The answer to this question must be in the literature somewhere but I haven't been able to track it down or figure it out for myself. Let A and B be bicategories. By a functor from A to B I shall mean a morphism of underlying reflexive globular sets that preserves all operations and identities on the nose and satisfies the standard coherence conditions. A psuedo-functor is a morphism that preserves operations and identities only up to 1 cells that are equivalences in B ( a 1-cell f is an equivalence if there is a 1-cell g such that fg and gf are both defined and isomorphic to the respective identitity 1-cells). (These 1-cells must of course satisfy additional, standard coherence conditions.) An equivalence from A to B is a functor that is essentially surjective on 0-cells (every 0-cell of B is equivalent in B to a 0-cell in the image of F) and which induces an equivalence of 1-categories between A( x,y ) and B ( Fx, Fy ) for all 0-cells x, y in A. A psuedo-equivalence from A to B is a psuedo-functor that has these same properties. Question 1 : If F is a psuedo-equivalence from A to B does there exist an equivalence G from A to B? (references/counterexamples?) Question 2: Same as question 1 but with A and B strict 2-categories. Qestion 3: If the answer to question 1 is yes, then can G be chosen to be equivalent to F in the bicategory whose 0-cells are psuedo-functors from A to B? I shall be very grateful for any guidance the list can provide on these questions. Carl Futia ------------------------------------------------------ SECOND ITEM: Subject: Correction to my last query I have to apologize for a silly error in my last request for help. The problem lies with the definition of psuedo functor. I should have said that a psuedo functor preserves operations up to a 2-cell isomorphism, not a 1-cell equivalence. Sorry! Carl Futia