BOUNCE categories@mta.ca: Approval required: (fwd)

[Bob Rosebrugh <rrosebru@mta.ca>]

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>From rrosebru  Sat Mar  4 22:36:10 2000
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Date: Sat, 04 Mar 2000 18:36:44 -0800
From: Dusko Pavlovic <dusko@kestrel.edu>
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To: Adam Eppendahl <ae@dcs.qmw.ac.uk>
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Subject: Re: categories: dom fibration
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> Can anyone explain why the codomain fibration cod: C^\rightarrow -> C,
> which requires pull-backs, gets loads of attention, while the domain
> fibration dom: C^rightarrow -> C, which works for all C, hardly gets a
> look in?  Is the dom fibration really such a poor relation?

heh, the amount of attention is not always proportional to the depth of
the issue. but in this case, i think, there are good reasons for
asymmetry.

the idea of fibred (or indexed) category theory over, say, a base S, is
that each category C is always given together with all categories C^I of
I-indexed families from C, where I are the objects of S. indeed, ordinary
categories are always given with their set indexed versions. we are
tacitly using C^2 to say "product". joining all such indexed versions of C
as the fibres, we get the fibred presentation of C, its "externalization".

but now, the base category S itself should come about as an object of
category theory over S as well. in ordinary category theory, we often
mention the category of sets. the base fibration cod: Ar(S) --> S is the
externalization of S itself as fibred over S. indeed, its fibres, the
slices S/I are the abstract categories of I-indexed families from S. when
S is Set, they are equivalent to S^I; but the latter may not exist for a
general S. the pullbacks in S correspond to reindexing within S. if there
is no reindexing, then S is a poor base for category theory, because it
cannot even reindex itself.

however, even the categories S that cannot reindex themselves, they can
always comprehend themselves (in a formal sense of categorical
psychology). this is expressed by the functor dom: Ar(S)-->S. whether cod
is a fibration or not, there is always the adjunction cod -| ids -| dom:
Ar(S) -->S, which is the categorical form of the comprehension scheme, as
studied by Lawvere and later others, who generalized the adjunction
requirements away, so that the conceptual link with the set theoretical
idea of comprehension got lost...

in any case, if you embed CAT_S--->FIB/S, then cod: Ar(S)-->S is the image
of S itself, while dom: Ar(S)-->S is not in the image of the embedding,
but a derived concept.

-- dusko