Re: Not invariant but good

[John Baez <baez@math.ucr.edu>]

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.

Let me expand on this a bit, first for properties and then for
structures.

Say we have a groupoid C.  A (possibly evil) property of objects
in C is a map

F: Ob(C) -> {F,T}

where Ob(C) is the class of objects of C and {F,T} is the set of
truth values.  I say the property is non-evil if it extends to a
functor

F: C -> {F,T}

where now we regard {F,T} as a discrete groupoid.

For example, suppose C is the groupoid of vector spaces
over your favorite field.

A typical non-evil property is "being 5-dimensional".  A typical
evil property is "having the empty set as its origin".  (In ZF set
theory, we can take any vector space and ask whether
its zero element happens to be the empty set.)

I think most mathematicians would be happy to see a theorem
that begins

Theorem: If V is a vector space that is 5-dimensional...

but somewhat surprised to see a theorem that begins:

Theorem: If V is a vector space with the empty set as its origin...

We instinctively feel that any theorem of the second sort
could have been phrased better: how could it really matter
that the origin is the empty set?

Now, structures.  Again, let C be a groupoid.  A (possibly evil)
structure on objects in C is a map

F: Ob(C) -> Set

The idea is that for any object c in C, F(c) is the set of
structures that can be put on that object.  I say the structure
is non-evil if it extends to a functor

F: C -> Set

If C is the groupoid of vector spaces, a typical non-evil
structure is a basis: here F(c) is the set of bases of the
vector space c.   A typical evil structure would be "a basis,
if the underlying set of c is the real numbers, but two bases
otherwise."

Again, I think most mathematicians would feel happy to work
with the non-evil structure, but somewhat uncomfortable
working with the evil one.  That's the feeling that this concept
of "evil" is trying to formalize.

Note that a non-evil structure on finite sets is what you call a
"species".  You wisely avoided studying the evil ones.

Best,
jb


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