Date: Wed, 9 Nov 1994 15:55:52 +0400 (GMT+4:00)
From: categories <cat-dist@mta.ca>
Subject: Fibrations<-->Adjunctions????
Date: Wed, 9 Nov 1994 19:10:37 +0100
From: Maura Cerioli <cerioli@disi.unige.it>
Hi,
I need to use the following result about the equivalence between the
existence of a fibration and the existence of a right adjoint to
an opportune functor between comma categories:
-----------------
Let A and B be categories and let U:A-->B be a
functor.
Let C denote the comma category A arrow (whose objects are the arrows
of A)
and
let D denote the comma category B over U (whose objects are arrows in B
from a generic object b into U(a), together with a).
Then U induces a functor V:C-->D, that basically is U (as both objects
and arrows in C are arrows in A too)
and is defined as follows:
* V(f:a-->a')=(U(f):U(a)-->U(a'),a') for each object
f:a-->a' in A;
* V(g,g')=(U(g),g') for each arrow
(g,g'):(f1:a1-->a'1)----->(f2:a2-->a'2) in A.
Then U is a fibration iff V has a right adjoint G:D-->C s.t.
G;V is the identity on D and the counit of the adjunction is
the identity natural transformation.
-----------------
Now, I know (or at least I believe) that the result holds, as
I have a proof for it; but a collegue said to me that this is
a known result and suggested to look at the works by Street
in the Sydney Category Seminar.
Unfortunately, being quite ignorant about 2categories, I find very
hard even to understand in which sense the result I want is there.
Does anybody know an easier reference, possibly not involving
2categories?
Thanks
Maura
Maura Cerioli
DISI - Dipartimento di Informatica e Scienze dell'Informazione
Viale Benedetto XV, 3
16132 Genova
ITALY
Date: Thu, 10 Nov 1994 18:53:32 +0400 (GMT+4:00)
From: categories <cat-dist@mta.ca>
Subject: Re: Fibrations<-->Adjunctions????
From: Jim Otto <otto@rodent.math.mcgill.ca>
To: categories@mta.ca
(the message being replied to is appended.)
dear Maura Cerioli
yes, the Street paper is hard to read. but he refers to a paper of
Gray's which has the result you mention. in fact Gray credits it to a
manuscript of Chevalley's which was lost in the mail.
while i didn't read your description of the result, the way i think of
it is as follows. you have a functor
p: E --> B
this induces a functor (with comma category B/p relative to id: B -->
B <-- E :p and with 2 the ordinal)
p.: E^2 --> B/p
which takes
`paths in E' to
`paths in B with the to end in E remembered'
then (modulo the axiom of choice) p is a fibration iff p. has a right
adjoint c. with identity couint. c. is then a cleavage. this unfolds
(viewing c. as terminal objects in comma categories) to the usual
elementary definition in terms of cartesian maps. Gray says p. is
rari, i.e. has a right adjoint right inverse. one can show that this
isn't weaker. as i recall, Street's definition is weaker in order to
be more invariant. (cod: B^2 --> B is an important example. then
c. is pull-back.)
actually i need fibrations in 2-categories of monoidal categories.
note that, in cat the 2-category of categories, E^2 is the comma
category E/E (relative to id: E --> E <-- E :id). further, still in
cat, E^2 is the cotensor 2 -o E. but in any 2-category, finite
weighted limits, including comma objects, can be built up from
cotensors with 2 and (2-) terminal objects and pull-backs.
to help you read Street, i now define cotensor with 2. 2-categories
have hom categories rather than hom sets. 2 -o E is defined by
isomorphisms
hom(W, 2 -o E) iso 2 -o hom(W, E)
(2-) natural in W. (sometimes such isomorphisms don't exist, but such
equivalences do.)
bon jour, Jim Otto
>Date: Wed, 9 Nov 1994 19:10:37 +0100
>From: Maura Cerioli <cerioli@disi.unige.it>
>
>Hi,
>I need to use the following result about the equivalence between the
>existence of a fibration and the existence of a right adjoint to
>an opportune functor between comma categories:
>-----------------
>Let A and B be categories and let U:A-->B be a
>functor.
>Let C denote the comma category A arrow (whose objects are the arrows
>of A)
>and
>let D denote the comma category B over U (whose objects are arrows in B
>from a generic object b into U(a), together with a).
>Then U induces a functor V:C-->D, that basically is U (as both objects
>and arrows in C are arrows in A too)
>and is defined as follows:
>* V(f:a-->a')=(U(f):U(a)-->U(a'),a') for each object
> f:a-->a' in A;
>* V(g,g')=(U(g),g') for each arrow
> (g,g'):(f1:a1-->a'1)----->(f2:a2-->a'2) in A.
>
>Then U is a fibration iff V has a right adjoint G:D-->C s.t.
>G;V is the identity on D and the counit of the adjunction is
>the identity natural transformation.
>-----------------
>Now, I know (or at least I believe) that the result holds, as
>I have a proof for it; but a collegue said to me that this is
>a known result and suggested to look at the works by Street
>in the Sydney Category Seminar.
>Unfortunately, being quite ignorant about 2categories, I find very
>hard even to understand in which sense the result I want is there.
>
>Does anybody know an easier reference, possibly not involving
>2categories?
Date: Thu, 10 Nov 1994 12:58:07 +0400 (GMT+4:00)
From: categories <cat-dist@mta.ca>
Subject: Re: Fibrations<-->Adjunctions????
Date: Wed, 9 Nov 1994 21:22:16 +0000 (GMT)
From: Dusko Pavlovic <D.Pavlovic@doc.ic.ac.uk>
The result goes back to Claude Chevalley's seminar. (One direction
does require the Axiom of Choice.) I think a simple formulation is in
J.Gray's paper on fibrations in the proceedings of La Yolla
conference, Springer 1965.
-- Dusko Pavlovic