Re: Fundamental Theorem of Category Theory?

Re: Fundamental Theorem of Category Theory?

[Matsuoka Takuo]

Dear categorists,

Although I know this thread is basically over, I would like to thank
Paul Taylor for (earlier than my previous post) pointing out that not
every subject of mathematics has or should have its single "fundamental
theorem". While I think one theorem can be more important than another,
what may more worth discussing (still not necessarily here on the list)
could be what fundamental theorems are wanted for the future.

Best wishes,
Takuo


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Re: Fundamental Theorem of Category Theory?

[claudio pisani]

The Yoneda Lemma is in fact a particular case of the reflections of Cat/X in discrete (op)fibrations over X (the reflection of an object x:1->X gives the slice X/x -> X, which corresponds to the representable X(-,x));
another particular case (X=1), gives the components of a category.
The above reflections are a consequence of the "comprehensive" factorization systems (final functors, discrete fibrations) and (initial functors, discrete opfibrations) on Cat.
It turns out that several aspects of category theory can be developed in any finitely complete category C with two factorization systems properly related (the main axiom is "reciprocal stability").
Thus category theory can be indeed founded on (a generalization of) the Yoneda Lemma; in particular, in this perspective, universal properties inside C depend on the universal properties which follow from the factorization systems.

Best regards

Claudio Pisani







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Re: Fundamental Theorem of Category Theory?

[Vaughan Pratt]

On 6/17/2009 3:45 PM, Steve Lack wrote:
> Hmm. Not sure if you mean you're allowing any full subcategory of
> [J^op,Set]; if so then you should drop the requirement that J-->C be fully
> faithful.
By "category of presheaves on J" I had in mind retaining J as part of it.

>> Am I missing something?  I was thinking that followed from density of J
>> in C.
>>
>
> No. The category Setf of finite sets has a fully faithful dense inclusion in
> to the (presheaf) category Set of all sets, but Set is not [Setf^op,Set].

Oops, right, I was mixing up cocomplete and cocompletion-of.  (Actually
I don't think in terms of either, I find it easier to think of
[J^op,Set] as the maximal dense extension of J up to equivalence, in the
sense that all dense extensions of J are full subcategories of it.)

Vaughan


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Re: Fundamental Theorem of Category Theory?

[Matsuoka Takuo]

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
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>


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Re: Fundamental Theorem of Category Theory?

[Steve Lack]

On 17/06/09 1:28 PM, "Vaughan Pratt" <pratt@cs.stanford.edu> wrote:

> Steve Lack wrote:
>> Your proposed characterization is actually a characterization of full
>> subcategories of [J^op,Set] containing the representables.
>
> Right, that's what I meant by "*a* category of presheaves on J" (as
> opposed to *the* category of all presheaves on J), the point of my
> analogy with Archimedean fields (as opposed to the field of all reals).
>

Hmm. Not sure if you mean you're allowing any full subcategory of
[J^op,Set]; if so then you should drop the requirement that J-->C be fully
faithful.

>> To get the whole
>> presheaf category you should add that C is cocomplete,
>
> Right, just as to get all of the reals one should say that the
> Archimedean field is complete.  For situations where one doesn't need
> the whole thing it is convenient to be able to characterize the
> categorical counterpart of an Archimedean field, with J in place of Q,
> as any full, faithful and dense extension of J.  Density serves to keep
> the extension inside [J^op,Set], just as it keeps Archimedean fields
> inside R.
>
>> and that homming out
>> of objects in J is cocontinuous (i.e. C(j,-) is cocontinuous for all j in
>> J).
>
> Am I missing something?  I was thinking that followed from density of J
> in C.
>

No. The category Setf of finite sets has a fully faithful dense inclusion in
to the (presheaf) category Set of all sets, but Set is not [Setf^op,Set].

Steve.


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Re: Fundamental Theorem of Category Theory?

[Fred E.J. Linton]

Once, long, long ago, I looked up the Yoneda paper 
then cited as source for the Y.L. 
Agreed: not there. 

But, in another Yoneda paper ("On Ext and exact sequences",
perhaps, I'm relying on memory alone, here), it *is* there,
not called Y.L., of course, but describing, as I recall, 
the connection between n.t.(hom(A, -), hom(B, -)) and 
hom(B, A) in the case that the hom-sets are the 
Ext equivalence classes (the only case of interest 
for that paper).

It didn't take much, either, to see the underlying 
Y.L. structure in the main proof there.

Cheers (and more detail, if called for,
once I'm back from Montrreal), 

-- Fred

------ Original Message ------
Received: Wed, 17 Jun 2009 09:22:42 AM EDT
From: "Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>
To: Makoto Hamana <hamana@cs.gunma-u.ac.jp>, <categories@mta.ca>
Subject: categories: Re: Fundamental Theorem of Category Theory?

> On Mon, 15 Jun 2009, Makoto Hamana wrote:
> 
> > 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.
> >
> When I lecture on category theory to first-year graduate students, I
> tell them there are two things they should remember about the
> Yoneda Lemma: it isn't a lemma, and it was never published by Yoneda.
> In this respect it resembles that bulwark of the British constitution,
> the Lord Privy Seal (who is none of the three things that his title
> claims).
> 
> Peter Johnstone
> 



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Re: Fundamental Theorem of Category Theory?

[Reinhard Boerger]

Dear all,

I strongly agree to Ellis Gould's quote of Bill Lawvere's remark on the
Yoneda Lemma:

> -- a mathematical theory corresponds "roughly to
> the definition of a class
> of mathematical objects"

One of the most important points in category theory are universal
properties. The existence of universal solutions is equivalent to the
representability of certain functors - at least under reasonable smallness
conditions. This is closely related to the Yoneda Lemma; therefore it is
really one of the fundamental theorems to me.


Greetinge
Reinhard



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Re: Fundamental Theorem of Category Theory?

[Vaughan Pratt]

Steve Lack wrote:
> Your proposed characterization is actually a characterization of full
> subcategories of [J^op,Set] containing the representables.

Right, that's what I meant by "*a* category of presheaves on J" (as
opposed to *the* category of all presheaves on J), the point of my
analogy with Archimedean fields (as opposed to the field of all reals).

> To get the whole
> presheaf category you should add that C is cocomplete,

Right, just as to get all of the reals one should say that the
Archimedean field is complete.  For situations where one doesn't need
the whole thing it is convenient to be able to characterize the
categorical counterpart of an Archimedean field, with J in place of Q,
as any full, faithful and dense extension of J.  Density serves to keep
the extension inside [J^op,Set], just as it keeps Archimedean fields
inside R.

> and that homming out
> of objects in J is cocontinuous (i.e. C(j,-) is cocontinuous for all j in
> J).

Am I missing something?  I was thinking that followed from density of J
in C.

Vaughan


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Re: Fundamental Theorem of Category Theory?

[Steve Lack]

Dear Vaughan,

Your proposed characterization is actually a characterization of full
subcategories of [J^op,Set] containing the representables. To get the whole
presheaf category you should add that C is cocomplete, and that homming out
of objects in J is cocontinuous (i.e. C(j,-) is cocontinuous for all j in
J).

Steve.


On 16/06/09 7:58 AM, "Vaughan Pratt" <pratt@cs.stanford.edu> wrote:

> Apropos of the Yoneda Lemma, is there some reason why it is usually
> stated on its own rather than as one direction of a characterization of
> categories of presheaves on J?  Unless I've overlooked or misunderstood
> something it seems to me that the Yoneda Lemma should state that C is a
> category of presheaves on J if and only if there exists a full,
> faithful, and dense functor from J to C.
>
> This should generalize the characterization of an Archimedean field as
> any dense extension of the rationals.
>
> Vaughan Pratt
>


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Fundamental Theorem of Category Theory?

[Ellis D. Cooper]

The archived file of this list
http://www.mta.ca/~cat-dist/archive/1992/92-08.txt
contains comments by Colin McLarty, Michael Barr, and Jim Lambek
about the Yoneda lemma.

Also, Peter Freyd gives an account of the connection (via Mac Lane
and Barry Mitchell) between Prof. Yoneda and the eponymous lemma.

Ellis D. Cooper



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Re: Fundamental Theorem of Category Theory?

[Prof. Peter Johnstone]

On Mon, 15 Jun 2009, Makoto Hamana wrote:

> 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.
>
When I lecture on category theory to first-year graduate students, I
tell them there are two things they should remember about the
Yoneda Lemma: it isn't a lemma, and it was never published by Yoneda.
In this respect it resembles that bulwark of the British constitution,
the Lord Privy Seal (who is none of the three things that his title
claims).

Peter Johnstone



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Fundamental Theorem of Category Theory?

[Ellis D. Cooper]

Dear Makoto,

At 11:08 AM 6/14/2009, you wrote:
>I don't know how much Yoneda Lemma is useful in other areas of
>mathematics, and
>I have wanted to know it.

The Yoneda Lemma came to mind partly because of M. Barr and C. Wells book
"Toposes, Triples and Theories." Its Preface recounts that in the sense of
Lawvere's insight -- a mathematical theory corresponds "roughly to
the definition of a class
of mathematical objects" -- toposes, triples, and theories are
beautifully connected fundamental notions.

Barr-Wells write that the Yoneda Embeddings Theorem, "the first of
several important consequences" of the Yoneda Lemma, "in one way or
another is used in
practically every mathematical argument in this book." (p. 27)

Perhaps subscribers to this list would care to comment on how
specific results in this book apply or relate to computer science,
other areas of mathematics, logic, or physics.

All the best,
Ellis D. Cooper




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Re: Fundamental Theorem of Category Theory?

[Vaughan Pratt]

Apropos of the Yoneda Lemma, is there some reason why it is usually
stated on its own rather than as one direction of a characterization of
categories of presheaves on J?  Unless I've overlooked or misunderstood
something it seems to me that the Yoneda Lemma should state that C is a
category of presheaves on J if and only if there exists a full,
faithful, and dense functor from J to C.

This should generalize the characterization of an Archimedean field as
any dense extension of the rationals.

Vaughan Pratt


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Re: Fundamental Theorem of Category Theory?

[Makoto Hamana]

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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RE: Fundamental Theorem of Category Theory?

[Hasse Riemann]

 
 
Dear Ellis
 

I also had this question when i started with category theory but
i was satisfied with the Yoneda lemma. Now thanks to your question
i know more theorems to answer this.

I don't think you can get a better answer than the replied suggestions.
 
However there is also higher category theory.
The interesting point would now be to generalize:
 
What are the coresponding theorems for strict/weak n-categories?
 
I plan to at least ask for and suggest a higher dimensional Yoneda lemma.
 
The other adjoints preserving limits theorem is also interesting to
generalize. But here as far as i know there is no concept of adjoint
for 3-categories and higher up. I am more uncertain as to limits,
but i have not seen limits in n-categories defined in the
graceful style of limits in 1-categories.
 
Best regards
Rafael Borowiecki
 
 

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Re: Fundamental Theorem of Category Theory?

[Miles Gould]

On Mon, Jun 08, 2009 at 07:44:40AM -0400, tholen@mathstat.yorku.ca wrote:
> You could make your choice more comprehensive: Freyd's General and
> Special Adjoint Functor Theorems give a more complete picture of the
> fundamental relationship between limit preservation and adjointness.

Indeed. I think there's an analogy to be made between these theorems and
the Fundamental Theorem of Calculus: one side is very simply stated, and
the other requires more care. Compare

* d/dx (integral f(x) dx) = f(x),
* integral (d/dx f(x)) dx = f(x) [up to constant offset...]

with

* all right adjoints preserve limits,
* all limit-preserving functors [satisfying some caveats...] are right adjoints.

Miles


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Re: Fundamental Theorem of Category Theory?

[tholen]

You could make your choice more comprehensive: Freyd's General and
Special Adjoint Functor Theorems give a more complete picture of the
fundamental relationship between limit preservation and adjointness.

Regards, Walter.

Quoting Miles Gould <miles@assyrian.org.uk>:

> On Fri, Jun 05, 2009 at 04:36:23PM -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?
>
> My suggestion would be the theorem that left adjoints preserve colimits,
> and right adjoints preserve limits.
>
...



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Re: Fundamental Theorem of Category Theory?

[Fred E.J. Linton]

On Sat, 06 Jun 2009 05:51:38 PM EDT, "Ellis D. Cooper" <xtalv1@netropolis.net>
asked:

> 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?

Perhaps that, yes; or, perhaps, the characterization of representable
functors.

Cheers, -- Fred




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Fundamental Theorem of Category Theory?

[Ellis D. Cooper]

Dear category theory community,

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?

Best,
Ellis D. Cooper

Ellis D. Cooper, Ph.D.
978-546-5228 (LAND)
978-853-4894 (CELL)
XTALV1@NETROPOLIS.NET



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