reverting religious terminology
With the help of Mike Shulman I have eventually understood how one arrives at a notion of "weak fibration". There is the following characterisation of P : XX -> BB being a fibration due to J. Gray : for every X in XX the functor P/X : XX/X -> BB/P(X) has a right adjoint right inverse. Of course "right inverse" is "evil" so let's replace it by the "non-evil" requirement that all P/X have a right adjoint which is full and faithful (replacing "counit is an identity" by "counit is an isomorphisms"). Working this out one sees that P is a weak fibration (i.e. a fibration in this weaker "non-evil" sense) iff for all X in XX and u : J -> P(X) there is a cartesian arrow phi : Y -> X with P(phi) isomorphic to u in BB/P(X). Just writing out the definition of "cartesian" one observes that it doesn't make reference to equality of objects. Thus, replacing "equal" by "isomorphic" in "for all u : J -> P(X) there is a cartesian arrow phi : Y - X with P(phi) equal to u in BB/P(X)" one obtains the above definition of weak fibration. So one can hardly deny that this is the right(eous) non-evil version of Grothendieck fibration. But for such weak fibrations one looses the important property that for every u : J -> I one can transport X over I to u^*X over J along u. This property is essential for category theory over an arbitrary base (topos). In other words whereas "evil" fibrations correspond to indexed categories BB^op -> Cat the weak ones do not. Moreover, indexed categories, i.e. pseudofunctors BB^op -> Cat, can be formulated in a "non-evil" way but one has to accept the bureaucracy of coherence conditions which does not show up when working with fibrations (and one also has to accept very big categories like Cat). Thus sticking to a "non-evil" discipline one comes to the conclusion that indexed categories are better than fibered categories. The latter are more elegant and easier to work with (one need not sweep under the carpet coherence issues all the time). This suggest to me that a better name for "non-evil" might be "puritan". This suggestion of "puritan" is meant as seriously as the suggestion of "evil". Thomas PS As pointed out by Andre Joyal when reasoning about the most "non-evil" category of homotopy types it seems to be essential to use concepts which are not stable under weak equivalence. Another instance of the principle "good" requires "evil". [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Mon, Oct 4, 2010 at 3:36 AM, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:
But for such weak fibrations one looses the important property that for every u : J -> I one can transport X over I to u^*X over J along u.
As I pointed out in my message to the list on September 16, all that one has to do to remedy this situation is consider "essential fibers" rather than strict fibers. In other words, the notion of "X over I" is itself evil and needs to be replaced by "X equipped with an isomorphism from P(X) to I". The category of all so-equipped Xs is called the "essential fiber" of P over I, and in a weak fibration there is indeed a functor u^* from the essential fiber over I to the essential fiber over J. In this way, any weak fibration also gives rise to an indexed category, and the 2-category of weak fibrations is biequivalent to that of indexed categories (whereas the 2-category of strict fibrations is strictly 2-equivalent to that of indexed categories). Also, if P is a strict Grothendieck fibration (indeed, an isofibration suffices), then its essential fibers are equivalent to its strict fibers, so the two constructions are compatible. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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Michael Shulman -
Thomas Streicher