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