Re: Fundamental Theorem of Category Theory?

[Matsuoka Takuo <moto@math.northwestern.edu>]

Dear categorists,

> I have asked Prof. Yoneda many years ago why Yoneda Lemma is
> called "Lemma", not "Theorem". He said that perhaps it was a
> bit about internal of category theory rather than insisting
> on applications to other mathematics. Doesn't Yoneda Lemma
> satisfy (c) in Mile Gould's post? I don't know how much
> Yoneda Lemma is useful in other areas of mathematics, and
> I have wanted to know it.

> | (c) it admits a huge variety of applications in "ordinary" mathematics.

I find this intersting, but I do not quite agree with Prof. Yoneda!

In order to challenge his claim, I would like to try making a list (which
I fear will not be "huge in variety") of some instances I know of in
mathematics where representable functors play central roles, and hope some
other people could do similar. While I know that I am not a particularly
well qualified person to write the part I am taking, I view this as a
great opportunity to share ideas from various different fields! (I hope
this is not off the topic of the list.)

- Given a category C of some mathematical objects, it is often equipped
with a "forgetful" functor C -> Set, so objects of C can be thought of as
sets equipped with some specific sort of structure. Let us call it a
C-structure. Then a C-structure on an object X of _any_ category can be
defined as a way to factorize the functor [ ,X], represented by X, through
the forgetful functor C -> Set. If C is the category of groups, then The
Lemma implies that giving a group structure on X is the same as giving
structure maps on X which are in analogy with the group operations for an
ordinary group. This readily generalizes for any sort of algebraic
structure, and this is related to Lawvere's notion of algebraic theories.
One can further replace the category Set with some other closed category
such as that of Abelian groups, using the language of enriched category
theory.

- Schemes in algebraic geometry can fruitfully be viewed as sheaves on the
opposite category Aff of that of commutative rings. Those schemes actually
represented by rings are called affine schemes. Thus, the category of
affine schemes is opposite to the category of rings, and is fully embedded
in the category of all schemes. The Yoneda lemma is a basic tool for the
study of schemes.

- Some presheaves on the category of (affine) schemes which fail to be
sheaves can more naturally be thought of as a groupoid-valued (rather than
set-valued) presheaves which can be represented by geometric objects
called algebraic stacks (which generalize schemes).

- Let G be a group (in a suitable category of "spaces"). In the theory of
principal bundles, the functor which assigns to a space X, the set of
principal G-bundles over X, modulo isomorphism, is represented (in the
homotopy category of spaces) by the so called classifying space BG of G.
That is, BG "classifies" principal G-bundles. Then The Lemma implies a
fundamental theorem that characteristic classes for bundles are the same
as cohomology classes of the classifying space.

- Similarly, one can consider the classifying stack of a group scheme
(i.e. scheme with group structure), in particular a finite group, G.

- Every spectrum, in the sense of stable homotopy theory, represents a
so-called generalized cohomology theory, and vice versa. The Lemma then
gives a way to compute natural operations between theories. The results of
computation of the algebra formed by operations on the "ordinary"
cohomology theory (with coefficients in a prime field), known by the name
the Steenrod algebra, is the input of the Adams spectral sequence, which
in turn computes (in principle) the stable homotopy groups of spheres,
which is of central interest in the field.

- On the category of commutative ring spectra, which are 'by definition'
spectra with commutative ring structure, the (covariant) functor
classifying characteristic classes, or "orientations", in the associated
multiplicative (because of the ring structure) generalized cohomology
theories is represented by the so-called Thom spectrum. Quillen pointed
out that the variant MU of Thom spectrum, classifying Chern classes, or
orientations for complex vector bundles, corresponds to the moduli stack
of formal groups (i.e. the stack classifying formal groups) thus
discovering a deep connection between homotopy theory and algebraic
geometry. MU has since been a key object in stable homotopy theory.

- One of the greatest recent achievements in algebraic topology is the
construction of a spectrum called tmf, the topological modular forms. It
is the global section of a certain sheaf of commutative ring spectra over
the moduli stack of elliptic curves. From this sheaf, one can recover the
Adams-type spectral sequence associated to tmf. According to Lurie, this
sheaf is actually the structure sheaf of the moduli stack classifying
"oriented elliptic curves" over commutative ring spectra, or, to be in the
correct variance, over derived affine schemes, in the world of derived
algebraic geometry. This extremely beautiful viewpoint enlightens the
meaning of Quillen's discovery just mentioned.

The disputed proposition (whether it is a theorem or a lemma) or its
appropriate generalization applies to any of these situations.

Another family of examples of representable functors would be supplied by
those represented by "dualizing objects" appearing in various contexts.
However, at this moment, I only have a vague idea of how the Yoneda lemma
would imply something useful in this situation. I think experts out there
are well in order to help me with this!

Concerning the discussion on the "fundamental theorem" of category theory,
it might worth remarking that preservation of limits by right adjoints
(and its dual) are a corollary of the more fundamental fact that adjoints
compose, granted uniqueness of the adjoint functor. The last is notably
one of the important consequences of The "Lemma".

Also, in addition to the claimed prominent applicability in mathematics,
the Yoneda lemma has remarkably neat and witty statement:
"Every presheaf represents itself."

Best wishes,
Takuo

On Mon, 15 Jun 2009, Makoto Hamana wrote:

> Dear Ellis,
>
> On Fri,  5 June 2009 16:36:23 -0400, Ellis D. Cooper wrote:
> | There are Fundamental Theorems of Arithmetic, Algebra, Calculus, and
> | indeed, many more.
> | My question is, What would be candidates for the Fundamental Theorem
> | of Category Theory?
> | Yoneda Lemma comes to my mind. What do you think?
>
> I have asked Prof. Yoneda many years ago why Yoneda Lemma is
> called "Lemma", not "Theorem". He said that perhaps it was a
> bit about internal of category theory rather than insisting
> on applications to other mathematics. Doesn't Yoneda Lemma
> satisfy (c) in Mile Gould's post? I don't know how much
> Yoneda Lemma is useful in other areas of mathematics, and
> I have wanted to know it.
>
> On Sat,  6 June 2009 23:22:52 +0100, Miles Gould wrote:
> | My suggestion would be the theorem that left adjoints preserve colimits,
> | and right adjoints preserve limits.
> | This may not be the deepest theorem in category theory, but
> | (a) it's pretty darn deep,
> | (b) it describes a beautiful connection between two fundamental notions
> | in the subject,
> | (c) it admits a huge variety of applications in "ordinary" mathematics.
>
> Best Regards,
> Makoto Hamana
>
>
>
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>


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