Having been cast in the role of Brutus -- saving the Republic of Category Theory from the crowning of a Fundamental Theorem -- I would like to say that it's "Not that I lov'd Caesar less, but that I lov'd Rome more". Takuo Matsuoka, whom I thank for the undeserved compliment, gave an excellent list of some of the applications of the Yoneda embedding as a way of enriching categories in various mathematical disciplines. The first of these was about schemes in Algebraic Geometry, which he has been explaining to me privately, despite my long-term mental block on this subject. Some of the following was part of my response to him, but I don't want to put him under pressure to be the spokesman for a subject that, as I discovered later, is not actually his own. So, this posting is addressed to algebraic geometers, synthetic differential geometers, and others, to give a sketch of some of the categorical techniques that have been developed for general topology by domain theorists over the past two decades. Whilst their subjects manifestly have a richer history than domain theory did, we now have a few new tricks that they seem to have missed. However, I shall start with some textbook material. The Yoneda Lemma is to category theory what Cayley's Theorem is to groups. An object of a category can be REPRESENTED by the collection of all INCOMING morphisms (from all of the other objects), just as a group can be represented by its permutation action on itself. This representation is faithful for the tremendous but trite reason that an object is represented by its own identity map. Here is a syntactic example: a type X in a lambda calculus is represented by all of its terms, possibly involving free variables. One such term is the single variable x:X, which is the identity in the category. The other objects of the category are the other types, or, given that there may be many free variables, the CONTEXTS, which are lists of typed variables. In proof theory the letter Gamma is used for a typical context. This yields a presheaf whose value at Gamma is the set of terms Gamma |- a:X of type X, using only the variables from Gamma, subject to the equivalence defined by the theory. It defines a contravariant functor in which the morphisms act by substitution. This example is spelt out in detail in my book, "Practical Foundations of Mathematics", in particular in Section 7.7, which describes categorical methods for proving normalisation, conservativity, consistency, etc. I use the letter Gamma for a typical object of ANY base category (not just for a syntactic context but also for a topological space or affine variety) over whose categories we may want to consider presheaves. In so far as there was a previous convention, it was to use "U" (Juergen Koslowski suggested to me that this stood for "Umgebung" = neighbourhood), "I" (for "index") or "-". Since a presheaf is a family of SETS subject to some bookkeeping, the huge advantage of the Yoneda embedding is that we can bring the power of SET THEORY to bear on categorical problems. (Rest assured, you will see that I have not gone over to the Dark Side.) For example, let f,g:X==>Y be parallel maps in a category C. The object X is represented by the SETS C(Gamma,X) of incoming maps, so we can consider the SUBSET ---- f ----> E(Gamma)={x|f.x=g.x} >---> C(Gamma, X) C(Gamma,Y) ---- g ----> of maps x:Gamma-->X that have equal composites with f and g. Notice that these are the data for the universal propery of an equaliser. Indeed, the equaliser E>-->X==>Y exists in C iff the presheaf E(-) is REPRESENTABLE, ie it is of the form C(-,E) for some object E of C, which is unique up to unique isomorphism. For this reason I also use the letter Gamma as the test object FROM which maps come in any universal property. I would urge students to work through this method for exponentials too, first writing down either their universal property or their introduction and elimination rules. In topos theory, another interesting presheaf is given by the set Sub(Gamma) of "sieves" on an object Gamma. This presheaf is representable iff the category C is a topos, and the representing object is the subobject classifier Omega. I this case, I invite you to work backwards from the definition of the subobject classifier to find out what a "sieve" is. In general, the Yoneda embedding preserves universal properties such as limits that are given by maps FROM a generic object Gamma. However, it completely DESTROYS the colimits in C, giving it new ones instead. Indeed, we can think of a presheaf as a "formal" colimit diagram. Now sheaf theory was invented to "PATCH" things together, that is, to add COLIMITS such as pushouts. But usually we don't want to scrap the old colimits altogether, because some of them were "already correct". So, we specify which ones we want to keep, and such a specification is called a GROTHENDIECK TOPOLOGY. Then, instead of using ALL presheaves, we just keep the ones that respect our chosen "correct" colimits, and call them SHEAVES. In particular, we are quite often happy with the existing COPRODUCTS in the base category. In particular, these will be stable and disjoint in any topos, but not the same under one Grothendieck topology as under another, so if the ones that we have in the original category already have this property then they are probably correct. Nowadays, a category with nice finite coproducts is called EXTENSIVE; topological spaces, locales and affine varieties all enjoy this property. Therefore, the colimits that we want to change are usually the epis, coequalisers pushouts and filtered (=purely infinite) ones. Presheaves may be adapted directly to the task of adding colimits of a particular kind. For example, Peter Johnstone describes the "IND" construction that adds filtered colimits in Section VI.1 of "Stone Spaces". In the dual category, this is how we obtain profinite objects, which often come with a compact Hausdorff but totally disconnected topology. A Grothendieck topos is therefore a truly magnificent beast, but it is a SET THEORY, whereas sheaves were originally introduced to study algebraic topology and algebraic geometry. I want to consider some ways in which it might be adapted to the construction of a category that is actually like whichever subject we wanted to study. For various reasons we consider that the category of affine varieties (the formal opposite of the category of commutative rings) and the traditional category of topological spaces are not rich enough for the requirements of their subjects. For example, they are not cartesian closed. So we introduce SCHEMES and (for example) EQUILOGICAL SPACES to do a better job. (Recall that I don't actually know what a scheme is, though I have my guesses, and the reason for spelling out all of this general category theory is to elicit a definition that I can understand, in return for some new techniques that algebraic geometers might find useful.) It is not altogether surprising that the new categories are represented as subcategories of the category of presheaves on the old categories. Since the NEW objects are conceived as "patched together" from old ones, they are, by design, faithfully represented by the incoming maps from (all of) the OLD objects. Steve Vickers' posting of 9 June concerned presheaves on the category of locales. (Locales have very strong formal analogies with affine varieties, being the formal opposite of a category of algebras.) My posting on the same day about "Aspects of locale theory" is also relevant to this Rather more people have studied subcategories of presheaves on the category of topological spaces. Pino Rosolini gave an excellent survey of these categories in his 2000 paper on "Equilogical spaces and filter spaces". You can obtain this and several other papers that provide many of the details of what I am about to say from his webpage www.disi.unige.it/person/RosoliniG/biblio.html Analogously to the "ind" construction above, Dana Scott's equilogical spaces formally adjoin quotients of equivalence relations to topological spaces, whilst Reinhold Heckmann's construction of "equilocales" does the same for locales. A Grothendieck topos is a set theory, but what in the construction makes it so? It is the requirement on the "correct colimits" (Grothendieck topology) that they be stable under all PULLBACKS. The SHEAFIFICATION functor that reflects presheaves to sheaves also preserves all finite limits. If we just want a CARTESIAN CLOSED category, rather than a topos, it is enough that the data and reflection functor be preserved by PRODUCTS, rather than general pullbacks. The categories that I shall describe all contain the base category but are contained in the smallest category of sheaves, so it becomes irrelevant whether we consider sheaves as intermediaries or go straight from presheaves to the smaller category. Along with several other people in the 1990s, Rosolini and I studied ways of defining and constructing categories like these, under the heading of SYNTHETIC DOMAIN THEORY (SDT). This was inspired by synthetic differential geometry, and I think Dana Scott is responsible for the name. However, most of the other people who worked under this banner were interested in realisability toposes rather than sheaf toposes, which was part of the reason why I changed to a different name. Key to this is the object SIGMA. Like Omega, this classifies subobjects, but now just OPEN ones in topology, or RECURSIVELY ENUMERABLE ones in computation. Rosolini identified the abstract properties of a class of monos (or "dominion") and its classifier ("dominance") in his PhD thesis. My "Euclidean principle", Fx & x = FT & x, puts it in an algebraic form -- see "Geometric and Higher Order Logic" on my webpage at www.PaulTaylor.EU/ASD/. However, if algebraic or differential geometers want to pick this idea up, they should not be misled by the idea of classifying subobjects of any kind. It is the algebraic property that matters. As a ring, Sigma should be the ring of polynomials in one variable. As an affine variety, it is the geometric incarnation of the ground field. This means that, for an affine variety X, the set of maps X->Sigma is (the underlying set of) the corresponding ring. Similarly, in topology it is the set of open subsets, ie the corresponding frame. The idea of ASD is that this "set" of maps should itself be a space, but I don't know what this is for algebraic varieties. Sigma is the spider at the centre of the web. Considered as a space, it carries the relevant algebraic structure (a lattice or a ring), and maps X-->Sigma determine the structure of other spaces X. (Notice that this is the opposite way from the maps Gamma-->X that motivated the category of presheaves.) In synthetic domain theory we devised various properties, involving maps X-->Sigma, that would say whether the presheaf X was to be regarded as a "domain". I am going to stick to those that can be expressed in pure category theory and generalised to other subjects. The weakest one, a kind of T0 property, was that X >---> Sigma^Y, ie that the presheaf X be a subobject (family of subsets) of some exponential. (Recall that you constructed exponential presheaves as an exercise earlier!) A stronger one is that X >---> Sigma^Y ====> Sigma^Z as an equaliser of such exponentials. Usually there is a reflection functor from all presheaves to the "domains", and this preserves products. I forget exactly what was in each of Rosolini's papers, but he studied the analogy between this reflection functor and sheafification, and also what happens in presheaf categories over various interesting concrete categories. Even with this modification of the notion of sheaf topos, the Yoneda embedding and Grothendieck toposes have at best been given the status of a "constitutional monarchy", so it is time to introduce some genuinely republican ideas. My programme "Abstract Stone Duality" moves away from the models based on toposes to a direct axiomatisation of the subject that we actually want to study. This axiomatisation consists of two parts -- some pure category theory and some specifically topological axioms on top of it. The reason why I believe ASD could be adapted to other subjects is that the topological axioms say little more than that Sigma is a lattice, so this could be replaced by a ring. MOST of the work is done by the underlying category theory, and this will be even more strongly so in the developments of the theory that I am currenly studying. In ASD, I concentrate on powers of Sigma, rather than of general objects, and other structure that can be defined in these terms. The version of my theory that is reasonably well established starts from the hypothesis that the adjunction Sigma^(-) -| Sigma^(-) be MONADIC. The result of this is an account of computably based locally compact spaces and computable continuous functions. I applied this to elementary real analysis, and obtained a computable construction of the Dedekind reals in which [0,1] is compact --- contrary to the received wisdom of Russian Recursive Analysis that this is impossible, and to Bishop's constructive analysis, which develops a lot of the subject in a "can do" fashion but without using compactness of [0,1]. The problem with this theory is that, rather than enlarging the traditional category to a cartesian closed one that is embedded in a category of presheaves, it cuts down to locally compact spaces. However, I am currently writing up a new technique that replaces the monadic assumption. In one manifestation, it gives an account of all sober topological spaces and continuous functions. However, since it says that products preserve epis, it doesn't work for locales, because of the counterexample that I gave in my "aspects of locale theory" posting. On the other hand, this DOES work for affine varieties -- over a field, because the same problem with locales arises if you try to work over a general commutative ring. (It comes down to whether modules are FLAT, ie tensor product with them preserves monos.) I talked about "illegitimate" presheaves in my "categories" posting on "aspects of locale theory", and Steve Vickers said something on the same topic. Set-theoretic issues aside, presheaves are "heavyweight" gadgets in that you have to give data pertaining to every object of the base category, so a scheme is defined by its relationship to ALL rings. I am offering a much more "lightweight" construction. In this, to give an object of the new category (a "scheme", maybe) would involve only a handful of objects and morphisms of the original one (rings). This "lightweight" construction will give an account of equilogical spaces (with some of their more set-theoretic features removed). I would like to know what the relationship is between the analogous result that I have for algebraic varieties, and whatever the official definition of SCHEME is in algebraic geometry. My category of "schemes" (if that is what it is) would be embedded in the presheaf category and closed therein under finite limits and exponentials. It would contain all affine varieties, along with their products, equalisers, pullbacks, coproducts and epis but not coequalisers (coproducts, coequalisers, pushouts, products and monos but not equalisers of rings). The (possible) difference from the existing notion of scheme is that I would give you the SMALLEST category of presheaves that has these properties. Along with this would come a "synthetic" and computable axiomatisation similar to the one that I give for elementary real analysis in www.paultaylor.eu/ASD/lamcra/elemcalc Finally, why is the Yoneda Lemma not a Theorem? Because Lemmas do the work in mathematics, whilst Theorems, like royalty, just take the credit. Thank you for listening. Paul Taylor www.PaulTaylor.EU [For admin and other information see: http://www.mta.ca/~cat-dist/ ]