From: Topos8@aol.com
To: categories@mta.ca
Subject: Equivalences and psuedo-equivalences (two items)
Date: Sat, 21 Feb 2004 08:46:40 EST [thread overview]
Message-ID: <a9.51beea9c.2d68bb40@aol.com> (raw)
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
next reply other threads:[~2004-02-21 13:46 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2004-02-21 13:46 Topos8 [this message]
2004-02-24 22:34 Steve Lack
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