Abstract

If a category B with Yoneda embedding Y : B \Gamma! CAT(B op ; set) has an adjoint string, U a V a W a X a Y; then B is equivalent to set. The authors gratefully acknowledge financial support from NSERC Canada. Diagrams typeset using M. Barr's diagram macros. 1 Introduction The statement of the Abstract was implicitly conjectured in [9]. Here we establish the conjecture. We will see that it suffices to assume that B has an adjoint string V a W a X a Y with V pullback preserving. A word on foundations and our notation is necessary. We write set for the category of small sets and assume that there is a Grothendieck topos, SET, of sets which contains the set of arrows of set as an object. The 2-category of category objects in SET, which we write CAT, is cartesian closed and set is an object of CAT. Thus, for C a category in CAT, CAT(C op ; set) is also an object of CAT and we abbreviate it by MC; (it was written PC in [8].) Substitution gives a 2-functor M : CAT coop \Gamma...

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