October 1997 - Categories - categories.org.au

abstract algebraic geometry
by Zhaohua Luo 20 Dec '97

20 Dec '97

The following report is based on my paper CATEGORICAL GEOMETRY. The plan is to generalize (and simplify) Diers's theory of Zariski categories (presented in his book [D]). A more detailed report (in LaTex) is available upon request. Comments and suggestions are welcome. Zack Luo ABSTRACT ALGEBRAIC GEOMETRY by Zhaohua Luo (1997) It is well known that most geometric-like categories have finite limits and finite stable disjoint sums. These are lextensive categories in the sense of [CLW]. We introduce the notion of an analytic category, which is a lextensive category with the property that any map factors as an epi followed by a strong mono. The class of analytic categories includes many natural categories arising in geometry, such as the categories of topological spaces, locales, posets, affine schemes, as well as all the elementary toposes. A large class of analytic categories is formed by the opposites of Zariski categories in the sense of Diers [D]. The notion of a Zariski category captures the categorical properties of commutative rings. Many algebraic-geometric analysis carried by Diers for a Zariski category can be done for a more general analytic category in the dual situation. We show that the notion of a flat singular epi developed in [D] can be applied to define a canonical functor from an analytic category to the category of locales, which is a framed topology in the sense of [L1] and [L2]. This topology plays the fundamental role of Zariski topology in categorical geometry. 1. Unipotent Maps and Normal Monos Consider a category C with a strict initial object. Two maps u: U --> X and v: V --> X are "disjoint" if the initial object is the pullback of u and v. If S is a set of maps to an object X we denote by N(S) the sieve of maps to X which is disjoint with each map in S. The set S is called a "unipotent cover" on X if N(S) consists of only initial map. We say S is a "normal sieve" if S = N(N(S)). A map is called "unipotent" if it is a unipotent cover. A mono is called "normal" if it generates a normal sieve. If C has pullbacks then a mono is normal iff any of its pullback is not proper unipotent. The class of unipotent (resp. normal) maps is closed under compositions and stable, and any intersection of normal monos is normal. Geometrically a unipotent map (resp. normal mono) plays the role of a surjective map (resp. embedding). 2. Framed Topologies Consider a functor G from C to the category of locales. A mono u: U --> X in C (and G(u): G(U) --> G(X)) is called "open effective" if G(u) is an open embedding of locales, and any map t: T --> X in C such that G(t) factors through G(u) factors through u. If u is open effective then u or U is called an "open effective subobject" of X, and G(u) or G(U) is an "open effective sublocale" of G(X). We say G is a "framed topology" on C if an object X is initial iff G(X) is initial, and any open sublocale of G(X) is a join of open effective sublocales. If {U_i} is a set of open effective subobjects of X such that G(X) is the join of {G(U_i)}, then we say that {U_i} (resp. {G(U_i)}) is an "open effective cover" on X (resp. G(X)). The collection T(G) of open effective covers is a Grothendieck topology on C. We say G is "strict" if its Grothendieck topology T(G) is subcanonical. 3. Divisors Here is a general method to define framed topologies. A class D of maps containing isomorphisms is called a "divisor" if it is closed under compositions, and its pullback along any map exists which is also in D; we say D is "normal" if any map in D is a normal mono. If D is a divisor, a sieve with the form N(N(T)), where T is any set of monos to X in D, is called a "D-sieve" on X. One can show the set D(X) of D-sieves on X is a locale and the pullbacks of D-sieves along a map induce a morphism of locales. Thus each divisor D determine a functor L(D) to the category of locales. If D is normal then L(D) is a framed topology, called the "framed topology" determined by D. 4. Extensive Topologies. Recall that a category with finite stable disjoint sums is an extensive category. An extensive category C has a strict initial object. An injection of a sum is simply called a "direct mono". An intersection of direct monos is called a "locally direct mono". The class of direct monos is a normal divisor E(C), called the "extensive divisor". The extensive divisor E(C) determines a framed topology, called the "extensive topology". It generalizes the Stone topology on the category of Stone spaces. For any object X we denote by Dir(X) the set of locally direct subobjects of X, viewed as a poset with the reverse order. If any intersection of direct monos exist in C, then Dir(X) is a locale for any object X, and Dir is naturally a functor from C to the category of locales, which is equivalent to the extensive topology. Special cases of extensive topologies were considered by Barr and Pare [BP] and Diers [D1]. 5. Analytic Topologies An "analytic category" is a lextensive category with epi- strong-mono factorizations. In the following we consider an analytic category C. One of the most important notion introduced by Diers to categorical geometry is that of a flat singular map. We consider the dual notion. A mono v: V --> X is a "complement" of a mono u: U --> X if u and v are disjoint, and any map t: T --> X such that u and t are disjoint factors through v. A complement mono is normal. A mono v: V --> X is called "singular" if it is the complement of a strong mono u: U --> X. A map f: Y --> X is called "coflat " if the pullback functor C/X --> C/Y along it preserves epis. The main point here is that any pullback along a coflat map preserves epi-strong-mono factorizations. A coflat singular mono is called an "analytic mono". A coflat normal mono is called a "fraction" (thus any analytic mono is a fraction). A fraction plays the role of local isomorphism in algebraic geometry. The class of coflat maps (resp. analytic monos, resp. fractions) is closed under compositions and stable. The class of analytic monos is a normal divisor A(C), called the "analytic divisor". The analytic divisor A(C) determines a framed topology, called the "analytic topology". It generates the usual Zariski topology on affine schemes. We say C is "strict" if its analytic divisor A(C) is strict. 6. Reduced and Integral Objects The analytic topology can also be defined algebraically, using reduced and integral objects, as in the case of affine schemes. An object is "reduced" if any unipotent map to it is epic. A non-initial object is "integral" if any non-initial coflat map to it is epic One can show easily that any quotient of a reduced (resp. integral) object is reduced (resp. integral) (i.e. if f: Y -- > X is an epi and Y is reduced or integral then so is X). A unipotent reduced strong subobject of an object X is called the "radical" of X. It is the largest reduced and the smallest unipotent strong subobject of X, thus is uniquely determined by X. An analytic category is "reduced" if any unipotent map is epic. An analytic category is reduced iff its strong monos are normal. An analytic category is "reducible" (resp. "spatial") if any non-initial object has a non-initial reduced (resp. integral) strong subobject. If any intersection of strong monos exist in C then the full subcategory of reduced objects is a coreflective subcategory; if moreover C is reducible then any object has a radical. 7. Spectrums A strong mono is called "disjunctive" if it has an analytic complement. An object is disjunctable if its diagonal map is a disjunctable regular mono. An analytic category is called "disjunctable" if any strong mono is disjunctable. An analytic category is "locally disjunctable" if any strong subobject is an intersection of disjunctive strong subobjects. A locally disjunctable reducible analytic category in which any intersection of strong subobjects exist is called an "analytic geometry". Let C be an analytic geometry. If X is an object we denote by Loc(X) (resp. Spec(X)) the set of reduced (resp. integral) strong subobjects of X, where Loc(X) is regarded as a poset with the reverse order. Then Loc(X) is a locale with Spec(X) as the set of points. If C is spatial then Loc(X) is a spatial locale. Since any quotient of a reduced (resp. integral) object is reduced (resp. integral), Loc (resp. Spec) is naturally a functor from C to the category of locals (resp. topological spaces). The functor Loc is equivalent to the analytic topology on C. If C is spatial then Spec determines Loc, thus in this case we simply say that Spec is the analytic topology on C. The space Spec(X) is called the "spectrum" of X. A spatial analytic geometry C together with the topology Spec is a metric site defined in my paper [L1]. Any object in C is separated (i.e. its diagonal map is universally closed). In fact Spec is the smallest separated metric topology on C. The metric completion of a strict spatial analytic geometry plays the role of "schemes" in categorical geometry. 8. Zariski Geometries A cocomplete regular category with a strict analytic opposite is a Zariski category in the sense of Diers if it has a strong generating set of finitely presentable objects including the terminal object which are disjunctable in its opposite. The opposite of a Zariski category is a strict spatial analytic geometry, whose analytic topology coincides with the Zariski topology defined by Diers. We introduce a (simplified) geometric version of a Zariski category. A strict locally disjunctable analytic category is called a "Zariski geometry" if it is a locally finitely copresentable category with a finitely copresentable initial object. Any Zariski geometry is a strict spatial analytic geometry with coherent spectrums. Most of the theorems proved by Diers in [D] for a Zariski category can be extended to any Zariski geometry. 9. Examples (a) An analytic category is "coflat" if any map is coflat (or equivalently, any epi is stable). In a coflat analytic category any epi is unipotent, any singular mono is analytic, any normal mono (thus any analytic mono) is strong, and any integral object is simple. (b) In a reduced coflat disjunctable analytic category, the notions of strong, normal, analytic, singular, and fractional mono are the same. (c) Any elementary topos is a coflat disjunctable analytic category; its analytic topology is determined by the double negation; a topos is reduced iff it is boolean; a reducible Grothendieck topos is an analytic geometry. (d) The category of locales is a reduced analytic geometry; its analytic topology is the functor sending each locale to the locale of its nuclei. (e) The category of topological spaces (resp. posets) is a reduced coflat disjunctable spatial analytic geometry; its analytic topology is the discrete topology. (f) The category of coherent spaces (resp. Stone spaces) is a reduced spatial analytic geometry; its analytic topology is the patch topology. (g) The category of Hausdorff spaces is a strict reduced disjunctable spatial analytic geometry; its analytic topology is the Hausdorff topology. (h) The opposite of the category of commutative rings is a Zariski geometry; its analytic topology is the Zariski topology. References [BP] Barr, M. and Pare, R. Molecular toposes, J. Pure Applied Algebra 17, 127 -152, 1980 [CLW] Carboni, C. Lack, S. and Walters, R. F. C. Introduction to extensive and distributive categories, Journal of Pure and Applied Algebra 84, 145-158, 1993. [D] Diers, Y. Categories of Commutative Algebras, Oxford University Press, 1992. [D1] Diers, Y. Categories of Boolean Sheaves of Simple Algebras, Lecture Notes in Mathematics Vol. 1187, Springer Verlag, Berlin, 1986. [L1] Luo, Z. On the geometry of metric sites, Journal of Algebra 176, 210-229, 1995. [L2] Luo, Z. On the geometry of framed sites, preprint, 1995.

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Topos-abelian categories that are neither
by Vaughan Pratt 01 Nov '97

01 Nov '97

Toposes and abelian categories are striking for the number of elementary properties they have in common (monic+epi = iso, mono/epi is a unique factorization system, etc. etc.) and the paucity of their common models, namely just the final category. This observation prompts the following questions. 1. Are there any other pairs of large and/or useful classes of categories whose respective theories have so much in common yet whose models have so little in common? 2. What can be said of the class of those categories having all the elementary properties common to toposes and abelian categories? In particular does it contain anything other than toposes and abelian categories? And if so, does this outcome change when the language is extended to say second order logic? To the extent that both toposes and abelian categories share much pleasant structure, the models of the intersection of their theories, for a suitable choice of language, would seem to be a nice class in its own right. Vaughan Pratt

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pratt cat
by Peter Freyd 31 Oct '97

31 Oct '97

Vaughan wrote: To the extent that both toposes and abelian categories share much pleasant structure, the models of the intersection of their theories, for a suitable choice of language, would seem to be a nice class in its own right. There are several variations on the topic. I'll look first at the theory obtained by starting with finite completeness, coterminator, pushouts for pairs of maps one of which is monic, pushouts for kernel pairs then adjoining all universal Horn sentences in these predicates that hold for both the category of abelian groups and the category of sets (or, if prefered, hold for all abelian categories and all pre-topoi). We will see that this theory can be finitely axiomatized and that the representations of its models into the categories of abelian groups and sets are collectively faithful. This will be the "first task. The representation will be done by finding a single faithful representation into the product of an abelian category and a topos and using known theorems thereon. By a representation I here mean a functor that preserves finite limits, coterminators, pushouts of pairs of maps one of which is monic and pushouts of kernel pairs. (It follows that a representation preserves epimorphisms, finite coproducts and images.) If we expand the theory to include all positive AE sentences it remains finitely axiomatizable, and the only models are those categories that are a product of an abelian category and a pre-topos. This will be the "second task". After that, I'll add some more essentially algebraic structure so that we can talk about power-objects using universal Horn sentences and obtain again the finite axiomatizability and the fact that the the only models are those categories that are a product of an abelian category and a pre-topos. This will be the "third task". Note that each of axioms below is true for both abelian categories and topoi. Until we reach axiom VAE, all can be turned into universal Horn sentences if the completeness conditions listed above are turned into an essentially algebraic theory. V1) Effective regular category. (Yes this can be stated as universal Horn conditions on pullbacks and the special pushouts mentioned above.) V2) 0 -> 1 is monic. Note that it follows that all maps from 0 are monic. In the diagrams below all non-horizontal lines should have arrows added in downwards direction. V3) If A -> C is monic and A / \ B C \ / D is a pushout then it's a pullback, V4) and so is 0xA / \ 0xB 0xC \ / 0xD. Note that binary coproducts therefore exist and are disjoint. V5) If B -> 0xC is epic and A / \ 0 B \ / 0xC is a pullback then it's a pushout. Two definitions: let 0 x X / \ 0 X be a product diagram. If the map to 0 is an iso, I'll say that X is type-T; if the map to X is an iso, I'll say that it's type-A. Note that (using V2) an object is type-A iff it has a map to 0. The type-A objects are closed under binary products and coproducts, subobjects and quotient objects. The full subcategory of A objects is coreflective, with AX = 0xX as coreflector. Most important, it is an abelian category. V5) If the rhombi in A B |\ /| D C E \ | / F are pullbacks and if D E \ / F is a coproduct diagram, and if all the objects are type-T then A B \ / C is also a coproduct diagram. Note that an object is type-T iff every type-A object has a unique map thereto. Type-T objects are closed under binary products and coproducts, subobjects and quotient objects. (For coproducts use V3.) The full subcategory of type-T objects forms a reflective subcategory: the reflector, T, is defined via the pushout diagram: 0xX / \ 0 X \ / TX One now has what's needed to show that the full subcategory of type-T objects is a pre-topos. Note that the functor A preserves pullbacks and pushouts of pairs of maps one of which is monic (V4). Add V6) T preserves pullbacks. (It clearly preserves pushouts.) Only one more axiom is needed for the first task: V7) If f is a map such that Af and Tf are both isos then f is an iso. For the second task add: VAE) For every X there's a map TX -> X such that AX TX \ / X is a coproduct diagram. Note that TX -> X is a coreflector (making V6 redundant). One can force the map to be unique by further requiring that TX -> X -> TX is the identity map. And note that the type-A objects can not be reflective: if 1 has a map to any type-A object the entire category collapses. For the third task, I'll add the following structure: functions on objects P,E,l, r such that lX:EX -> PX and rX:EX -> X and V8) <lX,rX>:EX -> PXxX is monic and PX, EX are type-T. I need one more operation, denoted with an upcase lambda, here written as /\. Given a pair of maps f:R -> Y, g:R -> X where R is of type-T, then /\(f,g): Y -> PX is defined and V9> The composition of relations /\(f,g) (lX)' rX Y ----------> PX -------> EX ---------> X is equal to f' g Y -------> R --------> X (where ' is being used to denote the reciprocation operator on relations). V10) If R f / \ g Y EX h \ /lX TX is a pullback and R of type-T, then /\(f,(rX)g) = h. Note that in the full category of type-T objects, P yields power-objects with E, l, r naming the universal relations. (V10 provides the uniqueness condition.) In any abelian category the only type-T object is the zero object, which forces P=E=0 for abelian categories. It's routine that both A and T preserve the new structure. We can now remove the existential from VAE. Define TX -> X as the image of rX. The third task is finished with: V11) For every X AX TX \ / X is a coproduct diagram. These axioms will, no doubt, be much simplified.

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LL Workshop in Utrecht
by Workshop Problems Advances in the Semantics of Linear Logic 28 Oct '97

28 Oct '97

************************************************************** * * * WORKSHOP * * PROBLEMS AND ADVANCES IN THE SEMANTICS OF * * LINEAR LOGIC * * * ************************************************************** * Utrecht, 28 and 29 november 1997 * ************************************************************** On friday 28 and saturday 29 november there will be a workshop, hosted by the Mathematical Institute of Utrecht University, dedicated to recent work and work in progress in the semantics of linear logic: games, coherent and hyper coherent spaces, Chu spaces, phase semantics, bi-completion of categories, models for light linear logic et cetera. The meeting will take place in the center of the town of Utrecht. There will be a number of talks by invited speakers, including Jean Yves Girard, Samson Abramsky and Vaughan Pratt. We welcome your participation, as well as proposals for communications on work related to the subject of the workshop. For all information, please contact the organizers at: pasll(a)math.ruu.nl You can also look at URL: http://www.math.ruu.nl/pasll/ Vincent Danos Harold Schellinx Equipe de Logique Mathematique Mathematisch Instituut Universite Paris VII Universiteit Utrecht

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logic jobs at Carnegie Mellon
by Steve Awodey 24 Oct '97

24 Oct '97

CARNEGIE MELLON UNIVERSITY Department of Mathematical Sciences Visiting Positions in Logic The Carnegie Mellon Department of Mathematical Sciences seeks visitors in logic for one or more semesters (possibly one or two years) of the 1998-99 and 1999-2000 academic years. We seek persons who share interests with the logic faculty and can teach at the undergraduate and graduate level. Information about our department and Carnegie Mellon's interdisciplinary Ph.D. program in Pure and Applied Logic can be obtained from: http://www.math.cmu.edu/ http://www.cs.cmu.edu/afs/cs/project/pal/www/pal.html Applicants should send a vita, list of publications, and a statement describing current and planned research to: Appointments Committee, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213. Recent Ph.D's should also arrange to have at least three letters of recommendation sent to the same address. Carnegie Mellon University is an Affirmative Action/Equal Opportunity Employer.

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PSSL65 - Second Announcement
by Carsten Butz 21 Oct '97

21 Oct '97

Peripatetic Seminar on Sheaves and Logic, PSSL 65 SECOND ANNOUNCEMENT The 65th meeting of the Peripatetic Seminar on Sheaves and Logic will take place in Aarhus (Denmark) over the weekend of 1-2 November 1997. As usual, we welcome talks on category theory, sheaves, logic, and related areas. To register, please fill out the form below and return (preferably by email) to Carsten Butz (butz(a)brics.dk, postal address below) as soon as possible. We maintain a WWW-page with information for the seminar, although this is only in addition to the standard procedures (which means: registration, confirmation, hotel and travel information will be sent out by email or snail mail, as usually). The URL is http://www.brics.dk/~butz/PSSL65/pssl65.html and http://www.brics.dk/Activities/97/PSSL65/index.html (mirror). The home page contains links to travel information; we point out that there are direct flights from London (Heathrow) and Amsterdam to Aarhus Airport. Also, the airport of Billund has many international connections (including direct flights from Amsterdam, Berlin, Birmingham, Brussels, Paris, Frankfurt, London and Manchester). Both airports are connected to Aarhus City by a good bus service. >>From Central Europe, Aarhus can by reached by train via Hamburg/Flensburg. The following people (from outside Aarhus) have already indicated that they will take part in the seminar: R. Bvrger M. Jibladze P.T. Johnstone J. Koslowski E. Palmgren D. Pataraia J. Power S. Soloviev We have options for pension/hotel rooms in the range of Kr 300 - Kr 500. On Friday evening there will be a get-together in a downtown pub. Lectures will start on Saturday at 9.30 am. More detailed information about hotels etc. will be sent out by surface mail to those who registered for the seminar. Regards, Carsten Butz and Anders Kock --------------------------------------------------------------------- Looking forward to meeting you in Aarhus, Carsten Butz and Anders Kock Postal address: Carsten Butz BRICS Computer Science Department Aarhus University Ny Munkegade, Building 540 8000 Aarhus C Denmark email: butz(a)brics.dk (PSSL65 email address) WWW: http://www.brics.dk/~butz/PSSL65/pssl65.html http://www.brics.dk/Activities/97/PSSL65/index.html ---------------------------------------------------------------- I want to attend the 65th PSSL in Aarhus, Denmark. Name: Address: : : : email: WWW (if available): I wish to give a talk entitled: I wish to reserve accommodation for the following nights (single/double): ------------------------------------------------------------------

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Path Algebras
by A.L.Heyworth 17 Oct '97

17 Oct '97

I am trying to find a definition of a path algebra, does anyone have an easy definition? Is a path algebra with one "object" an ordinary (K-)algebra? Anne Heyworth (University of Wales, Bangor)

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Weak higher dimensional categories
by ajp＠dcs.ed.ac.uk 16 Oct '97

16 Oct '97

Those people interested in Tom Leinster's paper on multicategories and weak higher dimensional categories might also be interested in recent, closely related work by Claudio Hermida at McGill. I do not think there is a paper available yet, but he has given talks at Vancouver and at the recent meeting of the Canadian Math Society in Montreal, so there are probably slides available.

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Operads, multicategories
by Tom Leinster 13 Oct '97

13 Oct '97

An advertisement for an article, available by electric transmission from http://www.dpmms.cam.ac.uk/~leinster. ABSTRACT Notions of `operad' and `multicategory' abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term (S,*)-multicategory, subject to certain conditions on S and *. Different choices of S and * give some of the existing notions. We then describe the algebras for an (S,*)-multicategory, and finish with a selection of possible further developments. Our approach enable concise descriptions of Baez and Dolan's opetopes and Batanin's operads; both of these are included. Tom Leinster

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Induction and functors preserving Cartesian closure
by burns 11 Oct '97

11 Oct '97

In my last mailing, I talked about my attempt to develop vector spaces from CCC's with finite sets and real arithmetic. Now it's time for some questions. (1) Assume a category with domains I, standing for a finite set; and R, standing for the reals. If it's a CCC, we can get the biproduct structure, by currying the Kronecker delta, i.e. the equality function on I. Even to express very simple statements about vectors, one now needs a legitimate way to write _summation over_ I. E.g. if we have a full set of injectors eta : I -> X where X = [I->R] we want to write an arbitrary vector as Sigma x(i) (eta i) i e I This is shorthand for something like compose : R x R x ... x R -> X where the R's correspond 1-1 with I. It's the domain I itself (rather than any element of it) which determines that finite product. We know what we mean, by summation over I; but it seems to me that (1) we are relying on induction over I, and therefore on first, last and successor arrows to I, the whole Peano development; and (2) that this is not explicit in the CCC definition. Yes, it is stated that a CCC has all finite products. But to make this formal, we need a formal means of expressing the inductive step; the equivalent, perhaps of a FOR loop in algorithms. Is there a standard way to address this? (2) To make the issue more pointed, let's step back from our category, and name it X(I,R). Now, define a category of finite sets, say Y, in which I1, I2, ... are domains, and each one can stand in for I, so that we have X(I1,R), etc. We would like to say, there is a category V(Y,R), of all such categories; and define a functor V(-,R): Y => X(Y,R) : I |=> X(I,R) We should be able to say that VECT(R) is a proper subcategory of V(,-R), because each f : Y -> Y maps onto V(f,R) : X(Y,R) -> X(Y,R) : X(I,R) |-> X(f(I),R) But there is more to V(-,R) than that. For one thing, there will be a unique arrow in V(-,R), expressing summation (or in general what they call "fold", or APL, "reduce"), fold : [RxR ->R] x Y -> [[Y -> R] -> R] such that fold <(+),I> : [[I -> R] -> R] : (i |-> a(i)) |-> Sigma(i) a(i) The fold arrow expresses _reduction by a real binary function, over an inductable set_, producing a unique arrow in each X(I,R) that does the job. Now if this makes me uneasy, it's possibly just "abstraction vertigo". As far as I can make out, fold is justified iff in Y, each domain I is equipped with arrows: first : 1 -> I last: 1 -> I next : (I-last(I)) -> I That means that in Y itself, one can define: First: Y -> (1->Y) : I |-> (first : 1 -> I) etc. Something is slipping here, and I think its that the notation is being overloaded when "I" refers both to a domain in Y, and to the actual finite set it denotes. It seems to me there much be a well-tried way of doing this, but my reading into categories in language specification haven't disclosed anyone spelling it out. (3) Now for a major issue. I managed to define my biproduct in X(I,R) by defining X(I,R) as a CCC, with real arithmetic, equality on I, _and only that_. Which seems to me a notable finding, and one to be followed up. In introducing functors on X(I,R), and further, functors on X(Y,R), I want to deduce the consequences of X(I,R) being Cartesian closed. Therefore I want to develop just those functors which map one CCC to another. Such functors will only be coherent, so to speak, if they map values to values, products to products, and exponentials to exponentials. That is, if they preserve Cartesian closure. What has been said about the properties of such functors? I'm following one lead, John W. Gray, _A Categorical Treatment of Polymorphic Operations_ in _S-V Lecture Notes in Computer Science 298, Mathematical Foundations of programming Language Semantics_. Let's just say about this, that the development of functors between exponential domains is more complex than I had expected. Enough for the present.

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