Re: Not invariant but good

[selinger@mathstat.dal.ca (Peter Selinger)]

Hi John,

thanks for trying to move the discussion away from terminology and
back to actual mathematical matters. In order to focus on the math and
not on the terminology, let me today use the word "XXXX" instead of
"evil".

I don't think the notion used in your examples is general enough. For
example, fix some groupoid C, and consider the property of an object
x: "x is isomorphic to exactly 3 objects of C".  To me, this is
clearly XXXX, because it is not invariant under equivalences of
C. Yet, according to the definition you used in this email, it extends
to a functor C -> {F,T}, and therefore is non-XXXX.

For a property P of objects x of a category C, "being invariant under
isomorphisms of objects in C" is strictly weaker than "being invariant
under equivalences of C". Proof: Clearly, any isomorphism of C can be
mapped to an identity of some category C' by some equivalence of
categories.  Therefore, any property that is invariant under
equivalences of categories is invariant under isomorphisms of
objects. The above example shows that the converse is not true.

I think for XXXXness of structures, a similar refinement is needed.
To me, the intuitive concept of XXXX for structures is "cannot be
transported along equivalences such that the equivalence becomes
structure preserving". To say it more explicitly: if category C has
the structure, and category C' is equivalent to C (as a category),
then C' can be equipped with a structure in such a way that the
equivalence (both directions) is structure preserving.

This is a fairly subtle concept, not least because it depends on the
precise 2-category in question (to fix what "equivalence" and
"structure preserving" means). For example, whether the structure of
being "strictly monoidal" is XXXX or not depends on what one means by
"structure preserving" (e.g., strict monoidal or strong monoidal
functors). The natural transformations need to be specified too, so
that one can define "structure preserving equivalence".

I tried to give a more general and precise 2-categorical definition on
the categories list on January 3, 2010, but I am not sure I got it
quite right. I think it was Mark Weber who also pointed out, around
the same time, that one person's XXXX concept is another person's
non-XXXX concept - in a different 2-category.

The fact that being XXXX depends on an ambient 2-category means that
it is not a moral judgment, and people should not be offended by
it. Some perfectly useful things can be XXXX sometimes, and some
perfectly useless things can be non-XXXX. For example, even a
not-very-natural property like "there are exactly 3 objects isomorphic
to x" can be non-XXXX when viewed in the right 2-category. For
example, this is the case in the 2-category of categories, functors,
and identity natural transformations.

Last comment. Thomas Streicher brought up the example of a fibration
P: XX -> BB as a concept that was XXXX but very useful. But I don't
think this concept is actually XXXX. Certainly if one thinks of the
fibration as a *structure* on BB, then this transports very nicely
along equivalences. Namely, given any equivalence BB <--> BB', one can
find a fibration P' : XX' -> BB' which is equivalent, as a fibration,
to P. Right?

So I don't think it is correct to identify the concept of XXXX with
"having to talk about equality". Rather, it should be defined in some
2-categorical way. See also Mike Shulman's post from January 4, which
discussed this distinction in more depth.

-- Peter


John Baez wrote:
>
> Dear Andre -
>
>> Many good things in mathematics are depending on the choice
>> of a representation which is not invariant under equivalences,
>> or under isomorphisms. Modern geometry would not exists
>> without coordinate systems.
>
> I agree.  I think you're arguing against a position that nobody
> here has espoused.
>
> A coordinate system is a structure, not a property.  In my
> earlier email I said a *property* is evil if it's not invariant under
> equivalences.  But I'd say a *structure* is evil if it's not *covariant*
> under equivalences.  Coordinate systems are covariant under
> equivalences, so they're not evil.
>

...


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