* Re: Three questions about fibrations
2000-06-15 17:02 Three questions about fibrations Michael Abbott
2000-06-16 4:43 ` Robert A.G. Seely
@ 2000-06-16 19:12 ` Claudio Hermida
1 sibling, 0 replies; 3+ messages in thread
From: Claudio Hermida @ 2000-06-16 19:12 UTC (permalink / raw)
To: categories
Michael Abbott wrote:
> I am wondering if anyone can give references for three remarks Wesley Phoa
> makes in the chapter on fibrations of his paper "An introduction to
> fibrations, topos theory, the effective topos and modest sets".
I think I can provide some meaningful answers to these questions, but of
course, one should always take such (throwaway) remarks in an informal
exposition with a pinch of black pepper.
>
> 1. Essentially Algebraic Theories
>
> In the footnote on page 7 Phoa comments:
> "[fibrations] are the models for a first-order, 'essentially
> algebraic' theory".
>
> I'm not quite sure what he means here, and this sounds like it must be a
> standard and well known connection. I'd be glad of a reference.
>
Fibrations as finite-limit theories: This follows from the analysis in
[S], that exhibits fibrations as (adjoint) pseudo-algebras in any 2-category
with comma-objects. In particular, for any category B with pullbacks, the
2-category Cat(B) of internal categories in B admits them.
>
> 2. Splitting Fibrations
>
> At the bottom of page 14 Phoa observes:
> "Every fibration .. is equivalent to a split fibration (there is an
> elegant proof due to John Power). However, it's not clear how to extend
> this result to more complicated structures. This is the coherence problem
> ..."
>
> Now I know that any fibration is equivalent to a split fibration via the
> "fibred Yoneda lemma" (Borceux, "Handbook of Categorical Algebra 2", 8.2.7
> and Jacobs, "Categorical Logic and Type Theory"), but I don't think that's
> the only splitting available, and I'm not sure that this correspondence
> helps very with coherence questions.
> I am aware of another, different, equivalent splitting, and I wonder
> if there are any references. In particular, can anyone guess what reference
> by John Power Phoa was referring to?
> In particular, I'd be very interested in any other observations on
> the "coherence problem".
>
Split fibrations: Since fibrations are pseudo-algebras, it follows from Power's
coherence theorem [P] (which applies in this situation) that every fibration is
equivalent (over its base category) to a split one. The construction there does
not appeal to Yoneda. Yet, given that fibrations are actually properties, that
is adjoint pseudo-algebras, one can give a neat explicit description of the
associated split one as follows:
Given p:E->B, consider the free fibration over it, E/p -> B. The unit
\eta: E -> E/p takes X |-> X,id_(pX). The right adjoint r: E/p -> E amounts to
a choice of cartesian lifting: (Y,u:I->pY) |-> u*(Y). Thus we get a comonad
\eta o r: E/p -> E/p. The resulting Kleisli category (E/p)_(\eta r) is fibred
over B and gives the corresponding split fibration. The equivalence is given by
the composite E -\eta-> E/p -J-> (E/p)_(\eta r).
The above construction is a special case of the theory in [H], specially
section 11. The paper and the references there in provide further material on
coherence.
>
> 3. Generalising the Definition
>
> In the footnote on page 12, in reference to the definition of a fibration,
> Phoa says:
> "If one really wants to take .. 2-categorical issues seriously, one
> needs .. a more sophisticated definition of 'fibration'."
>
> I can make some promising looking guesses. Any references?
>
Generalised fibrations: The only meaningful generalisation (in the categorical
context) proposed so far in the literature is that of [S1], fibrations "up to
equivalence". However I can see no impediment whatsoever at taking 2-category
theory seriously with the standard (Grothendieck) notion of fibration, so I
cannot guess what the author means.
References:
[S] R. Street, Fibrations and Yoneda Lemma in a 2-category, Category Seminar,
LNM 420, 1973.
[S1] R. Street, Fibrations in bicategories, Cahiers Top.Geom.Diff.Cat, 21,
1980.
[P] A.J.Power, A general coherence result, JPAA, 57(2):165-173, 1989.
[H] C. Hermida, From coherence structure to universal properties, (to appear
JPAA) (available at http://www.cs.math.ist.utl.pt/s84.www/cs/claudio.html)
Claudio Hermida
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