reverting religious terminology

reverting religious terminology

[Thomas Streicher]

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".


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Re: reverting religious terminology

[Michael Shulman]

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


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Re: reverting religious terminology

[Thomas Streicher]

Sorry for forgetting about your posting from a couple of weeks ago. I can
see that working with essential fibres escapes my criticism.
Still I feel uneasy about it because it messes up things to such an extent
that I can't imagine to rewrite even a basic introductory section of a text
about fibrations in terms of weak fibrations. Of course, one could do it using
the language  of intensional type theory and then interpreting it in the
groupoid model. I couldn't say in advance whether formalizing basic theory of
fibrations in intensional type theory is possible. Experience tells us that
there pop up unpleaseant suprises in even simpler situations.
So my question is what do you (or other readers of the list) think why a
non-evil account of fibrations could be useful after all besides for
ideological reasons. Generalising to 2-categories like toposes and geometric
morphisms - which is useful - is not an example because there one still may
stay strict and reduce everything to Grothendieck fibrations as you have
explained convincingly.
From http://ncatlab.org/nlab/show/Grothendieck+fibration I take that weak
fibrations are necessary only when considering fibrations in bicategories
that are not 2-categories. Why not take the paradigmatic case of the bicategory
Dist of distributors. Has this exampe been worked out in detail.

Thomas


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Re: reverting religious terminology

[Michael Shulman]

On Mon, Oct 4, 2010 at 12:38 PM, Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
> From http://ncatlab.org/nlab/show/Grothendieck+fibration I take that weak
> fibrations are necessary only when considering fibrations in bicategories
> that are not 2-categories.

Yes, the only reasons I know of for caring about weak fibrations are
(1) if you're in a bicategory, or even just in a strict 2-category
which lacks the property that weak fibrations can be strictified to
strict ones, and (2) to assuage any worries (ideological or otherwise)
one might have about the notion of strict fibration not being
covariant under equivalence.

One bicategory which comes to mind which is not a strict 2-category,
and in which I would certainly want to think about internal
fibrations, is the bicategory of internal categories and anafunctors
in some topos.

> Why not take the paradigmatic case of the bicategory Dist of
> distributors. Has this exampe been worked out in detail.

I don't know; I certainly haven't seen it done.  If it hasn't been
done, someone should work it out; it might be interesting.

Mike


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