Re: Not invariant but good

[Michael Shulman <shulman@math.uchicago.edu>]

Dear Thomas,

I did not intend to "generalize" anything, only to restate the
definition.  There are at least four equivalent definitions of
fibration in a 2-category that I know of:

- the "representable" one which I gave,
- having an adjoint one-sided inverse to a certain morphism between
comma objects
- being an algebra for a certain monad on a slice 2-category
- the one in the Elephant

All the definitions have two versions: a "strict" one a la
Grothendieck (which only makes sense in a strict 2-category) and a
"weak" one a la Street (which makes sense in any bicategory).  All the
strict notions are equivalent to each other, and all the weak notions
are equivalent to each other.  The idea of the equivalence of the
first three is essentially contained in Richard's remark, while a
proof of the equivalence of the Elephant's version with the
representable one can be found in Peter Johnstone's article
"Fibrations and partial products in a 2-category."

Weak fibrations are closed under composition with equivalences, while
strict ones are not.  In Cat and probably other well-behaved
2-categories, being a weak fibration is the same as being the
composite of a strict fibration and an equivalence, and so it ought to
surprise no one that in such cases it is sufficient to consider strict
fibrations.  It's generally only in the bicategorical world that weak
fibrations become important.

All the definitions can also be described either as "properties"
(such-and-such thing exists) or as "structure" (equipped with
such-and-such thing), since in all cases the such-and-such is unique
up to unique isomorphism when it exists.  This is the situation also
referred to as "property-like structure."  But perhaps you were
originally referring instead to the structure required on the
2-category itself?  It's true that the latter three definitions
require existence of some limits in the 2-category, while the
representable version does not.

Finally, in Cat with choice, the strict notions are all equivalent to
the usual Grothendieck fibrations, while the weak ones are equivalent
to Street's.  (Street actually originally gave his definition in a
general bicategory, and only later specialized it to Cat.)

Does this clarify what I meant?

> Nothing against anafunctors but it is an exaggeration to say that in absence
> of choice the usual notion of functor is not well-behaved.

Perhaps "not well-behaved" was a poor choice of words since it implies
a value judgement, but I think it is correct to say that in the
absence of choice, category theory becomes very unfamiliar unless we
replace functors with anafunctors.  For instance, if we insist on
using only functors, then a category with finite products does not
necessarily become a monoidal category, as there is no "product"
functor from A×A to A.  Also, without choice the 2-category Cat using
functors is not even a regular 2-category, let alone a 2-topos.  Since
the defining characteristics of Set include that it is a well-pointed
1-topos, it seems unlikely to me that one will be able to get much of
anywhere with a version of Cat that is not a 2-topos.

Mike


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