the full generality of this phenomena two constructions for modifying soft adjunctions are described: an "attaching procedure" and a "lifting procedure". These two procedures are shown to be inverses of each other in an suitable sense. A special type of "commutative monad" structure on an object of a 2-category is described and it is shown that this structure can be modelled by the category of finite ordinals and all functions in the same way that the ordinary monad structure can be modelled by the category of finite ordinals and non-decreasing functions. By a standard argument for such a situation where there are 2-category objects with structure a certain "strict" adjunction" is obtained; that is, an adjunction which is essentially the same as an adjunction in ordinary category theory. Starting with a slightly extended version of this strict adjunction examples of soft adjunctions are obtained by means of the attaching and lifting procedures. The relationship between the the various soft adjunctions which arise is investigated.

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