(This continues the "no fundamental theorems please" thread under a more apropos heading.) On 6/24/2009 8:17 AM, Steve Vickers wrote:
Vaughan Pratt wrote:
... The Yoneda Lemma usually states that J embeds (meaning fully) in [J^op,Set], ... The usual statement is significantly stronger than that (see e.g. Mac Lane, Mac Lane and Moerdijk, or Wikipedia). It says that, for contravariant functors F: C -> Set, the elements of FX are in bijection with transformations to F from the representable functor for X. Your statement can be deduced by considering the particular case where F too is representable.
Quite right, I should have used something other than "lemma" for that weaker statement. Misplaced force of habit. Googling "Theorem 5.1 (Yoneda)" returns section 5, on full completeness, of Phil Scott's nice Handbook of Algebra chapter. Phil introduces that section with "The most basic representation theorem of all is the Yoneda embedding: Theorem 5.1 (Yoneda) If J is locally small, the Yoneda functor Y: J --> [J^op,Set], where Y(j) = J(-,j), is a fully faithful embedding." (Phil's A is my J, which I'll use throughout as the base whether small or not in order to make it easier to compare theorems.) If "theorem" is good enough for Phil it's good enough for me. For definiteness let's name the theorems as follows. YT (Yoneda Theorem): [J^op,Set] fully extends J. YL (Yoneda Lemma): For any functor F: J^op --> Set and object j of J, F(j) is in bijection with [J^op,Set](J(-,j),F). (This replaces C in Steve's version with J^op.) DYT (Dense Yoneda Theorem): C is equivalent to a category of presheaves on a *small* category J if and only if C densely extends J. Here a category of presheaves on J means a full subcategory of [J^op,Set] containing the functors J(-,j) for all objects j of J (that is, "on J" qualifies not just the individual presheaves but the whole category of them, as in "free group on X"(*)). C densely extends J just when there exists a full, faithful, and dense functor K: J --> C. (If someone has a use for a broader notion of dense extension I wouldn't object to saying "fully densely extends.") For now assume "dense" is defined as in X.6 of CWM, namely that every object of C arises as a colimit of FP for the evident functor P: (K,c) --> J, (K,c) being the comma category whose morphisms are of the form Kj-->c, j in ob(J). (I'll consider equivalent ways of defining density in a followup message.) This has the effect of representing each object c of C as the presheaf C(K-,c): J^op --> Set (namely C(Kj,c) is the set of morphisms of (K,c) of the form Kj-->c) and each homset C(c,d) as [J^op,Set](C(K-,c),C(K-,d)) (that is, Nat(C(K-,c),C(K-,d)) in the CWM notation Steve is using), via the adjunction defining "colimit." The homsets [J^op,Set](J(-,j),F) used by YL (and hence YT) are always small even if J isn't. DYT however deals with arbitrary homsets [J^op,Set](F,G), which may be large when J is, whence DYT's requirement that J be small. (Steve Lack would be the person to ask what form DYT might take for large J.) The applicability of YL and YT to large J makes them incomparable with DYT. However if we require J to be small then we have YT < YL < DYT in the sense that each of them represents entities by using progressively more of [J^op,Set]. YT uses just the image of the Yoneda embedding to represent J, YL takes one step beyond YT by using all the homsets from that image to an arbitrary presheaf in [J^op,Set] to represent carriers of algebras with unary operations (aka presheaves), and DYT takes one more step than YL by using homsets between more general presheaves of [J^op,Set] to represent *all* homsets of C. Replacing "equivalent to" by "representable as" in DYT makes more explicit the sense in which DYT is unambiguously a representation theorem for certain abstractly defined categories, namely dense extensions of a small category, which are of greater generality than the categories contemplated in YT or YL.
There can be no doubt that this strong Yoneda Lemma is vitally important when calculating with presheaves - for example, it shows immediately how to calculate exponentials and powerobjects. If F and G are two presheaves, then the exponential G^F is calculated by
G^F(X) = nt(Y(X), G^F) (by Yoneda's Lemma) = nt(Y(X) x F, G) (by definition of exponential)
I would only buy that reasoning for small J (otherwise why should G^F be a functor to Set?), where YL isn't benefiting from the additional generality of YL over DYT. Is there any other reason to prefer YL to DYT? It doesn't seem a very natural candidate for theorem-hood (perhaps that's why it's relegated to the status of a lemma). It's hard to argue for it as the fundamental theorem of CT when it's sandwiched in between YT and DYT, both of which *do* come across as real theorems (the reason I'm comfortable calling both of them theorems). Anyone remember the history of why YL rose to the top?
I don't think you can get it and its useful consequences from your weaker statement, even if you start strengthening yours in the way you suggest by supplying converses.
Don't underestimate the power of the converse. YL < DYT because it's only a partial converse, DYT is the whole thing.
Another closely related and important result, though not known as Yoneda's Lemma as far as I know, is that the presheaf category over C is a free cocompletion of C, and the Yoneda embedding is the injection of generators.
The colimit-based definition of dense functor shows that this is pretty much equivalent to DYT and therefore stronger than YL in what I take to be the sense in which you consider YL stronger than YT. (Is there a stronger sense of "stronger," e.g. a natural nonstandard model in which YT is true and YL false, or YL true and DYT false?) Michael Shulman's comment about the enriched case is apropos here: defining cocompletion for conical (co)limits isn't sufficient when V doesn't permit them (for want of diagonal functors). This limitation can be avoided with left Kan extensions as treated in Chapter 4 of Kelly, "Basic Concepts of Enriched Category Theory." Kelly arrives at the notion of free cocompletion 15 pages into Chapter 4 and long after indexed (nowadays weighted) colimits (Chapter 3) as a satisfactory generalization of conical colimits. (Incidentally, of what use are non-free cocompletions? Is there any reason not to define "cocompletion" to make it free? I seem to recall people being happy to drop "free" in this context. Who ordered "free"?) This post is quite long enough already so I'll stop it here with the interesting question (to me anyway), what is the earliest point in the development of (ordinary or unenriched) category theory at which one can introduce either cocompletion or density in order to have functors that are dense, full, and faithful? Can either be usefully introduced before functor categories, for example? And can this be done with sufficient generality to carry over essentially unchanged to the enriched case? I imagine this would reduce to just avoiding conical colimits. Vaughan Pratt (*) But not as in "free beer on Stallman." [For admin and other information see: http://www.mta.ca/~cat-dist/ ]