Limits, colimits, and adjoints: I go along with this: it is the result of general category theory that I have used most in studying colimits of forms of multiple groupoids, for homotopical applications. It really does come under `categories for the working mathematician'. I have also been attracted in the same vein by fibrations and cofibrations of categories: see a recent paper in TAC. I well remember a remark of Henry Whitehead in response to a visiting lecturer saying: `The proof is trivial.' JHCW: `It is the snobbishness of the young to suppose that a theorem is trivial because the proof is trivial.' (There was and is no answer to that!) (His example was Schroder-Bernstein.) The leads to the interesting question of what makes a theorem nontrivial? Good discussion topic for the young (at heart). Ronnie Brown [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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

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/ ]

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

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

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

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

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

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/ ]

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

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