isomorphisms

isomorphisms

[Eduardo J. Dubuc]

isomorphims appear not only in examples but are essential also in the theory.
For example Grothendieck defines limits and colimits of categories (as
universal pseudocones) in SGA4 by means of an isomorphism of categories. Same
for toposes.

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Re: isomorphisms

[Colin McLarty]

2010/5/29 Eduardo J. Dubuc <edubuc@dm.uba.ar>

Expresses my main point when I quoted Grothendieck on equivalence and
isomorphism.

> isomorphisms appear not only in examples but are essential also in the
> theory.
> For example Grothendieck defines limits and colimits of categories (as
> universal pseudocones) in SGA4 by means of an isomorphism of categories.
> Same for toposes.


When AG says "none of the equivalences we meet in practice are
isomorphisms" he has in mind lots of examples that I will not even try
to survey.  (For a really simple one, the category of sheaves defined
as espaces etales on on a topological space versus the category of
sheaves defined as suitable functors on the site of open subsets.)

But when he defines functor categories, or derived categories, and a
lot of other things like that, he defines them up to unique
isomorphism over the data.  A topos E will often be defined only up to
equivalence.  But, given E, its derived category is defined up to
unique isomorphism and one constantly uses the fact that various
induced functors are isomorphisms.

AG's practice constantly distinguishes isomorphisms from equivalences,
and thus distinguishes identity of objects from isomorphism of them.

best, Colin


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Re: isomorphisms

[Michael Shulman]

It seems to me that there is an important distinction here that is not
being emphasized.  Isomorphisms of categories can be *technically*
quite useful.  Knowing that a given equivalence of categories is an
isomorphism, rather than merely an equivalence, can certainly make
things much simpler, or even make things possible that we didn't know
how to do before.  Many examples of this sort have been mentioned.
Another that should be added to the list is the theory of strict
2-categories, 2-limits, 2-adjoints, and so on, all of which is defined
using ordinary enriched category theory over Cat, and hence involves
many isomorphisms of hom-categories.

However, I find that in most or all of these examples, one is not
actually interested in the fact of an isomorphism of categories for
its own sake.  There is no "real meaning" in the fact that two
categories are isomorphic, rather than equivalent; generally it's a
technical accident of how we chose to define them.  It may be a very
*convenient* technical accident, but it is an accident nonetheless.
If we chose our definitions differently, or worked in a different
foundational system (such as one where the notion of "isomorphism of
categories" cannot even be defined), some or all of our isomorphisms
of categories might change to become only equivalences, but I don't
believe that any of the real content of category theory would go away;
it would just become harder to prove.

Mike


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Equality and fibration

[Joyal, André]

Dear category theorists,

I have not been able to define the notion of Grothendieck fibration
without using the equality relation between the objects of the base category. 
Can you?

André


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Re: Equality and fibration

[Prof. Peter Johnstone]

On Mon, 31 May 2010, Joyal, André wrote:

> Dear category theorists,
>
> I have not been able to define the notion of Grothendieck fibration
> without using the equality relation between the objects of the base category.
> Can you?
>
> André
>

No, because an equivalence of categories is not a Grothendieck fibration
in general. However, there is a weaker version of the notion where one
replaces equality by the existence of a (specified) isomorphism; there
are some comments about this in my paper "Fibrations and partial
products in a 2-category" in Applied Categorical Structures 1 (1993),
141--179.

Peter Johnstone


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Re: Equality and fibration

[Peter LeFanu Lumsdaine]

On Mon, May 31, 2010 23:26, Joyal, André wrote:

> Dear category theorists,
>
> I have not been able to define the notion of Grothendieck fibration
> without using the equality relation between the objects of the base
> category. Can you?

In a dependent language (eg ML, FOLDS, etc), this can be done.  The twist
is that it is _not_ just some extra algebraic structure on a functor p: \E
--> \C; for an arbitrary such p, as you say, the lifting conditions can't
be specified without referring to equality of objects.  Rather, it first
of all requires the objects and arrows of \E to be dependent types over
the objects of \C; then, the "equalities" needed become part of the
_typing specification_ of a lifting.  This refinement of the definition
complicates the theory a little, but is a quite natural requirement, I
think.  (Just as in this setting, a fibration of types is no longer an
arbitrary map f:A --> B, but rather a dependent type A over B.)

=====
Precisely, you can define it as follows:

Say \C is a category (i.e. a type C_0, and a dependent type C_1(a,b), for
a,b: C_0, an "equality" type on C_1, and appropriate composition/identity
operations).

The a (cloven) Grothendieck fibration \E over \C is:

a dependent type E_0(a), for a: C_0;
a dependent type E_1(a,b;i,j), for a,b: C_0, i:E_0(a), j:E_0(b)
(we can write just E_1(i,j), since a,b are implicit in the types of i,j);

operations making this a category over E_0;

and an operation which, given an arrow a,b:C_0, f:C_1(a,b), and a lift
j:E_0(b) of the target, returns liftings i:E_0(a) and \bar{f}:E_1(i,j),
with p(\bar{f}) = f, and appropriately cartesian.  (The definition of
cartesian similarly doesn't need to refer to equality of objects, since
they're included in the typings.)
=====

If we were dealing with higher categories, and hence didn't have equality
on C_1, then we would require E_1 to be a dependent type not just over
a,b:C_0 and i:E_0(a), j:E_0(b), but also over f:C_1(a,b), so the condition
that \bar{f} is a lift of f would again be part of its typing.  This is
reminiscent of latching/matching objects etc...

Cheers,
-Peter.


-- 
Peter LeFanu Lumsdaine
Carnegie Mellon University





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