D IAGRAMMES M ARIE B JERRUM A Sketch theoretical survey Towards a typology of mathematical structures Diagrammes, tome 61-62 (2009), p. 1-63 © Université Paris 7, UER math., 2009, tous droits réservés. L’accès aux archives de la revue « Diagrammes » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A Sketch theoretical survey Towards a typology of mathematical structures Marie Bjerrum April 2005. SK 00I8 R I0 SK m l  m  P P 09 n  O N 1 : o o  p 1 M q L b∗ b∗ L+ 1 : M  q L r 1 ; K  s 1 J < t  1 J 2 q t 0 = I 2  -$5 S uiSqz 1I I1 =q01 2 1 S S k  R C [ C [ @ FG 8 J G W @ G X l 8 2 Q Q > 7 m   t  m 3 69 P P t 9 J K ?o n  s  O 3 6: L :  r N M 3p o @ 6   ~ q ∗ ∗ qM 3 : qM M A   c C id d 6: } α∗ β ∗ C+ α β q N3 O p 5 L A ;r } 5; o o K  4 P P s < J t | m 5n< 4  Q Q B  J t 5 = 4 R SC!  }{ l m =  )1 ukm op 4 2 4 0∗ q K = u βS 2 I S IK2\ β 0∗ 4 =  5 ;:u> B CS G 2W [ Id8 J B J > > 4 2  5 t t J t 9 J K 4qu ? 5s t  K s  3 ? 5 L L  : r 6 @r M ~M M q@ : ~ M 3 3q q A p p q A j j 1 2 6 2 1 N N }   } p A o pO P  A 6  } ; o o O 3P3 } o 6 P Q B P B  | m n< Q | m n 7 2  Q R C Q R C  m = { l m 2 S!)    uk}{ l  op ∗ t S KK 3 I [ \ ] ] ^ ^ _` `q3a q2b qb1 c c1:u@ SG 3W ^ dIm[ [ \ ] ] ^ ^ =_ ` ` a 2 2b b c S IK K3 t∗ J > t 1 J t ? K 1 s  L r 1 @ M ~ q 1 τ∗ M w w q A k k 2 1 V τ∗ V+ 2 1 N } p 1 O P A o } 0 P Q B 0  n o  | m m Q R C0 { l 0 op S K4 I Q Q S4 S I K4 Masters Thesis for the Cand. Scient. degree in mathematics at the University of Copenhagen. Abstract We look at the advantages of adopting the sketch theoretical point of view when considering mathematical structures, i.e. when one considers mathematical theories as categories of models of a sketch and models as certain functors. We hereby get a formalization of the notion of a type of mathematical structure/theory, as well as a direct interplay between the sketch describing a theory (syntax) and the properties described (semantic). This leads to a fruitful application of the generalized associated sheaf theorem for certain strict types suggesting a general diagrammatic method for proving and discriminating, treating proofs as syntactic factorizations in the category of sketches, and discrimination as semantic investigations by comparison of model categories. Using this application we then work out two examples of how to typify classical mathematical structures according to where and how they are defined. Contents 1 Preface and Acknowledgements 4 2 Introduction 5 3 Preliminaries 3.1 Categories, Functors, natural transformations. . . . . . . . . . . . . . . . . . . 3.2 Projective and inductive limits, or limits and co-limits. . . . . . . . . . . . . . 7 8 10 4 Sketches 4.1 Models and sketchability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 12 13 5 A category of sketches 5.1 Sketching sketches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Sketches as models of a projective sketch . . . . . . . . . . . . . . . . . . . . . 19 20 27 6 Theoretical types 6.1 Types and syntax: proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Types and semantics: discrimination . . . . . . . . . . . . . . . . . . . . . . . 28 30 32 7 Examples 7.1 The classical type tclf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Two sketches of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 A type confusing the two sketches of fields . . . . . . . . . . . . . . . . 7.3 Two sketches of monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 A type confusing the two sketches of monoids . . . . . . . . . . . . . . 34 35 48 54 58 67 8 Conclusion 81 9 Bibliography 83 4 1 Preface and Acknowledgements Preface and Acknowledgements The style of the present work is mostly influenced by the french school of category theory initiated by the french mathematician Charles Ehresmann (1905-1979) and then further developped by his students. My work is more directly related to this later work based on his ideas. I in particular want to thank Laurent Coppey (now retired french mathematician, Paris VII) for introducing me to sketches and category theory in general and for making the initial suggestions for my work in his article about sketches and types (Coppey, L. [1992]). Also for introducing me to René Guitart (Paris VII), to whom I owe my warmest acknowledgement for writing me a guideline presenting basic ideas about types, basic definitions and notation, and then for having been my unofficial (but indispensable) supervisor throughout the work. Finally I would like to thank Carsten Butz (IT-University of Copenhagen) for agreeing to supervise this Masters thesis and thereby have made it possible for me to write in a subject rather foreign to the mathematics department at the University of Copenhagen. 5 2 Introduction The general motivation for such work, as here presented, has already been elegantly phrased by a great name in the history of mathematics, David Hilbert (1862-1943): The question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into several branches, whose representatives scarcely understand one another and whose connections become ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge we are still scaresly conscious of the similarity of the logical devises, the relationship of the ideas in mathematical theory and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science.[...]Every real advance [in mathematical science] goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. (Quoted from ”Mathematiche probleme”(1901), English translation BAMS 8,p. 478-79, by M.W. Newson). Category theory, including sketch theory, provides a natural language (a logic (Guitart, R. [1981])) able to analyse contemporary mathematical work at an abstract level; taken out of specific contexts. Thus, at least some part of mathematical activity struggles against the above feared (and horrible) fate. This thesis looks at how sketches, in describing mathematical theories/structures, can give direct diagrammatic ways of formulating usual mathematical problems on an abstract categorical level, and how this method makes it possible to consider different aspects of structures within a general frame (a category), using one and same (non set-theoretical) diagrammatic language (category theory). Now why is this interesting? I here present seven mottos, that could tempt someone to adopt the sketch theoretical point of view: 1 Sketches furnish a nice and direct way of formalizing the notion of a type of mathematical theory/structure. 2 Sketching is proving and proving is sketching; proofs become objects in a category, simply defined by their relations to other objects. (Coppey, L. [1992]) 3 Everything usually described by first order logic, can be described, more directly in terms of limits and co-limits, i.e. every first order theory is sketchable. (Guitart, R., Lair, C [1982]) 4 We can apply the generalized associated sheaf theorem and profit from universal properties to talk about freely generated theories of a sketch and suggest general methods for proving and discriminating. 6 Preliminaries 5 The application of the generalized associated sheaf theorem then also leads to joining the MacLane motto that ”everything is Kan extensions”. 6 By sketches, we get a double usage of universal properties: to study syntactic questions when specifying theories, but at the same time, to determine necessary semantics. Hence syntax and semantics becomes of the same, purely diagrammatic, nature. 7 Describing structures by sketches makes possible a formalization of how to discriminate between different ways of defining the same concept, avoiding false confusions of set theoretical definitions. (Coppey, L. [1992]) These mottos will all more or less directly find support in this thesis. Now the more precise goal is to present a formalization of the notion theoretical type and then lay forth a basic understanding of what role these types play in the on the one hand syntactic and on the other semantic aspects of considering mathematical theories, emphasizing the double aspect of types in proving and discriminating. The presentation is followed by two examples, the first one a rather simple example concerning two ways of sketching a field and the second example concerns two sketches of monoids. They are both examples of two different sketches with the same set theoretical models and we are then interested in finding out in what type of categories the two given sketches have the same models (equivalent model categories). This question turns out to have many faces, touching all of the above mottos. But our main objective is again two see how syntax and semantic become two sides of the same diagrammatic procedure. Working out these two examples according to ideas indicated by René Guitart (partly based on the article Coppey, L. [1992]) has been my method of conduction. So the reader is invited to think of this thesis as an introduction to a general formalization of what mathematicians mean when they use the concept a type of structure, by working out two examples suggested in different articles (Coppey, L. [1992], Guitart, R. [1988]). The main content consists of: an introduction of a category of sketches (section 5.) that will serve as the main frame, wherein the notion theoretical type is to be defined. Then we look at the syntax and semantic of sketches by discussing proving and discriminating in the the category of sketches limited to cases where we can apply the generalized associated sheaf theorem, which then leads to a general problem (section 6.). At last we present the two examples (section 7.) following this general problem. 3 Preliminaries The reader is supposed to be familiar with category theory. But since the style of this paper is very much influenced by the french school, I nevertheless, choose to briefly introduce some absolute basic concepts. 3.1 Categories, Functors, natural transformations. •Oriented Graph: We say that S = (ObS , ArS , c, d, i) is an oriented graph if and only if: - ObS is a set (set of objects). 3.1 Categories, Functors, natural transformations. 7 - ArS is a set (set of arrows) and we write S(A, B) or Hom(A, B) for the subset of arrows from A to B. - c : ArS / ObS is the map sending an arrow to its target (co-domain). - d : ArS / ObS is the map sending an arrow to its source (domain). / ArS is a map that sends an object C ∈ S to its corresponding - and id : ObS identity arrow 1C , such that: ObS i u d / ArS id c : c · id = d · id = 1ObS . •Multiplicative graph: S · = (S, CoS , k) is said to be a multiplicative graph, if and only if - S is an oriented graph, - CoS is the set of composable pairs of arrows ( i.e. a subset of all pairs of consecutive arrows {(g, f ) ∈ ArS × ArS | d(g) = c(f )}), - k : CoS / ArS is a partial composition (multiplication) of arrows k(g, f ) = g ◦ f , given subjected to the following axioms: Unitarity: ∀g ∈ ArS : (id · c(g), g), (g, id · d(g)) ∈ CoS et k(id · c(g), g) = k(g, id · d(g)) = g Position: ∀(g, f ) ∈ CoS : d(g ◦ f ) = d(f ), c(g ◦ f ) = c(g) i.e. an oriented graph with a table of equations concerning a partial composition. •Category: A category C is a multiplicative graph where CoC is the set of all consecutive arrows and where the composition k is associative. The notion ”category” makes sense even if Ob et Ar are not sets. But then one of course has to be conscious of all use of ordinary set theory. In this thesis there will be certain size limitations, which will be discussed when appropriate. / D is a •Functor : If D and C are multiplicative graphs, a (covariant) functor F : C / ObD et F1 : ArC / ArD such that: pair of applications F = (F1 , F0 ): F0 : ObC ∀(g, f ) ∈ CoC : (F (g), F (f )) ∈ CoD , F (g ◦ f ) = F (g) ◦ F (f ); ∀C ∈ ObC : F (1C ) = 1F (C) noting (abusively) F in stead of F0 or F1 . / D , then a natural transformation •Natural transformation: Take functors F, G : C / G is defined by a family of maps ( τC : F (C) τ :F / D the following diagram commute: all arrows f : C . F (C) F (f ) τC  F (D) / G(C)  τD G(f ) / G(D) / G(C) )C∈Ob C such that for 8 Preliminaries If for all C ∈ C τC is an isomorphism, we say that F and G are naturally isomorphic and we write F ∼ = G. We say that two categories C and D are equivalent, and write C ∼ D, if there / D and G : D / C such that F ◦ G ∼ exist functors F : C = 1D , G ◦ F ∼ = 1C • The join of the graph K with the graph I, is denoted IK and contains: -two subgraphs (supposed disjoint) isomorphic to I and K (and hence identified with these). / I such that -In addition, for all pairs (I, K) ∈ ObI × ObK there is an arrow λKI : K / I 0 ∈ ArI and such that λK 0 I · κ = λKI j · λKI = λKI 0 for all K ∈ ObK and for all j : I / K 0 ∈ ArK . for all I ∈ ObI and for all κ : K i κ ' 0 I and K = K So if for example I = I graph: IK i  I 0 ql . K K λK 0 I κ where all composites commute. For a graph-homomorphism f : K f : IK K 0 then we get IK as the multiplicative λKI I nr I ) λKI 0  . K0 . λK 0 I 0 / K we associate the ”natural” graph-homomorphism / IK0 (without changing its name) defined by: f (i) = i for f (κ) = f (κ) i ∈ ArI for f (λKI ) = λf (K)I for κ ∈ ArK I ∈ ObI , ( where the second f is f : K K ∈ ObK . / K0 ) We will also use the notation I − for the joint graph I1 (1 is the graph of one object and only the identity arrow) and the notation I + for the joint graph 1I , ·=  ===  ==  ==     I @ · ^=  ===  ==  ==    I i.e. just joining a vertex to a graph I, such that I becomes the basis of a cone or co-cone. 3.2 Projective and inductive limits, or limits and co-limits. Let I be an oriented graph, C a category and ϕ : I graphs). / C a functor (between oriented 3.2 Projective and inductive limits, or limits and co-limits. 9 •Projective cones or just ”cones”: a projective cone (in C) with basis ϕ is a couple (A, q) / ϕ(I) )I∈Ob of arrows in C, satisconsisting of an object A from C and a family ( qI : A I fying / J ∈ ArI : ∀i:I ϕ(i) ◦ qI = qJ i ( i.e. all triangles in the cone commute. If I = { I I J }, we can illustrate this as follows: C A ϕ I ( J # qJ qI  ϕ(I) ϕ(i) ,  ϕ(J) ϕ(I) Let Λϕ denote the set of projective cones (in C with basis ϕ). •Inductive cones or ”co-cones”: it is the dual notion, i.e. (B, ι) is a pair consisting of an / B )I∈Ob such that ∀ i : I / J ∈ ArI , we have object B and a family ( ιI : ϕ(I) I ιJ ◦ ϕ(i) = ιI and analogous we note Vϕ the set of co-cones with basis ϕ. •Limits: A cone (P, p) with basis ϕ is said to be a limiting cone (or limit of ϕ), written lim ←−ϕ, if it fulfils the following universal property: ∀(A, q) ∈ Λϕ ∃! h : A / P : (A, q) = (P, p) ◦ h; P pI   ∃!h tf _ X {A %   qI p  qJ  J ϕ(I) ϕ(i) ,   ϕ(J) where (P, p) ◦ h is to be understood as the cone (A, p ◦ h) obtained by composing all arrows in p with h. Likewise (dually) (S, s) is said to be a limiting co-cone with basis ϕ, written − lim →ϕ, if it fulfils / the following universal property: ∀(B, ι) ∈ Vϕ ∃! k : S B : (B, ι) = k ◦ (S, s). / C , we say that C has I-limits [resp. I• If lim lim ←−ϕ, [resp. − →ϕ] exist, for all ϕ : I co-limits]. / C there has been chosen a limit • If C has I-limits [resp. I-co-limits] and if for all ϕ : I l (ϕ) [resp. − l (ϕ)] of ϕ, we say that (C, ← l ) [resp. (C, − l )] is a category with chosen I-limits ← − → − → [resp. I-co-limits]. 10 4 Sketches Sketches We here introduce the basic notions concerning sketches. • a projective sketch is a pair (S, P) with S a multiplicative graph, called the underlying graph of the sketch, and P is a set of cones in S, called the set of distinguished cones. • A mixed sketch is then a triple (S, P, I) with a supplementary set I of distinguished co-cones. / σ 0 is a graph morphism such that any •Sketch Morphism: a sketch morphism s : σ distinguished cone (co-cone) in σ is mapped to a distinguished cone (co-cone) in σ 0 . i.e. if a / σ then s(c) will be a distinguished cone with distinguished cone c in σ has basis B : I basis s ◦ B. 4.1 Models and sketchability We here introduce the sketch theoretical notion of a model and how a sketch then can be viewed as describing/sketching a theory. / C from • A model of a sketch σ = (S, P, I) (I perhaps empty), is a functor R : S the underlying graph to a category C that transforms all distinguished cones and co-cones in σ into limits and co-limits in C. • A category C is called sketchable in a category D if there exist a sketch σ, such that we have an equivalence of categories Mod(σ, D) ∼ C. / D and arrows all natural where Mod(σ, D) is the category with objects all models σ transformations beween models, we then view C as the theory sketched by σ in D. •For a sketch morphism t : σ → σ 0 we will call forgetful functor the corresponding functor between model categories: Ut : Mod(σ 0 , Set) / Mod(σ, Set) R /R◦t  Generalised Associated Sheaf Theorem For applications later we briefly state a general result concerning the existence of a left adjoint to a forgetful functor: Generalised associated sheaf theorem: Given a sketch morphism t : σ → σ 0 between two projective sketches, then if σ 0 is small the forgetful functor Ut has a left adjoint.  There is also a version of this result called Kennison’s Theorem (stated in Barr and Wells’s ”Toposes, Triples and Theories”, Springer 1985). The version here stated is from the french 4.2 Sketching 11 school of Charles Ehresmann and his students and the proof goes by a transfinite induction using Kan extensions. The way of really understanding what a sketch is, is by sketching and sketching is what this thesis is all about. We start with some quite elementary examples which are also indispensable later on. 4.2 Sketching Sketch of multiplicative graphs: when we consider the definition of a multiplicative graph, we see that there are the following four sets in play: objects, arrows, consecutive arrows and composable arrows, which indicate that a graphic version of ”what a graph is” is to be modelled in Set and should contain four Objects: S1 to be realized as the set of objects, S2 to be realized as the set of arrows between these as well as the identity arrows. This is the oriented graph part which can be sketched as follows: S1 g w d id / S2 d · id = c · id = 1S1 . c Then we add an object S3 to be realized as the set of all consecutive arrows, the subset of all pairs of arrows (f, g) where the domain of f matches the co-domain of g. We get S1 g w d id / S2 p1 w S3 g c p2 where p1 , p2 will be the restrictions of the projections from the arrow product. If S3 plays the wanted role, it should when modelled, by a model with underlying multiplicative graph S, / ArS with d·g = c·f there exist posses the universal property that for all arrows f, g : A / ArS ×Ob ArS such that p1 h = g, p2 h = f , where ArS ×Ob ArS a unique arrow h : A S S is to be understood as the fibered product over ObS , the pullback of d, c modelled. This means that we need to distinguish the following cone: S3)  ))   ))) p1  ))p2 ))   ))   s1  S21 S2 11 1 c d 11    S1 s1 just naming the composites. And at last we need an object S30 to be realized as the subset of the object of consecutive 12 Sketches arrows for which there is defined a composition. We get S1 g d w c p1 w / S2 C O [g id p01 = S3 k p02 p2 j S30 p1 j = p01 , p2 j = p02 just further restricting. Now to make sure that S30 is realized as a subset of consecutive arrows, we distinguish the following cone  3     1S 0 S30 S30 ?? 1 0 ?? S3 ?? ? 0 j S3 @@ ~ @@ ~ ~ @ ~~j j @@ ~   ~ S3 meaning that j is realized as a monomorphism (injection in Set). Now the composition is / S3 subject to some axioms (unitarity, position), giving us the need for maps j1 , j2 : S2 intuitively sending an arrow to (1c(f ) , f ) resp. (f, 1d(f ) ), i.e. we have equations: p1 · j2 = 1S2 j · j10 k · j10 j · j20 = j2 p1 · j1 = id · c, / S0 : 3 and then j10 , j20 : S2 p2 · j1 = 1S2 , unitarity: = j1 , p2 · j2 = id · d = 1S2 = k · j20 position: d · k = d · p02 · j, c · k = c · p01 · j. All in all we get the underlying multiplicative graph of the sketch of multiplicative graphs: s1 d id  w S1X g sd w / S2 sc 6 E O Y gh c j20 p01 s01 Equations:  A SF 3 p2 k p02 j10j2 # S30 { p1 · j = p01 p2 · j = p02 j · j10 = j1 j · j20 j1 p1 = j2 p2 · j1 = 1S2 p1 · j2 = 1S2 j c · i = d · i = 1S1 d · p01 = c · p02 = s01 p1 · j1 = id · d = sd p2 · j2 = id · c = sc k · j10 = 1s2 = k · j20 4.2 Sketching 13 And distinguished projective cones: S3*  **   *** p1  **p2 **   **   s1  S23 33 3 d 33    S30 ,  **   *** ** 1S 0 1S 0  3  ** 3  **  j *  S2 S30 c S1 22 22 j 22    S30 j S3 both indexed by the graph · → · ← · . So if call this sketch g we have Mod(g, Set) ∼ = Graph. Sketch of categories: We can continue the above sketch of graphs to obtain the sketch of categories by making all consecutive arrows composable (S3 = S30 ) and adding a fourth object to be modelled as consecutive triples of arrows and then equations to describe the associativity of the composition. We get: s8 s7 q3 q2 s5 s4 s1 Underlying graph : d id    } w S1 g sd c   }w / S 2 o sc 6 Y \ a gh q1 j1 p1 k k1  w A S3 ] g p2 j2 w1 S4 w2 s2 s3 k2 s6 Distinguished projective cones : S3)  ))   ))) p1  ))p2 ))   ))  s1   S21 S2 11 1 c d 11    S1 S4& 4 4  && 44   &&& 44 q1 4 q3 q2 & && 444   44 & s7  4   &&&s8  &  S21 S2  S21 && 11 11 &  &  1 1 c d 11  c d 11 &&     S1 S1 where the second and new cone is the cone assuring that, in any set theoretical model, S4 will be mapped to the the set of consecutive triples of arrows (up to isomorphism). The arrows 14 Sketches of the sort si are just naming the composites. Equations: c · i = d · i = 1S1 p2 · j2 = i · c = sc p2 · j1 = 1S2 p1 · j1 = i · d = sd p 1 · w1 = q1 d · p1 = c · p2 = s1 p1 · j2 = 1S2 p1 · w2 = q2 p1 · k1 = k · w1 = s2 p2 · k1 = q3 p 2 · w1 = q2 p1 · k2 = q1 p 2 · w2 = q3 p2 · k2 = k · w2 = s3 c · k = c · p1 = s 4 k · j1 = k · j2 = 1S2 d · q1 = c · q2 = s7 d · q1 = c · q3 = s8 d · k = = d · p2 = s 5 k · k1 = k · k2 = s6 We will come back to the intuitive meaning of the arrows, in the underlying multiplicative graph, later when we in the examples make use of this sketch. Though a careful reading of the equations should suffice. When sketching, we will often need to point out that certain arrows in the underlying multiplicative graph should be realized as monomorphisms (as we saw in the sketch of graphs) or epimorphisms. To avoid too much unnecessary repetition we introduce the notation m / D to mean that m is distinguished as potential monomorphism, i.e. the followC ing cone is distinguished: C@ @@ 1 ~ @@C ~ @@ ~ ~ ~ m C@ C @@ ~~ @@ ~ ~~m m @@  ~~ 1C ~~ D meaning that any model/realisation will render this diagram a pullback, and hence as is well known and easy to see, the map m sketchs a monomorphism. ε / B to distinguish the fact that ε is to be realized Likewise we introduce the notation A as an epimorphism, meaning that we distinguish the following co-cone: B ~> O `@@@ 1 @@B @@ ~~ ~ ~ ε B `@ >B @@ ~~ @@ ~ ~~ε ε @@ ~~ 1B ~~ A then any model will render the diagram a push-out diagram and hence again, as is well known and easy to see, the arrow ε sketchs an epimorphism. 15 Furthermore, when we illustrate a sketch by a diagram we usually present a generating sample with an explanation. For example we normally don’t include arrows signifying composites (arrows such as the si ’s in the above). 5 A category of sketches In this section we show that the category of sketches is sketchable with a projective sketch, when we limit the size of the sketches by some large cardinal λ. The category of sketches thus arrived at, as the model category of set theoretical models of a projective sketch, will then serve as the frame for our actions. We shall present the projective sketch of sketches and then look at how we then think of a mixed sketch as a set theoretical model of a projective sketch (within the size limitation). As mentioned, the category of mixed sketches is sketchable projectively, at least, when we limit the size of the sketches by some cardinal λ, meaning that the sets of distinguished cones and of distinguished co-cones are smaller than λ, plus all index-graphs I should be λ-small, i.e. ArI smaller that λ. We will call the category of mixed λ-small sketches with sketch morphisms Esqλ (Esq as in the french word for sketch ”esquisse”),I could have chosen the notation ”Sketchλ ”, but it seemed natural to me to leave some french trace concerning the main ingredient. Let λ = (Sλ , Pλ ) denote the projective sketch of mixed λ-small sketches for some arbitrary cardinal λ. λ is constructed to be modelled in Set, so the size limitation is a way of avoiding paradoxical notions such as ”the set of all sets”, i.e. we wish the result: for all cardinals λ there exists a sketch λ such that there is an isomorphism: Mod(λ , Set) ∼ = Esqλ Remark that we wish isomorphism and not just equivalence, we return to this in the next section. First the sketch of Esqλ will be constructed. 5.1 Sketching sketches I will go through the construction of λ according to the way it is presented by C. Lair and L. Coppey in their ”lecon 3. Esquisses” (Coppey, L., Lair, C. [1984]). In fact it is almost a direct translation of their presentation. But it is indispensable to the context of this paper to have this construction at hand, and in matching notation. So take λ a cardinal and J, J0 two sets of multiplicative graphs such that: and for all I ∈ J, I 0 ∈ J0 : |J|, |I|, |J0 | < λ |I 0 | < λ then we set out to sketch the category of all small {J, J0 }-mixed sketches, meaning J-projective and J0 -injective. To simplify notation and better understand the idea of the construction we first look at the sketch of small {I}-projective sketches, for some multiplicative graph I ∈ J. Let us denote 16 A category of sketches this (generating) sketch I = (SI , PI ). Now the underlying multiplicative graph SI will at least contain a copy of the sketch of multiplicative graphs, destined to determine the underlying graph of any {I}-projective sketch (any model of I in Set). So the only thing needing specification is the choice of {I}-projective distinguished cones, and to this we need: − - a special object; ”object of projective I-cones, thus denoted S I , where S is meant to model the underlying multiplicative graph of an arbitrary {I}-projective sketch. m / I − where the domain CI will be modelled as the subob- a special arrow CI S ject of projective I-cones, to be distinguished; m is the potential monomorhism that distinguishes cones. − Now S I is destined to describe the set of functors from I − to S in some realisation R of I so S (= RSI ) is the underlying multiplicative graph of RI . m is destined to be the injection − choosing the distinguished cones among the objects of S I , or more precisely choosing bases of cones to be distinguished. Thus we need distinguish the fact that m is to be realised as a monomorphism (injection in Set) and this is indicated in the manner described in the introduction, by writing m . − Now we take a closer look at how the object S I is to be connected to the sketch of graphs, to complete I . We will do this with an arbitrary graph I. The point is to sketch the object of functors from I to S (functors/homomorphisms of multiplicative graphs). The set S I is determined by the existence of three arrows: ObI / ObS ; I / SI ArI / ArS ; i / si (i0 , i)  / ti0 ,i CoI / CoS ; satisfying: ∀i ∈ ArI : 0 ∀(i , i) ∈ CoI : c(si ) = Sc(i) d(si ) = Sd(i) t(i0 ,i) = (si0 , si ). If k denotes the partial composition of S, we have: kt(i0 ,i) = si0 ◦ si = si0 ◦i and of course S1I = 1SI . In this way S I can be viewed as a certain subset of the following product: I P = ObOb × ArSArI × (CoS )CoI S 5.1 Sketching sketches 17 So we can take the restrictions of the canonical projections from P evaluated at some object, some arrow and some composable pair, meaning that we get a triple of projections for all triples (I, i, (i0 , i00 )) commuting with some arrows in ObS i d id v p1 / ArS o v h CoS k p2 c So in Set the cone will look like this: qI SI ...... /P q(i00 ,i0 ) qi ...... c  uv v ObS i  ...... p1   }v / ArS o id )  k h CoS p2 d evI π1 / ObObI / ObS (where π1 is the canonical then qI is the restriction of the map P S projection of the product P and evI is the evaluation in I of a map). More precisely the cone will be indexed by the multiplicative graph §§(I): Ob§§(I) := disjoint union of ObI , ArI and CoI Ar§§(I) := pairs (z, x) y x ∈ Ob§§(I) and z an arrow in ObI e c id x p1 / ArI o k f CoI p2 d such that z(x) is defined; then x is the domain and z(x) the co-domain. §§(I) ∗ §§(I) := pairs ((z 0 , x0 ), (z, x)) such that z 0 ◦ z is defined and z(x) = x0 then (z 0 , x0 ) ◦ (z, x) = (z 0 ◦ z, x). This can be illustrated by the following diagram of ”generating samples”: . . ObI c(i) d(i) s (c,i) i0 ◦ i (id,I) I . . i0 k * 1I . CoI (p2 ,(i0 ,i)) is (d,i) } . ArI q (k,(i0 ,i)) (i0 , i) (p1 ,(i0 ,i)) . 18 A category of sketches Now the above graph-homomorphism conditions can be described in terms of compositions of arrows: ∀i ∈ ArI : 0 ∀(i , i) ∈ CoI : c ◦ qi = qc(i) d ◦ qi = qd(i) p1 ◦ q(i0 ,i) = qp1 (i0 ,i) = qi0 p2 ◦ q(i0 ,i) = qp2 (i0 ,i) = qi ∀I ∈ ObI : k ◦ q(i0 ,i) = qi0 ◦i id ◦ qI = qid(I) = q1I . In fact S I is, when modelled in Set, the vertex of the limit with image of basis ObS i c id v / ArS o p1 v k h CoS p2 d since if one take any fI , fi , f(i0 ,i) such that the diagram XD DD {{ DD { { DD f { DD (i0 ,i) fI {{{ DD fi {{ DD { DD {{ { p1 c DD { }{{v  ! v k id / ArS o CoS ObS i h p2 d commutes (with respect to the basis §§(I)), then there is a unique h : X (S I , q) ◦ h = (X, f ), defined by: h(x) = I 7−→ fI (x), i 7−→ fi (x), (i0 , i) 7−→ f(i0 ,i) (x)  / S I such that So we need distinguish a cone indexed by §§(I) to get our sketch I , which will then be achieved by taking the sketch of graphs and add the following: − - the objects C and S I ; the arrows C qI  S1 SI  − qi S2 q(i0 ,i)  S3 m / − and   I ∈ ObI − i ∈ ArI − where  0 (i , i) ∈ I − ∗ I − SI - equations: ∀i ∈ ArI − : ∀(i0 , i) ∈ I − ∗ I − : ∀I ∈ ObI − : c ◦ qi = qc(i) d ◦ qi = qd(i) p1 ◦ q(i0 ,i) = qi0 p2 ◦ q(i0 ,i) = qi k ◦ q(i0 ,i) = qi0 ◦i id ◦ qI = q1I 5.1 Sketching sketches 19 - distinguished cone: SI 0 − π(i0 .i) . πI πi  * 0 (i0 , i) ·g i ( · §§(I − )− i0 ◦ i . . . I q(i0 ,i) qi !  · i ; · I0 " qI p1  t c to / id S1 j S2 j k p2 d  S3 . . Hence we have established that I is a {· → · ← ·, §§(I − )}-projective sketch. − From the generality of the construction of projective cone-diagrams S I (or just cone di+ agrams), we see that introducing an object representing inductive cone-diagrams S I (or just co-cone diagrams) amounts to the same. Hence the general sketch λ , is a matter of notation: to the sketch of multiplicative graphs we add − -Objects S I , CI -Arrows { CI m I / 0+ ∀I ∈ J and S I , CI 0 − S I }I∈J and { CI 0 ∀I 0 ∈ J0 m I0 / + S I }I 0 ∈J0 -Cones as above indexed by §§(I − ), §§(J + ) for each I ∈ J, J ∈ J0 . A (generating) diagram of the underlying graph of the sketch λ is thus: Underlying graph: s1 j1 p1 d id v SP 1 bh sd c qi qI CI m I / SI s01 / S2 vh sc 6> [7h j0 77 0 1 77 pp12 77k j 77 2 p02 7  0 j2 .' S 0 > 3 q(i,i0 ) ? SM 3 j 20 A category of sketches Distinguished projective cone: S33 33 33p2 3 p1  S23 s1 S2 33 3 d 33    c S1 − and for all I ∈ I, J ∈ I0 the projective cones with vertexes S I and S J Equations: p1 ◦ j = p01 p2 ◦ j = p02 j ◦ j10 = J1 j ◦ j20 = j2 p2 ◦ j1 = 1S2 p1 ◦ j2 = 1S2 + as described above. c ◦ i = d ◦ i = 1S1 d ◦ p01 = c ◦ p02 = s01 p1 ◦ j1 = i ◦ d = s d p2 ◦ j2 = i◦ = sc k ◦ j10 = 1s2 = k ◦ j20 plus equations from the above additions concerning ”choice” of distinguished cones, i.e. for all I ∈ I, J ∈ I0 : ∀i ∈ ArI − ∪ ArJ + : c ◦ qi = qc(i) d ◦ qi = qd(i) ∀(i0 , i) ∈ CoI − ∪ CoJ + : p1 ◦ q(i0 ,i) = qi0 p2 ◦ q(i0 ,i) = qi k ◦ q(i0 ,i) = qi0 ◦i ∀I ∈ ObI − ∪ ObJ + : id ◦ qI = q1I Now we obviously obtain an isomorphism of categories Rλ : Esqλ ∼ = / Mod(λ , Set) hence λ is sketchable in Set projectively, as claimed. Having done the construction of λ , we can rather easily justify that λ itself can be viewed as a member of Esqλ : we have seen that λ is a {· → · ←}∪{§§(I − )}I∈J ∪{§§(J + )}J ∈J0 -projective sketch, so the question is whether we can consider{· → · ← ·} ∪ {§§(I − )}I∈J ∪ {§§(J + )}I 0 ∈J0 to be included in J. We have: |Ob§§(I) | ≤ |ObI × ArI × CoI | < λ3 |Ar§§(I) | ≤ |{z, x)|x ∈ Ob§§(I) , z ∈ {id, c, d, p1 , p2 , k}}| < λ × 6 |§§(I) ∗ §§(I)| ≤ |Ar§§(I) × Ar§§(I) | < λ2 so as long as we suppose λ to be infinite, there shouldn’t be a problem, as well as there is no problem in supposing this. 5.2 Sketches as models of a projective sketch 5.2 21 Sketches as models of a projective sketch We now look at how the above isomorphism Rλ will allow us to consider an object in Esqλ (a mixed λ-small sketch) as an arrow i in Esqλ . We saw that λ is a member of Esqλ , so if we abusively consider the category Set as a member of Esqλ ; as the mixed sketch with underlying multiplicative graph Set and distinguished cones and co-cones all λ-small limits and co-limits in Set, then the above isomorphism of categories, saying that we can jump between sketches as objects in Esqλ and corresponding models of the sketch of mixed λ-sketches into Set, can be viewed (locally) as taking place inside Esqλ . This is abusive since we don’t pretend to consider Set as a model of λ in Set, which would obviously lead us into trouble. We rather consider Set as a very big sketch into which we would like to model sketches (the real members of Esqλ ). We join Set to the category Esqλ remembering that Esqλ with Set as extra object is not not sketchable by λ . We can perhaps think of Set as an added semi-final object in Esqλ , since we always have a set theoretical model, i.e. always an arrow to Set (but it is rarely unique). Or just think of Set as the big and rich reference category, since we are often guided by set theoretical properties when sketching. But most importantly we just want to consider set theoretical models as arrows in Esqλ . In fact we can consider any λ-small category as a member of Esqλ , by taking the category as underlying multiplicative graph and certain limits and co-limits as distinguished cones and co-cones, within the given size-frame. By this view, the notation such as Setσ will be understood as ”hom-sets” in Esqλ , i.e all sketch morphisms from σ to Set; all set theoretical models of σ, that we until now have written Mod(σ, Set). The point is that sketch morphisms into a category (viewed as a sketch) form the special sketch morphism we call models. Thus whenever we have a λ-small category C and a sketch σ ∈ Esqλ we can write the category of models as a hom-set HomEsqλ (σ, C) or as category of functors C σ . 6 Theoretical types In this section we introduce the central notion theoretical type and then we go through the two main purposes of this notion, first on the syntactic side concerning the nature of proofs in sketches, and then the semantic side concerning the realization of proofs; when do we have proofs? But first we will make precise what we mean by a theoretical type. As mentioned in the introduction the idea of theoretical types is essentially just a formal way of distinguishing mathematical theories according to their structural properties. Since we here can view mathematical theories as sketches (categories viewed as sketches), we let the notion of type concern all sketches in Esqλ . Or one could consider this as an abstraction of the notion of a theory to a sketch, since we can consider sketch morphisms as realizations so in a way, any sketch is realizing some sketch (at least itself). The formalization presented here will thus take place in Esqλ , and we will think of types of theories, as a typification of sketches by extensions of their corresponding model of λ (via Rλ ) along a sketch morphism specifying the ”typical properties” in question. 22 Theoretical types We consider a morphism of λ-small projective sketches /τ t : λ we can think of t as a condition perhaps satisfied by a theory specified by some mixed sketch σ. Then the category of set theoretical models of τ in Setτ is called Typet , the category of t-types. •Theoretical type: We say that a sketch σ in Esqλ is of type t (or satisfies t) if the corresponding / Set factorizes through t by a sketch morphism Rλ (σ)t : τ / Set , model Rλ (σ) : λ i.e. the following diagram commute: /τ t λ6 66 6 Rλ (σ) 66   Set Rλ (σ)t •Strict theoretical type: Then we say that σ is of strict type t if the above factorization is unique. My work only considers these strict types, since they are much simpler to handle by way of the application of the generalized associated sheaf theorem, as we will see and they are in fact not so terribly restrictive. / µ of a sketch σ into a cate- • We call model of type t, or t-model, a model R : σ gory µ of type t. Now, any type t gives rise to the forgetful functor Ut : Setτ R  / Setλ ; Typet / Esq / R ◦ t; R(τ )  / R ◦ t(λ ) λ . Since t is a morphism between projective λ-small sketches we get, by the generalized associated sheaf theorem, that Ut has a left adjoint Lt : Setλ / Setτ and the unit of this adjunction / Ut ◦ Lt ηt : 1Setλ / Lt (Rλ (σ)) ◦ t ηt (σ) : Rλ (σ) / Set . We denote determines for each sketch σ ∈ Esqλ a realization Lt (Rλ (σ)) ◦ t : λ the corresponding sketch Tt (σ) (i.e. Rλ (Tt (σ)) = Lt (Rλ (σ)) ◦ t), Tt (σ) is clearly of type t since we have /τ λ6 t 66 6 Rλ (Tt (σ)) 66   Set Lt (Rλ (σ)) 6.1 Types and syntax: proofs 23 / Tt (σ) The isomorphism Rλ : Esqλ ∼ = Setλ permits us to write the unit as ηt (σ) : σ which we will refer to as the constructor of the theoretical t-type of σ. Since (Lt Rλ (σ), ηt (σ)) is often called the Ut -free structure generated by σ, we call Tt (σ) the freely generated theoretical t-type of σ (a sort of t-completion of σ). Clearly σ is of type t if and only if ηt (σ) is an isomorphism. Furthermore, if t is strict, it extends (set theoretical) models of λ uniquely and we have that Ut is an embedding, hence strict types Typet for t strict, are full sketchable subcategories of Esqλ . And, for all sketches µ, of a strict type t, we have the natural isomorphism Hom(Tt (σ), µ) ∼ = Hom(σ, µ) (F) by the universal property of the adjunction. Meaning in particular that two sketches have isomorphic freely generated t-types for a strict type t if and only if they have isomorphic model categories for models of type t. This means that asking if two sketches have the equivalent model categories for models of a certain strict type t, is the same as asking wether the two sketches have equivalent freely generated sketches (theories) of this type. There are two aspects of this application (supporting our mottos), one on the syntactic level: we get a description of theorems and proofs in Esqλ , and on the other on the semantical level: we get a way to discriminate in Esqλ telling us when a given theorem is valid; when does it have a proof. 6.1 Types and syntax: proofs We here look at the first aspect of the above application of generalized associated sheaf theorem concerning the purpose of strict types on the syntactic level. The application leads us to a purely diagrammatic conception of proofs, as well as envisaging that sketching is proving and proving is sketching. The isomorphism (F) allows us to consider a proof of a theorem in a theory specified by a sketch σ, as a progression towards a type. Meaning that if we wish to proof a theorem for all t-models of σ for a strict type t, it becomes a question of enlarging σ (sketching towards Tt (σ)) in a way such that the enlarged sketch σ 0 has the same t-models as σ and such that it is evident (!) that the theorem is valid in all t-models of σ 0 . We then say that σ 0 contains the general proof of a theorem relative to σ and the progression from σ to σ 0 is proving this theorem for all t-models of σ. More precisely: Let Θ be a theorem relative to a sketch σ, that we wish to prove for all t-models for some strict type t, then this proof π can be achieved in an environment (a sketch Tt (σ)πN , like σ 0 in the above) in between σ and Tt (σ) by N finite steps σ −→ Tt (σ)π1 −→ · · · −→ Tt (σ)πN −→ Tt (σ). Meaning that obtaining a proof in N steps, becomes a question of obtaining a factorization σ QQQ QQQ ( ηt (σ) Tt (σ)πN / T (σ) kk5 t k k k k such that Tt (σ)πN evidently(!) satisfies Θ. Because the above application of generalized associated sheaf theorem gives that all realizations of σ factor uniquely through a realization of 24 Theoretical types Tt (σ), thus also through a realization of Tt (σ)πN . Hence all realisations of σ satisfy Θ. Take a theorem Θ relative to σ such that Θ can be formulated via a sketch morphism / θ , for a model R : σ / µ of strict type t, in the following manner: R facυ:σ / µ such that R = Rυ υ. Then, torizes through υ, i.e. there exist a realization Rυ : θ establishing a proof amounts to constructing a sort of captivation of υ by ηt (σ); a factorization: ηt (σ) σ MM MMM υ M& / Tt (σ) t: t t tt tt t t θ RRRRR R) e Tt (σ)πN e is the evidence arrow, the finite progression from θ to Tt (σ)πN making sure that the sketch Tt (σ)πN evidently(!) satisfies the theorem Θ (indicated by the sketch morphism υ), for models of type t. /θ At last we simply put π = Tt (σ)πN and consider a proof, of a theorem via υ : σ relative to a sketch σ, as a factorization σ MM MMM υ M& ηt (σ) θ PPPP e PP' π / Tt (σ) y< yy y yy yy where the arrow e ◦ υ should be finite; the progressing from σ to π via θ should be achieved by a finite series of sketch mutations towards a sketch with the same free t-type. So proving is sketching: Progressing (sketching) towards a sketch (π) from a sketch (σ), via a sketch morphism (υ), in a restricted manner (factorizing ηt (σ)). This idea of general proof by syntactic progression should be more clear in view of the examples in section 7. below. / κ a conjecture relative to the We could now be tempted to call any sketch morphism σ sketch σ, since, by the above, any sketch morphism can be viewed as a potential theorem. It is just a matter of investigating what semantics are necessary in order to achieve a theorem, e.g if σ and κ have the same t-model categories for some strict type t, then any property evidently satisfied in all t-models of κ gives rise to a theorem relative to σ for t-models. Whether this is interesting of course depends on the sketch κ. We return to this in the example section. First we look some more at this other purpose of types, concerning the semantic level and the interplay between semantics and syntax in Esqλ . 6.2 Types and semantics: discrimination This section is about the other important purpose of theoretical types, that when we sketch to describe a certain theory we at the same time envisage the properties needed in order to realize our sketch (as when we sketch set theoretically in the previous sections). This leads to certain types, hence a category Typet for a strict type t is often arrived at as the category of possible semantics for a certain sketch. Or, perhaps more interestingly, arrived at as the type of semantic in which a theorem via a sketch morphism is valid, which amount to a type in which two sketches (σ and π in the previous section) have the same model categories. So we are interested in at least two situations: 6.2 Types and semantics: discrimination 25 I Given two different sketches of the same structure, find out what type of models ”ignore” this difference, i.e. give rise to equivalent model categories. / θ , find out what I Given a sketch σ and a theorem Θ via a sketch morphism υ : σ type of models ”ignore” the difference between σ and the sketch π containing a general proof of Θ. This leads to the introduction of two concepts. Confusion and discrimination of sketches: Given two sketches σ, θ, if there exists a / θ , such that third sketch η and a sketch-morphism υ : σ ηυ : ηθ R  / ησ /R◦υ is an equivalence of categories, then we will say that, via υ, η confuses σ and θ. If not we will say that, via υ, η discriminate between σ and θ. Remark: The discrimination will rather depend on the sketch η, than the morphism υ, meaning that the difference between two sketches rather depends on the structural richness/poverty of the type of category we wish to model in. Said differently the discrimination takes place on a semantical level, but then again it depends on υ how easy it is to determine whether η confuse or discriminate. We are thus interested in the following general problem. / τ such General problem: Given sketches σ and θ in Esqλ , find a strict type t : λ that all sketches of the type t confuses σ and θ. Then find at least one sketch ν discriminating σ and θ, meaning in particular that ν is not of type t; ν and Tt (ν) (its freely generated t-type) are non-isomorphic. Then one can amuse oneself in finding out what Tt (ν) resembles, either by calculating the Kan extension or by more ad hoc methods, to see how gravely it differs. A third situation permitted by the application of the generalized associated sheaf theorem and that might be interesting is: given two sketches to be compared (perhaps via sketch morphism indicating a theorem wishing for a proof) and a ”favorite” Semantics, then if our favorite semantics can be represented by a strict type t, we can check if models of this type confuse the two sketches, by comparing their freely generated t-types. If the two sketches have isomorphic t-types (or just equivalent when the types are categories), they are confused by the type t (we will have proofs of the theorem perhaps indicated, in all models of type t). This procedure involves calculating Kan extensions, which can be rather heavy (at least notationally) if the sketches are complicated. But in theory it is possible and perhaps fruitful and then again by the application of generalized associated sheaf theorem we perhaps don’t need to calculate the entire free t-type, if we using categorical gymnastics can find a sketch with same model categories as both of the sketches we wish to compare, for our favorite type (the example of monoids in the section of examples below, contains such a procedure). We hence get a general method for checking certain conjectures; assuring the existence of proofs in certain types of categories without going in to the concrete details of the concrete proofs. 26 7 Examples Examples We will give two examples of the general problem. First a rather simple one concerning the sketch of fields, where we will compare the standard sketch of a field, following the settheoretical definition, with the sketch of fields slightly modified, in order to get topological fields, when modelled in the category Top of topological spaces and continuous maps. Then we look at an example where we compare two sketches of monoids each following a different standard set theoretical definition of what is meant by a monoid. This example will show that the apparent difference in language for two ways of defining the same structure, is not always trivially formalized, meaning that we don’t see right away what the actual structural differences amount to, when we put on stronger glasses than just the set theoretical ones. In short, we present two examples where two different sketches of the same classical mathematical structure have the same set theoretical models, but not the same model categories for any type of model. We then, following the general problem stated in the previous section, determine a type confusing the two sketches in question and at last give an example of a category discriminating them. Before we present the examples we take a look at a basic type, that will be very helpful in the search of strict types confusing two given sketches, this counts in particular for the examples chosen in this thesis. 7.1 The classical type tclf In the search for sketches playing η’s role in the general problem above, a type of interest is the classical type of categories with all finite limits and co-limits, since if η is a category then η θ and η σ are (model-) categories (as required in general problem). Sketches of interest (sketches of classical mathematical structures) normally have, at least, a number of finite distinguished cones and (perhaps) co-cones. So claiming that η is a category with all finite limits and co-limits, is a sort of minimal claim to avoid pathologies. We will sketch this classical type, meaning that we will construct a sketch τclf such that the category of set-theoretical models of τclf is isomorphic to the category of categories with all finite limits and co-limits. Then we adjust the sketch in order for it to signify a strict type, i.e. in order for it to give rise to a strict type-morphism. This will turn out quite useful in the examples below, furnishing a sort of basic strict type. Construction of the sketch τclf : We set out to sketch the category of categories with all finite limits and co-limits, so obviously we depart from the sketch of categories, as constructed earlier (p.12). Now we wish to enlarge this sketch such that any model (category) will have all finite limits and co-limits. As in the construction of λ , we add elements representing the cones we want to distinguish, only this time we want these cones to be limiting cones, so further structure is added to specify the universal property for the, chosen, limiting cones. For a model of the sketch to have all I-limits for a chosen graph I, we need, as in construction of λ , to add the − object S I to the sketch of categories and then specify the limiting cones for all cones of base I − in the underlying multiplicative graph S. We consider the following five multiplicative 7.1 The classical type tclf 27 graphs with graph-inclusions: 0 αk ∅ . K2 K3 K4 e 1• i g • • 1 e !1 _ Y + m o q c b ` u a b •0/ M 4 1 O Q S U 2 8 W Y -   β 4 < • 6 >| • 9 A z x G ? v L . . . F rt P M T X m S Y [ _ b e t _ d i _τ ^ \ u0 [a , ~~ ~~ ~ ~~ 5 ~~ ~ u j ~~ g ~~ ~ ~~ β0 b  • • . and then construct the joint graphs, as introduced in the preliminaries (p. 7), of the graphs 0, 1, K2 , K3 , K4 with the graph I and get I0 , . . . , IK4 with graph inclusions inherited. Now we add the objects S I0 , . . . , S IK4 to the sketch of categories, in the same way as in λ , together with the induced arrows, i.e. we enlarge the sketch of categories with the diagram: t∗ L w C , β 0∗ { V , I S K4 S S , Q _ h m ~ ~  ∗ ∗ 9 8 J # τ m - I PH !1 P    } } 8 7 Iβ ∗ n . H  H P 7 6   | | n ∗ " / O G 6 | 5α H   o { 0 O 5 G F o z {5 0 N 4 F E 4 o y 1 F y z p4 E 3N  y 2 2 E D q01 q 3 2 M x u p x 3 C 2 1 w  xM qL D C 4 B 1 0 w q  5 L  B v w0 A vr /   6   Ku B /  vr @ / 7   u  ? . K A / 8  t su ? -   t J @%  .       yyts ys S I0 k + S I1 S1 I K2 IK3 l S2 S3 where the broken arrows represent (the rest of) the restricted (and evaluated) projection. Now, L is the arrow ”formally choosing” a limiting cone for each diagram of type I, C is the ”choice of comparison” arrow and V is the arrow designed to secure the uniqueness in the universal property of the limiting cones. This works by assuming the following equations to hold: (1) !∗1 L = 1S I0 (4) qu2 CL = idq01 L ∗ (2) α C = 1S I1 (5) τ ∗ V = 1S I3 ∗ ∗ (3) L!1 = β C (6) Cβ 0∗ = t∗ V. Further explication of the equations: 28 Examples  /1 / I1 (inherited from the unique arrow 0 Ad(1): !1 is just the injection of graphs I0  ∗ ∗ in the category of graphs), so !1 L = 1S I0 (!1 composition with ! from the right) means that for any graph indexed by I, L chooses a projective cone with this very graph as image of the basis. !1 Ad(2): α∗ C = 1S I1 makes sure that C takes a cone to a comparison diagram where it itself plays the role as left hand cone. Ad(3): L!∗1 = β ∗ C means that for all cones I we have that if we forget the vertex (go by arrow !∗1 ) and then take the ”choice of” limiting cone (by arrow L), then this is the same as taking the comparison graph (graph of type IK2 ) and then forget the left hand cone (cone with vertex at the source of the ”hook” u), by β ∗ . That is for any cone with basis indexed by I, taking the limiting cone with same basis is the same as taking the right-hand cone in the comparison graph where it plays the role as left-hand cone, so C compares any cone with the limiting cone of the same basis. Ad(4): qu2 CL = idq01 L makes sure that a chosen limiting cone can only be compared to itself, thus a limiting diagram is (by C) sent to the comparison diagram with hook the identity. Ad(5): τ ∗ v = 1S IK3 means that any graph of type IK3 (i.e. a comparison diagram with two hooks; two arrows from one vertex to another in cones with same basis) is the left part of a diagram of type IK4 , by the arrow V . Ad(6): Cβ 0∗ = t∗ V means that whenever there are two hooks between vertexes of cones with same basis, then V extends this diagram with a hook to the limiting cone of same basis. So if the right-hand cone in the diagram with two hooks is a limiting cone, then u is the identity, by (4), and hence the two hooks are identical (equal to u0 , see figures above). Concerning choice of co-limits, we do the same thing but the arrows reversed in the system of five graphs, i.e. with 0 αj ∅ . K2op K3op K4op e 1 •K h g •K S D •K S 1 e !1 _ Y + l o q db c b ` u a •00 M 3 5 2 P R T V 3 9 W Y β 5 = • • 7 {= : B y w H @ u L . . . G qs P N T X m T Y \ _ b e t _ d i β0 _τ ^ \ [ aZ , b u0 >• }} } }} }} } k 5 }}u g j }}} }} }} • . and construct the joint graphs 0I , 1I , K2op I , K3op I , K4op I and then have the same equations except that in the explications left and right should evidently be interchanged. 7.1 The classical type tclf 29 Remark: Sketching categories with all finite limits can be reduced to sketching categories with finite products (including terminal object) and equalizers, meaning that we only need ( the above addition to the sketch of categories for the graphs {0, 2, • 6 • } where 0 is the empty graph and 2 = {• •} the graph with two objects and only trivial arrows. So we actually get a finite sketch, with (generating) underlying multiplicative graph of τclf : t∗ L w C , β 0∗ { V , I S K4 l } # l | ∗ ; 9 K τ m - I QIb∗ P  m {  |  9 8 Jβ ∗ . mm z  P H 8 7 α∗I   {  m " / n P I G 7 z 6 y nn 0 O 5  H G 7 n y z n 0 O4 G F 6 o x 1 G x y o 5o E 3N  w 2F o 4 2 D q01 q 3 N p u w w p 3 D 2  wM p E D  3 2 v  4 C 1v pp M   u 5 q L D B 1 0   u uq  6   k1  jL1t C 0 uq qA / 7  q   s p1 L B 0 @ . 8 w1  sr rt d ? .   s K A! /   }     yyv sr r % y v v id / S2 h S4 eL 9 S3 h B SL 1V. ehe KL s }> = G R # ] a : _ H R \ ] _ c : K L K .  LJMc w2 ~  9 K p2 s |  9 /  M M}  I J 8 8  r { " q  j 1 0 7 k2 | I M M r7 2z I 1 H N q I q2 6 6 { z 2 Gq N N y 5 Hq 5 {2 3G G xN N 5 5  z p 2 3 F OO 4 G q01 qu 4 y w p O O3 4 E F 3 x w o  PP  5 D E 2 2 x v o  6u D 1 PPP 1   D w o  7  QQ C 0 D 0   v u 8 n  Q  t  B 0 QQ C 0 9  u mu + n  s C+  A/  V+ R B/  m L , , Kop + 1 op op S 0I l S I` hg S K3 I l S K2 Id S 4 I S I-0 k Q + b∗ S I1` hK S IK2 S IK3 l τ∗ β 0∗ β∗ α∗ t∗ ( for I ∈ {0, 2, • 6 • } and the six equations as explained above with both L, C, V and + + + L , C , V plus the equations from the sketch of categories, distinguished cones as in sketch op of categories plus all cones with vertexes S I0 , . . . , S IK4 and S 0I , . . . , S K4 I for the three graph types. Now for τclf to be a sketch of a theoretical type, there need to be a sketch morphism λ / τclf . which is evidently not possible to construct directly, since τclf does not have distinguished cones corresponding to all of the distinguished cones in λ , we only have the objects of the − + ( type S I , S I for I ∈ {0, 2, • 6 • }. But we can expand the sketch τclf to get a − + sketch τ̄clf (with same set-theoretical models). We add all missing objects S I , S J for ( ( • I ∈ J \ {0, 2, • 6 • }, J ∈ J0 \ {0, 2, 6 • }. We also add objects of I( limits and I-co-limits for I ∈ {0, 2, • 6 • } in the following way: 30 Examples -We introduce the object of isomorphisms, i.e. we add an object Iso(S), and arrows miso , n, h to the underlying sketch of graphs, as follows: S1 h p2 d v uo / S2 i O id c S3 ∼ = O S %& c k p1 m iso Iso(S) h ρ∗ / S R n σ where ρ∗ is composition from the right by · @·=  ===  =  ρ G· // · ·  and with miso distinguished as mono. By the definition of being an isomorphism, we need equations p2 hσn = miso , p1 hn = miso and khn = idcmiso , khσn = iddmiso . So we distinguish the cone: Iso(S)0 n n S3 o h  S ppp k  wppp S2 gNN NNN N id N  S1 m iso   hσ / S3 S QQQp1 h NNN m p hσ 2 m QQQ m NNN k  m QQ(  vmmmm ' S2 QQQ 7 S2 m p m p Q QQQ p mmm p m p Q m p Q( p id vmm c d S1 then, when modelled in Set, Iso(S) will be isomorphic to the image of the mono modelling miso , which will be the subset of arrows for which there exist an arrow both right and left inverse. − We can now introduce the object of I-limits ”I-limits(S)”, as the (potential) subobject of S I such that the comparison map is an isomorphism, i.e adding object and arrows to distinguish the following cone: I-limits(S) GG GG g GG GG G# yy m I-limits y yy yy y |y − S I OOO OOO qu CI O' S2 n nnn vnnn m iso Iso(S) Where CI is the comparison map, taking a I-cone to the diagram comparing it with the chosen I-limit of the same basis and qu is the projection of the hook between vertexes in the comparison diagram. 7.1 The classical type tclf 31 So we get the sketch τ̄clf from τclf by adding some cones (or a whole bunch, though still a set) to be distinguished. In this way we get an inclusion τclf  Set τclf ∼ Set τ̄clf  / τ̄clf introducing an equivalence of model categories , and a sketch morphism from λ to τ̄clf / τ̄clf tclf : λ to serve as our type of categories with finite limits, defined as follows: - the ”sketch-of-multiplicative-graph” part of λ is mapped to the ”sketch-of-categories” part of τ̄clf in the evident way, indicated thus: n = S3 a tclf j /2 S3 = a j10 * p01 p2 j20 S30 JP y k yy j1 p 1 k p 2 j2 j1 p 1 y   |yyy sd S2 sc p02 6 O h j0    / S2 sc 6 O h sd 2 c d id  S1 c d id   / S1  making all consecutive arrows composable (when modelled). - for I ∈ {0, CI − and m− / I 2, CI + m− / I ( 6 • } we send to I-limits(S) − SI m+ I - for I ∈ J \ {0, CI − • − SI / SI + CI + m+ I / + SI ( to 2, • / SI − m I-co-limits / SI + 6 • } we send 0 − limits(S) to and for I ∈ J0 \ {0, Ico-limits(S) to • 2, m I-limits m 0−limits / ( ! − S0 ∼ = S1 ∗ 6 • } we send 0 − co-limits(S) m 0−co−limits / ! + S0 ∼ = S1 / ∗ where !∗ is composition from the right with the unique arrow I − , I + 0− 0+ SI / − SI + / 1 to the graph 1 in the category of λ-small graphs and S , S will both be isomorphic to S1 when modelled, since 0− , 0+ are the two ways of joining the empty graph with the graph of one element, so diagrams in S of this sort is just all objects (the model of S1 ). 32 Examples When modelled in Set the set of distinguished 0-cones is the set of 0-limits, i.e. the set of terminal elements or, said differently the set of all singletons. The set of 0-co-cones is the ( set of initial elements {∅}. So the map mI in λ for I ∈ J \ {0, 2, • 6 • } destined to choose cones [co-cones] to be distinguished in a model of λ , is sent to the map in τclf − + modelled as the inclusion of terminal [initial] elements into the object of functors S I [S I ], meaning that in a model of τ̄clf the only limits [co-limits] distinguished by such functors is the constant functors on terminal [initial] elements. Now that we have a sketch morphism, we just need to assure that it is a strict type, i.e. has the property that a supposed sketch of type tclf will, viewed as a model of λ , factor uniquely through tclf . To show this we will take a more general look at what just happened, by naming concepts treated and state some properties. Furthermore, treating this question more generally will be of much help afterwards in our two examples. •Expansion by cones: for a sketch σ, I will call, an expansion by distinguished cones (or just cone-expansion) the sketch σ̄ obtained by, only, adding objects to σ as vertexes of cones to be distinguished (with all images of the bases in σ), and then only extra arrows and equations as part of the cones added. Proposition 1: If σ is a mixed sketch the sketch inclusion σ  equivalence of model categories Setσ̄ ∼ ς∗  ς / σ̄ induces an / Setσ . We remind the reader of the following basic theorem (MacLane, S. [1997]): / C are logically equivTheoremThe following properties of a functor S : A alent: (i) S is an equivalence of categories, (ii) S is part of an adjoint equivalence, (iii) S is full and faithful, and each object C ∈ ObC is isomorphic to SA for some object A ∈ ObA .  Proof of proposition 1: ς ∗ is evidently full and faithful. And each model R ∈ Setσ can be extended (uniquely up to iso) to a model R̄ of σ̄ such that ς ∗ R̄ = R, by taking the added cones to the chosen limits of the respective bases. Meaning if S 0 ∈ Obσ̄ \ Obσ is the vertex / σ , then R̄ takes S 0 to the vertex of the chosen of a distinguished cone with basis B : I limit in Set with basis R ◦ B. Since Set has ”chosen” limits for any (λ-small) index graph, we have indeed ς ∗ (R̄) = R.  •Set-model-epi: We will call set-model-epi a sketch morphism s : σ / σ 0 in Esq with the λ / Set satisfying ϕ ◦ s = ψ ◦ s property that for all models (sketch morphisms) ϕ, ψ : σ 0 we have ϕ ∼ = ψ. Notice that a strict theoretical type then is a set-model-epi with source λ . 7.1 The classical type tclf 33 Corollary 1:For a sketch σ the sketch inclusion σ   ς / σ̄ is a set-model-epi  / σ is a set-model-epi, then so is Corollary 2: if the sketch morphism t :  / σ̄ (by cone-expansion of the target). the expansion t̄ :   We are now ready to proof that we have constructed a classical strict type Proposition 3: tclf : λ theoretical type. / τ̄clf is a set-model-epi, i.e. tcl defines a strict f / Set such that ϕ ◦ tcl = ψ ◦ tcl , then we need Proof: Take any pair of models ψ, ϕ : τ̄clf f f ∼ to proof that ϕ = ψ, i.e. that there exists a natural isomorphism between ϕ and ψ. / Set is isomorphic to, what I will I will show this by showing that every model ϕ : τ̄clf / Set , which, in word, will be ”the model call, its corresponding reference model κϕ : τ̄clf enabling us to understand the model ϕ”, i.e. the set-theoretical model we ”thought of” when we constructed the sketch τ̄clf . Meaning that κϕ equals ϕ on the ”oriented-graph-part” of τ̄clf (the set of objects and the set of arrows between objects), i.e. κϕ : S1 g w d id c / S2  / v ϕS1 h d id / ϕS2 c we won’t write κϕ f for modelled arrows (f an arrow in the sketch-of-categories-part of τ̄clf ) since these are in the model κϕ exactly what they were named to become in a model, when we made the sketch. Then κϕ takes the distinguished cones in the ”sketch-of-categories-part” to the expected fiber products, i.e. the actual set of consecutive pairs of arrows and consecutive triples of arrows, 34 Examples which we illustrate: S33 S4 F FF xx FF x x FF x x FF q3 q1 xx x FF q 2 xx FF x x FF xx FF x x F"  |x S2 J S2 7 S2 JJ 77 tt  J t  JJ 77  tt JJ  c tt c d 77 t J d  t J  zt $  33 33 33 p2 33 33 33  p1  S2 C CC CC C d CC! S1 {{ {{ { { c { }{ S2 S1 S1 _ _ κϕ κϕ   ϕS2 ∗ 3ϕS2 p1  ϕS2B ϕS2 ∗ ϕS2 F∗ ϕS2 33 33 33 p2 33 33 33  BB BB B d BB! ϕS2 ϕS27 xx xx x x q1 xx x q2 x x xx x xx  |xx 77 77 d 77  | || ||c | }|| ϕS1 ϕS1 u uu uu u c u zuu FF FF FF FF q3 FF FF FF FF F" ϕS2 I II II II d II$ ϕS2  c   ϕS1 Then in the model κϕ all the arrows are exactly what they were thought to be: w1 : (f1 , f2 , f3 )  / (f1 , f2 ) w2 : (f1 , f2 , f3 )  / (f2 , f3 ) k1 : (f1 , f2 , f3 )  / (k(f1 , f2 ), f3 ) k2 : (f1 , f2 , f3 )  / (f1 , k(f2 , f3 )) p1 , p2 the restrictions of the Cartesian projections etc. Only we set the composition κϕ (k) to be determined by the composition ϕ(k) composed with the isomorphism of limit vertexes ϕS3 ∼ = ϕS2 ∗ ϕS2 . Now the objects of the sort S I will be sent to exactly the set of functors from I to S (all diagrams indexed by I) as introduced in the sketching of sketches earlier, one could say that κϕ is the model justifying the notation, which is why we will write κϕ S I = S I . The choice of 7.1 The classical type tclf 35 limit/choice of co-limit diagram in the sketch τ̄clf will then be sent to the diagram induced by ϕ via the isomorphisms ϕS I ≃ h diagram commute: / S I meaning that all natural squares in the following ϕt∗ ϕL ϕS I0 l ϕb∗  , v ϕC ϕS I1a i ≃  h0   Lϕ S I0 l b∗ , ϕV ϕS I K3 m   ϕβ ∗  ≃  h1 ϕα∗ ≃  h2 ≃  h3     Cϕ S I1     z ϕS I K2    , ϕβ 0∗ a ci , β∗ α∗ S ϕS I K4   ≃  h4    IK2 ϕτ ∗ -  S d β 0∗ I K3 l Vϕ τ∗ ,   S IK4 t∗  / (I 7→ fI (x), i 7→ fi (x), (i0 , i00 ) 7→ f(i0 ,i00 ) ) as mentioned in x the introduction of the objects of the sort S I . / S Ii ; where hi : ϕS Ii We now claim that the obvious isomorphism of objects between the models ϕ and κϕ is natural. This is almost obvious by definition of κϕ . But let us go through it just once: we / κϕ , and start by looking at the ”sketchname the claimed natural isomorphism h : ϕ of-categories-part” of τ̄clf . The only natural squares that do not totally obviously commute by force of being projections in a limiting cone, are ϕS4 hS4 ϕf / ϕS2 ∗ ϕS2 ∗ ϕS2 , for f ∈ {w1 , w2 , k1 , k2 } f  ϕS3 hS3  / ϕS2 ∗ ϕS2 but commutativity follows immediately by the equations in the sketch of categories, since if we take x ∈ ϕS4 and suppose hS4 (x) = (f1 , f2 , f3 ), then ϕp1 ϕw1 (x) = ϕq1 (x) = q1 hS4 (x) = f1 ϕp2 ϕw1 (x) = ϕq2 (x) = q2 hS4 (x) = f2 and hS3 is the unique map from ϕS3 to ϕS2 ∗ ϕS2 s.t. p1 hS3 = ϕp1 and p2 hS3 = ϕp2 , giving us that p1 hS3 ϕw1 (x) = f1 p2 hS3 ϕw1 (x) = f2 36 Examples so hS3 ϕw1 (x) = (f1 , f2 ) = w1 (f1 , f2 , f3 ) = w1 hS4 (x), same goes for w2 . And since, by definition of κϕ , we have khS3 = ϕk, we get ϕp1 ϕk1 (x) = ϕkϕw1 (x) = khS3 ϕw1 (x) = k(f1 , f2 ) ϕp2 ϕk1 (x) = ϕq3 (x) = q3 hS4 (x) = f3 which gives p1 hS3 ϕk(x) = k(f1 , f2 ), p2 hS3 ϕk(x) = f3 so hS3 ϕk1 (x) = (k(f1 , f2 ), f3 ) = k1 hS3 , same for k2 . So ϕ and κϕ gives isomorphic underlying categories. Now this natural isomorphism clearly (or by corollary 2.) extends to all the distinguished cones with vertex of the sort S I , and furthermore, by definition of κϕ , naturality extends to the ”choice-of-limit-diagrams” (p. 26). The only part of τ̄clf not mentioned yet concerns the object of isomorphisms Iso(S) and the objects I-limits(S) and I-co-limits(S), but these are all introduced as vertexes of distinguished cones, hence again by corollary 2. we still get an / κϕ is indeed a natural isomorphism. isomorphism. Thus h : ϕ Now to finish the proof of the proposition we need to justify that ϕ and ψ (agreeing on tclf (λ )) must have isomorphic reference models κϕ ∼ = κψ . But since ϕ and ψ are equal on the oriented-graph-part of τ̄clf their reference models will evidently be exactly the same except for the choice of limit maps Lϕ and Lψ , but they are clearly isomorphic since by uniqueness (up to isomorphism) of limits, they have isomorphic images (we are in Set) (same for choice of colimits). So we can without pain conclude that κϕ ∼ = κψ , which gives the natural isomorphism of ϕ and ψ through their reference models.  Remark: The essence of this proof is just pointing out how we have been sketching while thinking of certain (canonical) set-theoretical models 7.2 Two sketches of fields In this section we go through a simple example of the general problem (p. 24) where we compare two sketches of fields that have the same models in Set but not in the category Top of topological spaces with continuous maps. The construction of the first sketch of fields ξ1 is following the standard set theoretical definition of a field and will be sketched gradually in the order of its underlying structures. First the sketch of monoids, following the set theoretical definition Def. 1: A monoid (M, k, e) is a set M with a binary operation ”k” everywhere defined, associative, and with a distinguished unit element e with respect to the operation. 7.2 Two sketches of fields 37 Underlying multiplicative graph: S3 r2 r1 k1 k2 &  85 S 2 p1 v2 v1 q1 q2 q3 k p2    | 1 5S F ê p e  S0 Equations concerning binary operation k: p1 · r1 = q1 , p2 · r1 = q2 k · r1 = l1 , k · r2 = l2 p1 · k2 = q1 , p2 · k2 = l2 p1 · r2 = q2 , p2 · r2 = q3 p1 · k1 = l1 , just naming the composite p2 · k1 = q3 k · k1 = k · k2 associativity of k Equations concerning the unit element, or 0-ary operation e: ê = e · p p1 · v1 = 1S1 , p1 · v2 = ê, p2 · v1 = ê p2 · v2 = 1S1 k · v1 = k · v2 = 1S1 unitarity of e with respect to k. Distinguished projective cones: S 21 S0 .     p1 ∅ . .  S1 11 11p2 11  S1 S 31 q1 q 2 . .  S1 11 11q3 11   S1 S1 . We can then think of S 0 as 1, S 1 as the ”set of elements” (M ), S 2 as the ”set of pairs of elements” (M × M ) and S 3 as ”the set of triples of elements” (M × M × M ). The arrows p1 , p2 , q1 , q2 , q3 are thought of as the canonical projections, r1 , r2 partial projections 38 Examples ((x1 , x2 , x3 ) 7−→ (x1 , x2 ), (x2 , x3 )), k the ”addition of elements”, k1 , k2 can be thought of as: k1 : (x1 , x2 , x3 )  / (x1 + x2 , x3 ) k2 : (x1 , x2 , x3 )  / (x1 , x2 + x3 ) e is the unit element (0) with respect to k (addition) and this is assured by v1 , v2 thought of as / (x, 0) v1 : x  v2 : x  / (0, x) Now we add what is missing to obtain the sketch of abelian groups, i.e. arrows and equations representing the fact fact that all elements should have a left and right inverse, with respect to the composition ”addition”, and furthermore that the composition is additive. To this we add arrows: s (”taking an element to its inverse”), v10 , v20 (”taking an element x resp. to the pairs (−x, x) and (x, −x)”), and σ (a permutation). Giving us the underlying multiplicative graph: S3 r2 r1 k1 k2 &  8 > S 2 ` if σ q1 q2 q3 k p2 v10 v20 v2 v1 p 1    v yw 1 5S F i ê s p e  S0 Equations as in sketch of monoids plus: -new equations concerning binary operation k: p1 · σ = p2 , k·σ = k p2 · σ = p1 σ a permutation k is additive. -new equations concerning the inverses of elements: p1 · v10 = p2 · v20 p1 · v20 = 1S 1 = p2 · v10 k · v10 = k · v20 = e. Distinguished projective cones as above. 7.2 Two sketches of fields 39 We continue to get the sketch of a unitary ring, i.e. we add what is needed to describe a unitary multiplication, which is S3 r13 m31 m33 &  2 σ1 σ2 σ13 >S F w2 w1 m   SF 1 i û p u  S0 and further new equations: -concerning partial projections: p1 · r13 = q1 , p2 · r13 = q3 σ13 = m · r13 -concerning multiplication m and its distributivity (from left and right) on the ”addition” (composition k): p1 · m31 = σ1 , p1 · m33 = σ13 p2 · m31 = m · k2 , k · m31 = m · k2 , p2 · m33 = σ2 k · m33 = m · k1 -concerning 0-ary operation u: u · p = û p1 · w1 = û, p 1 · w2 = 1 S 1 , p2 · w1 = 1S 1 p2 · w2 = û m · w1 = m · w2 = 1S 1 40 Examples Where we can think of these arrows as: x 7−→ (1, x) w1 : x 7−→ (x, 1) w2 : (x1 , x2 , x3 ) 7−→ (x1 , x3 ) r13 : (x1 , x2 , x3 ) 7−→ x1 x2 σ1 : (x1 , x2 , x3 ) 7−→ x2 x3 σ2 : (x1 , x2 , x3 ) 7−→ x1 x3 σ13 : (x1 , x2 , x3 ) 7−→ (x1 x2 , x1 x3 ) m13 : (x1 , x2 , x3 ) 7−→ (x1 x3 , x2 x3 ) m33 : Finally we can talk about sketches of fields: First take the sketch of unitary rings, that we just constructed above. We need all symbols and their signification, essentially: k : S2 m : S2 e : S0 ê : S 1 u : S0 û : S 1 / S 1 formal law of addition / S 1 formal law of multiplication / S 1 formal law selecting the ”0” of the field / S 1 formal law ”constant on 0” of the field / S 1 formal law selecting the ”1” of the field / S 1 formal law ”constant on 1”. We then add the following piece of graph, wherein the new symbols are S ∗ , i, u0 , γ, γ 0 , γ1 , γ2 : S2 ] e p1 p2 m γ1 γ2   û p 1 5 SK S X g i γ0 u e  S0 SF ∗ i γ u0 with further new equations: γ 0 = i · γ, p1 · γ1 = i, p 2 · γ1 = γ 0 , i · u0 = u p1 · γ2 = γ 0 p2 · γ2 = i m · γ1 = m · γ2 = û 7.2 Two sketches of fields 41 and a single new distinguished co-cone: 1 S C [77 e  77i  7  S∗ S0 S ∗ thought of as all ”non-zero elements” and the arrows thougt of as: x 7−→ x−1 γ : x 7−→ (x, x−1 ) γ1 : x 7−→ (x−1 , x) γ2 : 7−→ x i : This sketch is indeed set-theoretical, because if we realize this sketch in the category of topological spaces we will not get the usual topological fields, i.e a field with a topology such that all the field operation maps are continuous. The problem is that a non-discrete topological space K is in general not the sum of its subspaces {0} and K ∗ (elements non-null), An example is the space R with the standard topology, then R is not the the direct sum of its subspaces {0}, R∗ , since 0 is not an isolated point in the standard topology. To get a sketch ξ2 giving us the topological fields when realized in topological spaces we need to modify the above sketch a bit (Guitart, R. [1988]), by distinguishing S 1 not as the sum S 0 + S ∗ , but as the target of an epi-mono with source this sum giving us instead: S2 \ a p2 p1 m    6= A SO 1R Y û γ1 γ2  i γ0 p e SO 0 aB u BB BB BB B SF ∗ i $ u0 S0 and instead, the distinguished co-cone: 0 DS [6  6666   S0 S∗ γ 42 Examples Then ξ2 gives us topological fields, when we model in Top (exercise!) / ξ1 We thus have two sketches of fields ξ1 and ξ2 with an obvious sketch morphism ξ2 (just collapsing S 1 and S 0 ) which clearly gives rise two an equivalence of model categories whenever we are in the classical type τ̄cl with the extra property that all epi-monos are isos. 7.2.1 A type confusing the two sketches of fields In order to get a strict type confusing ξ1 and ξ2 , we now sketch the extra property that all arrows that are both epi and mono, are isomorphisms. We will call this property ∗f ields and the extended sketch we then denote τ̄cl∗f ields . f We can introduce objects Pull(S) of pullback diagrams in S and Push(S) of pushout diagrams in S in the same way as the objects I-limits(S), I-co-limits(S) were introduced previously, i.e as certain distinguished subobjects: a ~ • AAb ~ Pull(S) m pull / A @@ S • Push(S) / B c }> • `AAd } m push ~}} C `AA S • >D }} . This enable us to introduce objects ”Epis(S)” and ”Monos(S)” as the subobjects of Push(S) and Pull(S) where the ”pushed out” or the ”pulled back” part of the diagrams are the identity maps, recalling the definition of epis and monos by certain diagrams being pushouts resp. pullbacks: ε epi and m mono if and only if we have pushout and pullback: • @@ ~~~~ @@@@@@ ~ ~ ~ @@@@@ ~~~~ @ # { ~~~ •@ • @@ ~~ @@ ~ ~~m m @@  ~~ • • ~~;C [c @@@@@@ ~ ~ ~ @@@@@ ~~~~~ @@@ ~ ~ ~~ • _@ ?• @@ ~~ @@ ~ ~~ε ε @@ ~~ • 7.2 Two sketches of fields 43 We can then sketch the object of monos and epis by adding objects and arrows to distinguish two cones: Monos(S) Epis(S) emono eepi   Pull(S) qa mpull Push(S) qb mpull idqA mpull qc mpush qd mpush idqC mpush &  x &  x S2 S2 a ~ • AAb ~ A @@ ~}} B • where qa , qb , qA are projections in the distinguished cone with S as vertex, likewise for qc , qd , qC . Meaning that when modelled in Set we get that Monos(S) is the unique subset of Pull(S) (pullbacks in the underlying graph) equalizing the ”pulled-back” part (model of / • a monomorphism maps a, b) with the identity on A. This gives A = B and the map A by the universal property of the pullback. Then of course the object ”Epi-monos(S)” will be sketched by distinguishing the cone: Epi-monos(S) o ooo o o oo wooo PPP PPP PPP PPP ' OOO OOO O mepis OOOO OO' nnn nnn n n nnn mmonos vnnn Epis(S) Monos(S) S2 hence Epi-monos(S) will, when modelled in Set, be the subobject of Ar(S) of maps both epi and mono. At last to get the property we set out to sketch, namely that all epi-monos are isos, we distinguish cone Iso(S) uuu uuuuu u u u uu v~ uuu NNN NNN NNN NN' JJ JJ JJ miso JJJ J$ o ooo o o ooomepimonos wooo Iso(S) Epi-monos(S) S2 ”Epi-monos(S)” will then in Set be realised as isomorphic to the object ”Iso(S)”, because we distinguished the pullback of miso along mepimonos to be an epi and it will evidently be a 44 Examples mono, so since we are modelling in set it is an iso, which sketchs our property. Remark: One should here be careful not to confuse the fact that epi-monos are isos in Set, with the fact that we are sketching this property for models in Set. The theoretical type tcl∗f ields and its corresponding free generator f τ̄cl∗f ields is constructed by a finite sequence of cone-expansions of τ̄clf , so the sketch morf phism tcl∗f ield : λ f / τ̄cl∗f ields f will indeed by propositions 3 and corollary 2. (in section 4.1.1) be a strict type morphism (a ”set-model-epi” from λ ) So we have constructed a strict type confusing the two sketches of fields ξ1 , ξ2 . We already saw that Top discriminate between ξ1 and ξ2 so, following the general problem, the only thing left for us to do is to make some conjecture about the free tcl∗f ields -type generated by Top. f The free tcl∗f ields -type of Top: Lets see what happen if we force all epi-monos in Top to f be isos. For any topological space X (object of Top) the identity on X (as a set) gives, for any topology T on X rise to a continuous epi-mono (X, P(X)) / / / (X, T ) . where P(X) is the discrete topology on X. Hence if we force epi-monos in Top to be isomorphisms, we get that all topologies are the discrete topology. We thus conjecture Tt ∗f ields (T op) ∼ = Set. cl f Since tcl∗f ields is strict we can profit from the application of generalized associated sheaf theof rem, in the ways discussed in section 6. We conclude: / ξ1 indicate the theorem that fields in the sense of ξ2 are -The sketch morphism ξ2 co-products of their 0-subspace and non-0-subspace. valid in all categories with finite limits and co-limits where all epi-monos are isos. This is trivial since ξ1 is itself the general proof, it evidently fulfill this property. -If our favorite semantics is of type tcl∗f ields then there is no need for the more general sketch f ξ2 , when interpreting field-structures, we can stick to the simpler ξ1 . The other way around, we should not believe that the simplest (set theoretical) way of sketching a structure always meet our purpose, since we saw that the first sketch of fields ξ1 does not give us topological fields when modelled in Top. We should free our minds of set-theory and describe concepts by sketching them according to where they are needed. 7.3 Two sketches of monoids We now look at another example a little more complicated than the previous, concerning two sketches of monoids following two different set theoretical definitions of monoids. We will establish a strict theoretical type confusing these two sketches, as well as look at one example 7.3 Two sketches of monoids 45 of a category that discriminate between them. Two ways to define a monoid in the classical set-theoretical setting: Def. 1: A monoid (M, k, e) is a set M with a binary operation ”k” everywhere defined, associative, and with a distinguished unit element e with respect to the operation. Def. 2: A monoid (M, k) is a set M with a binary operation ”k” everywhere defined, associative, and for which there exist a unit element. Sketch of monoids following definition 1: This is exactly the sketch of monoids as constructed as first step towards the sketch of fields ξ1 in the previous section (p. 36), we will call it µ1 . Sketch of monoids following definition 2: Sketch of semi-groups: S3 r2 r1 k1 k2 &  S2 p1 q1 q2 q3 k p2    | S1 which is just the first sketch of monoids µ1 , deprived of everything concerning the 0-ary operation e, the given unit element. In stead of the 0-ary operation we add the object Φ representing the formula (conjunction) x · e = e · x = x as the vertex of a distinguished cone: Φ i h .  S2 k p1    S1 k·σ 46 Examples / S 2 is to be thought of as a permutation (i.e. p1 · σ = p2 , where σ : S 2 This gives us the sketch SΦ : p2 · σ = p1 ). S3 r2 r1 Φ k1 &/  S 2 i σ q1 q2 q3 i h k2 p1 k p2     | S1 Thus we have added the equations: h = p1 · i = k · σ · i = k · i (h just naming the composite). When modelled in Set this will exactly mean that Φ is the subset of pairs of elements that satisfy the unity-condition, i.e. the subset {(x, e) ∈ M × M |x · e = e · x = x}. But since we need the formula ∃e¬(∃x¬(x · e = e · x = x)), we introduce ¬Φ by distinguishing the discrete co-cone i Φ 2 S D \9 99j 99 ¬Φ following the set theoretical meaning of the complement of Φ in S 2 (i.e. in terms of formulas all pairs not fulfilling Φ). Remark: One should here keep in mind that the uniqueness of ¬Φ given Φ, is in no way guaranteed in an arbitrary model, since it is a special property of Set. This means that we are already envisaging certain types of models having the property that one co-projection in a binary co-product completely determines the other co-projection as subobject of the coproduct (sum). To express the central part of the definition 2. (i.e. the existence of a unit element as a property of the binary operation; not itself introduced as a 0-ary operation as in µ1 ) we need to distinguish certain factorizations of certain arrows: for a formula Θ (figuring in a sketch as a vertex of a distinguished cone, like Φ) we can express ∃yΘ as the ”image” of the projection forgetting the variable y. So to get ∃x¬Φ we add a factorization of the map p1 · j: 7 S 2 OOO nnn OOO p n n OO1O nn n n OOO n n n OO' nn p1 ·j / 1 ¬Φ PPP n7 S n n PPP nnn PPP nnn0  n  PPP 0 n p1 P' nn j j ∃x¬Φ and distinguish p01 as a potential epi and j 0 as a potential mono. As before (with Φ) we distinguish the ”complement” of ∃x¬Φ as ”complementary subobject” 7.3 Two sketches of monoids 47 in S 1 , by distinguishing the co-cone j 0 ?   S 1 aC CCi0 CC ∃x¬Φ ¬∃x¬Φ To be able to talk about the truth of our formula we add the object S 0 (= 1) a vertex of the empty cone (terminal object in the model) S0 .     . ∅ / S 0 . Thus we can as before get the formula ∃e¬(∃x¬Φ), as ”suband projection p : S 1 object” of 1, by adding the factorization: k5 S 1 QQQQ kkk QQQ p k k QQQ kk kkk QQQ k k k QQQ k k p·i0 ( ¬∃x¬Φ SS 6/ S 0 m m SSS mmm SSS mmm m S m S m m  p0 SSSS) mmm ι i0 ∃e¬∃x¬Φ and again p0 should be realized as epi and ι as mono. Our formula ∃e¬∃x¬Φ is to be true in any model, so we need distinguish ι as invertible, i.e. / ∃e¬∃x¬Φ and equation ι · ι0 = 11 and ι0 · ι = 1∃e¬∃x¬Φ . add an arrow ι0 : 1 All in all, we get the underlying multiplicative graph of µ2 : S3 r2 r1 Φ ¬Φ p01 k1 k2 &/  2 σ q1 q2 q3 r9 S i i r r r r rrr rrr j p1 k p2 h "     } / S1 9s s i0 sss s s s ss  j0 ∃x¬Φ ¬(∃x¬Φ) p0 p ι  ∃e¬(∃x¬Φ) &  S0 h ι0 48 Examples Cones and equations are the ones gradually introduced in the above. Following the general problem we wish to find an equivalence ϕ : Setµ2 ∼ Setµ1 , derived / µ1 , i.e. such that ϕ = Setυ but obviously it is not posfrom a sketch-morphism υ : µ2 / µ1 in the strict sense, by lack of distinguished cones sible to get a sketch morphism µ2 and co-cones in µ1 . So we add to µ1 what is needed to get a sketch-morphism, in a way such that the new larger sketch µ¯1 has the same models in Set as both µ1 and µ2 . We thus get two sketch morphisms to µ¯1 in the following way µ2 υ / µ̄1 O incl µ1 and we are then interested in determining a general type in which, both of these two sketchmorphisms induce an equivalence of model categories and hence an equivalence: ModC (µ1 ) ∼ ModC (µ2 ) for all categories C of this type. Construction of µ¯1 : we just add and distinguish the cones and co-cones missing to get a mor/ µ¯1 . We thus add to µ1 objects X1 , X2 , X3 , X4 and distinguish co-cones phism υ : µ2 and cone: SE 2Y3 i1 X1 33 33i2 3 X2 SE 1Y3 i3 X3 X E 3Y3 33 33 3 33 33i4 3 X4 X3Y3 33 33 3 X2 X E 3 X1 i  S2 k p1    S1 k·σ 7.3 Two sketches of monoids 49 hence µ¯1 has underlying multiplicative graph: S3 X1 WWWWWW r2 r1 k1 k2 i 1 WWWWW WWWWW !  WW'+ gg936 S 2 i σ q1 q2 q3 g g g g g g g g ggg i2 ggggg X2 0 p1 v2 v1 k p2  p1 X3 WWWWWW i WWW3WW WWWWW      | WW+ 1 ggg3 S g F i ê g g g g g g gg ggggg i4 X4 gRRR p RRR e RRR RRR RRR ! RRR R)  0 S with equations and cones as µ1 plus u ◦ i4 =!, p1 ◦ i2 = i3 ◦ p01 , the three distinguished co-cones and the distinguished cone, above. / Set can be expanded to a µ¯1 has same models as µ1 in Set, since any model R : µ1 / Set . Because R̄X1 is determined as the vertex of a limit, then X2 can (up model R̄ : µ¯1 to iso) only be mapped to the complement of R̄X1 in R̄S1 × R̄S1 and R̄X3 will be determined by epi-mono factorization and at last R̄X4 will be determined in the same way as R̄X2 . So we in fact get an isomorphism of set model categories. Now this makes use of some properties particular for the category Set: the existence of complements and unique epi-mono factorization. But we are interested in identifying which general properties a category need to satisfy for µ1 and µ̄1 to have the same models therein. We want to arrive at a general strict type of categories that has the properties needed. We can systematize this procedure by checking the properties of (iii) in the standard theorem stated earlier concerning equivalences of categories (theorem 7.1). We thus investigate the properties of a category C needed for Mod(µ¯1 , C) to have the properties of (iii): -incl is an inclusion so incl∗ is evidently full. -incl∗ is faithful, i.e. for any R, R0 models of µ¯1 , the map HomMod(µ¯1 ) (R, R0 ) incl∗ / Mod(µ1 , C) / HomMod(µ ) (incl∗ R, incl∗ R0 ) 1 / R0 two natural transformations such that τ 6= τ 0 , then there is injective: take τ, τ 0 : R is at least one object S in Obµ¯1 such that τS 6= τS0 . If S is in µ1 we evidently have incl∗ τ 6= 50 Examples incl∗ τ 0 . If not we must have S = Xk for some k ∈ {1, . . . , 4} and by naturality Rik RXk 0 τX τ Xk τS j k  R0 Xk / RS j R0 i k R0 ik τXk = τS j Rik τ0 j  / R0 S j S and R0 ik τk = τS0 j Rik 0 implies R0 i τ 0 0 Now if all R0 ik ’s are monomorphism, we have τXk 6= τX k Xk 6= R ik τXk and hence k 0 0 ∗ ∗ 0 τS j Rik 6= τS j Rik which gives us τS j 6= τS j and hence incl τ 6= incl τ , i.e. incl∗ is faithful. We have that R0 i1 and R0 i3 are monomorphisms for C any category, and if C has the property that a co-projection in a sum is a monomorphism then R0 i2 and R0 i4 will be monomorphisms too. / C , we then wish to find an To show the last property of (iii) take a model of µ1 R : µ1 ∗ expansion of R to a model R̄ of µ¯1 such that incl R̄ ∼ = R. First we suppose the category C to be at least of type tclf , i.e. it has finite limits and co-limits, then R̄X1 is determined up to isomorphism (it is the vertex of a equalizer cone), so we need C to have properties assuring the existence of objects R̄X2 , R̄X3 , R̄X4 defining a model R̄ of µ¯1 . As already pointed out, the properties that all monomorphisms are co-projections in a binary sum and that all such co-projection determines the other up to isomorphism are desired. We will call the latter the sum-property. Now: -We can expand R to model X2 : if monomorphisms are co-projections in some sum (in C) R̄X2 will be determined by R̄X1 up op isomorphism by the sum-property. -We can expand R to model X3 : because of the way we (in µ2 ) sketched the notion of existence following the second definition of monoids any category serving as target for models will have some epi-mono-factorization property. So if we demand that C has epi-mono-factorization for all arrows (not necessarily unique) then the existence of R̄X3 will follow. -We can expand R to model X4 : again if all monomorphisms are co-projections in some sum, R̄X4 will be determined by the sum-property. We conclude that incl∗ is an equivalence of model categories if C is of the type tclf with the extra properties: the sum-property, all arrows have epi-mono-factorization, all monomorphisms are co-projections in a binary sum and all co-projections are monomorphisms. 7.3 Two sketches of monoids 51 / µ¯1 We now look at υ : µ2 r2 r1 Φ VVVVVV k1 k2 VVVV VVVV VVV&*  i hh4 S 2 i σ q1 q2 q3 h hhh ? h h h hhh j hhhh ¬Φ   p01 11S ׈(e) p1 k p2  ( ∃x¬Φ VVVV j 0 2 VVVV VVVV ;      } VVV* 1 iii4 S i0 iiiii i iiii ¬(∃x¬Φ) p0split D p 0 p ι ∃e¬(∃x¬Φ) c #  ι0 S0 / S3 υ S3 r2 r1 k1 k2 / X1 S SSS SSS SSS SS&)  i1 / k57 S 2 i σ q1 q2 q3 k i2kkkk 9 k k kkkk / X2 0 p p 1 k 2  p 1 v2 v1 / X3 S SSS SSS SSS      } SS) i3 / k5 S 1 ê kkk H i k k kk kkk i4 / X4 k FF FF p e FF FF F ! FFF FF #  -/ 0 S / ModC (µ2 ) is an equivalence, so again we check We are interested in when υ ∗ : ModC (µ¯1 ) the properties of (iii): We start with the last property of (iii) (assuming that C has the properties arrived at for incl∗ / C a model of µ2 . We then again wish the existence to be an equivalence): Take R : µ2 of an extended model R̄ of µ̄1 such that υ ∗ R̄ ∼ = R. If C has properties that permit, from R(µ2 ) the existence of arrows modelling the arrows e, ê, v1 , v2 (in µ¯1 ), we will have the extension searched for. If C has the property that all epimorphisms are split and we note εsplit the split morphism of an epimorphism ε in C, then R̄ can be defined as equal to R on everything hit by υ (meaning that we get υ ∗ R̄ = R) and then we define R̄(e) = R(i0 )R(p0 )split R(ι0 ) and R̄(v1 ) = 1RS 1 × R̄(ê), R̄(v2 ) = R̄(ê) × 1RS 1 . Then R̄ is obviously a functor extending R to µ̄1 , but it is not obvious that it defines a model. We need to proof that R̄(p1 )R̄(v1 ) = R̄(k)R̄(v1 ) = R̄(σk)R̄(v1 ). I had overlooked this in the first version of this paper, so I have not yet proven that we only need the properties arrived at so far in order to prove that R̄ is a model of σ¯1 . But it is obviously true for set theoretical models, since sums are disjoint unions in Set, the question is wether the sum property assures the connection between the two ways of sketching unitarity, I am not sure, but if we add the property that all arrows from the terminal object to a sum will facter through one and only one of its co-projections, then we get the connection searched for. We shall leave this at that since we have no particular use of this example other than to illustrate how the ”the mashinery” works. υ ∗ is faithful: same procedure as before (with incl∗ ) and it follows from the fact that υ is surjective on objects. 52 Examples υ ∗ is full: take any pair of models R, R0 ∈ ObMod(µ¯1 ,C) then / HomMod(µ ,C) (υ ∗ R, υ ∗ R0 ) HomMod(µ¯1 ,C) (R, R0 ) 2 is surjective: if we take a natural transformation τ : υ∗R / υ ∗ R0 / R0 such that υ ∗ τ̄ = τ . υ is then we need to find a natural transformation τ̄ : R surjective on objects, so it is enough to show that the naturality of τ extends to the arrows v1 , v2 , e, ê. As long as we are talking about models in a category C with all epis split there is no harm in adding the pair of potential split maps p0split , 11S × ê of p0 , p1 to the sketch µ2 (as above indicated by dotted arrows). Then we can set υ(11S × ê) = v1 and the naturality of τ will follow easily by the equations v2 = σv1 , ê = p2 v1 , ep = ê. 7.3.1 A type confusing the two sketches of monoids By the above systematic request for equivalent model categories, we can conclude1 that a sufficient2 type of category, in which the two sketches µ1 and µ2 have the same models, is a category with finite limits and co-limits and furthermore satisfying: the sum-property, all epimorphisms are split, all arrows have some epi-mono factorization, all monomorphisms are co-projections in some sum and all co-projections in a sum are monic. In fact the last property follows from the others: Lemma: A sufficient type for which µ1 and µ2 have equivalent model categries, is the type: categories with finite limits and co-limits, with the sum-property, where all epis split, where all arrows have epi-mono factorization and where all monos are co-projection in some sum. Proof: In a category of the type in the Lemma, all co-projections in a binary sum are monomorphisms: take a sum A + BeJ b a 9 A t ttt JJJ B and take an epi-mono factorization of, lets say, a /A+B OO 'O a llll5 l ε ' 5l m A OOO I then the diagram A + BeJ b m u: I 1 2 :uuu JJJ B up to the missing verification. remember that we nowhere proved necessity of all of these properties 7.3 Two sketches of monoids 53 is a sum. Because, take any diagram c ? C `AAAc0 I B / C such that then, since (A + B, (a, b)) is a sum, there exists a unique arrow h : A + B 0 ha = cε and hb = c . Now since a = mε we have that ha = hmε = cε which, by ε epimorphic, gives hm = c. So (A + B, (m, b)) is a sum, and hence by sum-property A ∼ = I, thus a is a monomorphism (same goes for b).  We will call the properties arrived at, the ∗monoids -properties. i.e. the properties: sumproperty, all epimorphisms split, all arrows have some epi-mono factorization and all monos are co-projections. The corresponding sketch of the type confusing the two sketches of monoids, will hence be noted τ̄cl∗monoids , and this sketch will be constructed in the followf ing, by adding the ∗monoids -properties to the sketch τ̄clf one by one: The sum-property: given a binary co-product (co-limit indexed by the discrete graph 2), if one co-projection is known then the other is known up to isomorphism. So in the sketch τ̄clf we need make the restriction, required by this sum-property, on the object destined to be the set of binary co-products: For I = 2 = {• •} we have our choice of co-limits, by the arrow L2 destined to choose a 2-co-limit of all pairs of objects in the underlying graph of the model. We will call sums all co-limits indexed by 2, so ”the object of sums” will be the object 2-co-limits(S) as introduced in the construction of τ̄clf . This is practical since the sum-property concerns all sums and not just the chosen ones, though the chosen ones are representative up to isomorphism. We have (in the model) the object of sums Sum(S), as the subobject of S %- such that the hook by the comparison map (the arrow u between the vertexes of the two co-cones compared) is an isomorphism, i.e the following cone is distinguished: Sum(S)     m sum       << << << <