Subject: categories: Equivalences and psuedo-equivalences (two items)

From: [email-masked]
Date: Sat, 21 Feb 2004 08:46:40 EST

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

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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