Approved: tcas

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From rrosebru Sat Mar 4 22:36:10 2000 Received: from zent.mta.ca (zent.mta.ca [138.73.101.4]) by mailserv.mta.ca (8.9.3/8.9.3) with SMTP id WAA07262 for <categories@mta.ca>; Sat, 4 Mar 2000 22:36:10 -0400 (AST) Received: FROM kestrel.kestrel.edu BY zent.mta.ca ; Sat Mar 04 22:37:02 2000 Received: from brant.kestrel.edu (brant.kestrel.edu [206.159.212.32]) by kestrel.kestrel.edu (8.9.3/8.9.3) with ESMTP id SAA10181; Sat, 4 Mar 2000 18:36:01 -0800 (PST) Received: from kestrel.edu (blackbird.kestrel.edu [206.159.212.151]) by brant.kestrel.edu (8.9.3/8.9.3) with ESMTP id SAA24249; Sat, 4 Mar 2000 18:35:58 -0800 (PST) Message-ID: <38C1C83C.A6DCAC6E@kestrel.edu> Date: Sat, 04 Mar 2000 18:36:44 -0800 From: Dusko Pavlovic <dusko@kestrel.edu> X-Mailer: Mozilla 4.5 [en] (Win98; U) X-Accept-Language: en,nl MIME-Version: 1.0 To: Adam Eppendahl <ae@dcs.qmw.ac.uk> CC: categories@mta.ca Subject: Re: categories: dom fibration References: <200003032128.VAA02592@bruno.dcs.qmw.ac.uk> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit

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