Category Theory. Course by Dr. Arthur Hughes, Typset by Cathal Ormond

Transcription

1 Category Theory Course by Dr. Arthur Hughes, 2010 Typset by Cathal Ormond

2 Contents 1 Types, Composition and Identities Programs Functional Laws Categories Deinitions Examples Functors Deinitions More Deinitions Examples o Functors Universal Properties Terminal Object Duality Initial Object Binary Product Examples o Binary Products Binary Sum Examples o Binary Sums More on Functors Covariant Hom Functor Covariant Hom Functor Subcategory Universal Morphism Natural Transormations Equivalence The Functor Category Yoneda Embeddings The Yoneda Lemma A Supplementary Deinitions 28 A.1 Function and Classes A.2 Structures

3 Chapter 1 Types, Composition and Identities 1.1 Programs A program (unction) applied to an argument x is denoted x or (x). We will develop some notation beore we continue: g x = (g x) <, g > x =< x, g x > { x i t = l [, g] < t, x >= g x i t = r We also deine the ollowing primitive unctions: id x = x outl (x, y) = x outr (x, y) = y inl x =< l, x > inr x =< r, x > zero x = 0 succ x = x + 1 The above notation is quite abstract, so we can think o them in amiliar terms by using set notation: I : A B, then x A x B. I : A B, g : B C, then g : A C. I : T A, g : T B, then <, g >: T A B. I : A T, g : B T, then [, g] : A + B T. We can also consider the above deined unctions in terms o set theory: id : A A

4 4 outl : A B A outr : A B B inl : A A + B inr : B A + B zero : 1 N succ : N N where 1 is the set containing one element, sometimes denoted { }. It is common to denoted such unctions by what are called commuting diagrams. For example, we denote the act that i : A B, g : B C then g : A C by the ollowing commuting diagram: 1.2 Functional Laws We have a set o laws that apply to all programs/unctions: Identity Law: I : A B, then id A = : A B Identity Law: I : A B, then id B = : A B Associativity Law: I : A B, g : B C, h : C D, then h (g ) = (h g) : A D I A : T A, g : T B then outl <, g >= : T A I A : T A, g : T B then outr <, g >= g : T B < outl A,B, outr A,B >= id A B : A B A B We can represent the above by commuting diagrams: A id A A B A B idb B A B h g g g C h D T A outl <,g> A B g outr B A outl A B id A B A B outr B outl outr where we have combined the 4 th and 5 th conditions in the second last diagram. We wish to sepa- We shall denote by (A B) the set o all unctions rom A to B. rate the ollowing two concepts: unctional programs and their laws the meaning o unctions as deined by their application. For example, given (x) = x 2 we wish to dierentiate between and x 2. We will usually consider simply and its properties. To do this, we use categories and unctions, instead o sets and mappings.

5 Chapter 2 Categories 2.1 Deinitions We deine a Category C to contain the ollowing data: 1. Obj(C), a class o objects. 2. Mor(C), a class called the morphisms o C. 3. dom, cod : Mor(C) Obj(C). For all Mor(C), we call dom() the deomain o and cod() the codomain o. 4. id : Obj(C) Mor(C). For all A Obj(C), id (A) = id A is called the identity morphism or A. 5. : Mor(C) Mor(C) Mor(C) a partial unction called composition. For, g Mor(C) we denote by g the composite o g ater subject to the ollowing conditions: dom(id A ) = A = cod(id A ) g Mor(C) cod() = dom(g) i g is deined, then dom(g ) = dom() and cod(g ) = cod(g) i dom() = A and cod() = B, then id B = = id A The associativity law holds on Mor(C) We denote by C[A, B] or C(A, B) the class o morphisms rom A to B. I or all A, B Obj(C), C[A, B] is a set, then these sets are called homomorphism (or simply hom) sets. We have the ollowind deinitions: A category C is called Small i Obj(C) is a set and or all A, B Obj(C), C[A, B] is a (hom) set. A category C is called Locally Small i or all A, B Obj(C), C[A, B] is a (hom) set. A category C is called Large i Obj(C) is not a set.

6 6 2.2 Examples The ollowing are large categories From Sets Set a category o sets, whose objects are sets and whose morphisms are mappings between sets. Pn a category o sets, whose objects are sets and whose morphisms are partial mappings between sets. Rel a category o sets and relations, whose objects are sets and whose morphisms are binary relations on the sets. Set a category o sets, whose objects are inite sets and whose morphisms are mappings. Set the category o pointed sets, whose objects are pairs o the orm (A, A ) where A is a set and A A and whose morphisms are mappings : A B such that ( A ) = B, called base point preserving. Set a category o sets, whose objects are sets which don t contain and whose morphisms are mappings : A { } B { } such that ( ) =, called - preserving Algrbraic Structures Graph the category o directed graphs, whose objects are directed graphs and whose morphisms are graph morphisms. Mon the category o monoids, whose objects are monoids and whose morphisms are monoid morphisms. Grp the category o groups, whose objects are groups and whose morphisms are group homomorphisms. Ab the category o Abelian Groups, whose objects are Abelian Groups and whose morphisms are group homomorphisms. Rng the category o rings, whose objects are rings and whose morphisms are ring homomorphisms. CRng the category o commutative rings, whose objects are commutative rings and whose morphisms are ring homomorphisms. Vect F the category o vector spaces, whose objects are vector spaces over the ield F and whose morphisms are linear transormations. Pre the category o preorders, whose objects are preorders and whose morphisms are monotone (order preserving) mappings. Pos the category o posets, whose objects are posets and whose morphisms are monotone mappings. M-Set the category o M actions, whose objects are actions on a ixed monoid M and whose morphisms are M-action morphisms.

7 Topological Spaces Top a category o topological spaces, whose objects are topological spaces and whose morphisms are continuous mappings. Top h a category o topological spaces, whose objects are topological spaces and whose morphisms are homotopy classes o continuous mappings. Top a category o topological spaces, whose objects are topological spaces with base points and whose morphisms are base point preserving continuous mappings. Isomorphism A morphism C[A, B] is called an Isomorphism i there exists some g C[A, B] such that g = id A and g = id B. We call g the inverse or sometimes denoted 1, and also say that A and B are isomorphic, denoted A = B Proposition I g 1, g 2 C[A, B] are inverses or C[A, B], then g 1 = g 2 Proo Proposition g 1 = g 1 id B = g 1 ( g 2 ) = (g 1 ) g 2 = id A g 2 = g 2 Identity morphisms are isomorphisms Proo This ollows directly rom the deinitions o isomorphism and identity Proposition The composition o two isomorphisms is an isomorphism. Proo Let and g be isomorphisms with inverses 1 and g 1 respectively. We ll show that (g ) 1 = 1 g 1. (g ) 1 = 1 g 1 (g ) 1 g 1 = id g ( 1 ) g 1 = id g id g 1 = id g g 1 = id the last statement o which is clearly true. The other direction, i.e. showing that ( g) 1 = g 1 1 ollows similarly. The ollowing are small categories. Given a preorder P = (P, ), the objects are elements o P and the morphisms are given by : x y exists i x y.

8 8 Given a set S, the objects are elements o S and the morphisms are simply the identity morphisms. Given a monoid M =< M,, u >, the objects are the single object and the morphisms are the mappings x : or all x M Given a graph G =< N, E, s, t >, the objects are the nodes in N and the morphisms are paths between nodes. 0 the empty category, graph with no objects and no morphisms, generated rom the empty graph. 1 the trivial category containing one object and the identity mapping, generated rom a graph with one node and no edges. 2 the category containing two points and three mappings (two identity mappings) generated rom the graph with two nodes and one edge. the category with one object and one mapping, generated rom a graph with one node and one edge. A unction Rwhich assigns to each pair A, B in a category C a binary relation R A,B on the hom class C[A, B] is called a congruence or relation on C i: or all A, B Obj(C), R A,B C[A, B]. is a relexive, symmetric and transitive relation on or all A, B, A, B Obj(C) and or all, C[A, B], g C[A, A ], h C[B, B ] we have R A,B (h g)r A,B (h g) Proposition Given any unction R which assigns to each pair A, B in a category C a binary relation R A,B on the hom class C[A, B], then there exists a least congruence R on C with R R. Quotient category Given a category C and a unction R which assings to each pair A, B in a category C a binary relation R A,B on the hom class C[A, B], then there exits a Quotient category C/R whose objects are objects o the category C and whose objects are the hom classes (C/R)[A, B] := C[A, B]/R A,B, where R A,B is the least congruence o C containing R. Dual Category Given a category C, we deine the Dual category, denoted C op, by Obj(C op ) = Obj(C) or all A, B Obj(C), C op [A, B] := C[B, A] dom op () = cod() and cod op () = dom() id op A = id A op g := g

9 9 Product Category Given categories C, D, we deine the Product category, denoted C D, by Obj(C D) = Obj(C) Obj(D) or all A, A Obj(C) and B, B Obj(D) we deine C D[< A, B >, < A, B >] := C[A, A ] C[B, B ] dom C D (<, g >) :=< dom C (), dom D (g) >, cod C D (<, g >) :=< cod C (), cod C (g) > id <A,B> :=< id A, id B > < g, g > <, >:=< g, g > Slice Category Given a category C and an element X Obj(C), then we deine the Slice category over X, denoted C/X, by Objects: pairs < A, > where A Obj(C) and C[A, X] Morphisms: mappings h :< A, > < A, > where h : A A is a morphism in C and = h

10 Chapter 3 Functors 3.1 Deinitions A (covariant) Functor F : C D between categories C and D consists o an object mapping F Obj : Obj(C) Obj(D) and a morphism mapping F : Mor(C) Mor(D) such that or all Mor(C), we have dom(f ()) = F Obj (dom()) and cod(f ()) = F Obj (cod()), or all A Obj(C), we have F (id A ) = id FObj (A) or all A, B, C Obj(C) and or all C[A, B], g C[B, C], we have F (g ) = F (g) F (). Usually, the subsript Obj is dropped when the meaning is clear. Equivalently, we have a (covariant) unctor F : C D consists o a unction F : Obj(C) Obj(D) and a amily o unctions F [A, B] : C[A, B] D[F (A), F (B)] induced by pairs < A, B > o objects o C such that or all A Obj(C), we have F [A, A](id A ) = id F (a) or all A, B, C Obj(C) and or all C[A, B], g C[B, C] we have F [A, C](g A,B,C ) = F [B, C](g) F (A),F (B),F (C) F [A, B]() 3.2 More Deinitions Given unctors F : C D and G : D E, we deine the Composite unctor G F : C E by the ollowing: A G(F (A)) B G(F ()) G(F (B))

11 11 Given a category C, the Identity unctor on C denoted id C : C C is given by A A B B We denote by Cat the category o small categories and unctors between small categories. We denote by CAT the category o categories and unctors between categories. A unctor F : C op D is called a Contravariant unctor. 3.3 Examples o Functors We have the inclusion unctor which, or example, maps Set to Pn or Pn to Rel. (x) i x A, x dom() : Pn Set with, where (x) = B i x A, x / dom() or B x = A mappings o the orm : A B. D :Set Pn with D, where dom(d) := {x A x A, (x) B } and (D)(x) = (x) or all x dom(d), where :< A, A > < B, B > U :Mon Set which is a orgetul unctor (it orgets the monoid structure and just gives a set). V :Cat Graph, a orgetul unctor given by: < Obj(C), Mor(C), dom C, cod C, id C, > < Obj(C), Mor(C), dom C, cod C > F V (F ) < Obj(D), Mor(D), dom D, cod D, id D, > < Obj(D), Mor(D), dom D, cod D > :Graph Cat, a ree unctor given by < N, E, s, t > < N, P (E), s, t, id, > h where we deine < N, E, s, t > < N, P (E ), s, t, id, > P (E) := {e 1,..., e n t(e i ) = s(e i+1 ), 1 i n} {id A A N}, where the identity element is interpreted as the empty word.

12 12 composition to be the concatenation or the paths which join head to tail. h : G H is a unctor between categories deined by: a N h n (a) N e 1 e 2 e n h (e 1 e 2 e n):=h e(e 1 ) h e(e n) b N h n (b) N I :AbMon Mon an inclusion unctor, which is the identity mapping on the objects o AbMon, and where AbMon[A, A ] Mon[A, A ] U :M-Set Set, a orgetul unctor. Given a monoid M =< M,, u >, we can also consider two unctors in the reverse order to the last example: Set M-Set, a ree unctor given by the ollowing commutative diagram A < M A, δ : M (M A) M A > :=id M B < M B, δ : M (M B) M B > where or all m, m M, a A, b B we deine δ (m, (m, a)) := (m m, a) and δ (m, (m, b)) := (m m, b) Set M-Set, a ree unctor given by the ollowing commutative diagram A < M A, δ : M (M A) M A > :=id M B < M B, δ : M (M B) M B > where or all m, m M, M A, g M B we deine [δ (m, )](m ) := (m m ) and [δ (m, g)](m ) := g(m m )

13 Chapter 4 Universal Properties 4.1 Terminal Object A terminal object in a category C is an object 1 such that or all A Obj(C) there exists a unique morphism rom A to 1, i.e. C[A, 1] contains one object. We ll denote this unique isomorphism by <> A Proposition I 1 and 1 are terminal objects o a category C, then there exists a unique isomorphism rom 1 to 1. Proo As 1 is terminal, there exists a unique <> 1 : 1 1. Similarly, as 1 is terminal, there exists a unique <> 1 : 1 1. Note that the only element in C[1, 1] is id 1 and the only element in C[1, 1 ] is id 1. However, <> 1 <> 1 : 1 1 and <> 1 <> 1 : 1 1, so <> 1 and <> 1 must be isomorphisms Proposition: Relection Law <> 1 = id 1 Proo <> 1 = id 1 i id 1 : 1 1, which is true Proposition: Fusion Law I C[A, B], then <> B =<> A Proo <> B =<> A i <> B : 1 1, which is true.

14 Duality Let S(C) be a statement about the objects and morphisms o a catagory C. Then we can orm, by reversing the direction o all the morphisms in S(C), another statement S op (C) = S(C op ) about C Proposition For all C Obj(CAT), S(C) is equivalent to or all Obj(CAT), S op (C). Proo For all C, we have S(C) S(C op ) S op (C). Similarly, or all C, we have S op (C) S op (C op ) S((C op ) op ) S(C). The question arises: what is the dual statement to the terminal object? 4.3 Initial Object An Initial Object in a category C is an object 0 such that or all A Obj(C) there exists a unique morphism rom 0 to A, i.e. C[0, A] contains one object. We ll denote this unique isomorphism by [ ] A. The irst ollowing three propositions now ollow by duality Proposition I 0 and 0 are terminal objects o a category C, then there exists a unique isomorphism rom 0 to Proposition: Relection Law [ ] 0 = id Proposition: Fusion Law I C[A, B], then [ ] A = [ ] B Proposition The empty category 0 is an initial object in Cat Proposition The trivial category 1 is a terminal object in Cat Proposition The empty set is an initial object in Set.

15 Proposition The singleton set { } is a terminal object in Set. 4.4 Binary Product A Binary Product o two objects A, B in a category C is speciied by an object A B o Obj(C) two projection morphisms outl: A B A and outr: A B B such that the ollowing diagram commutes T g!<,g> A A B outl outr B We say that a category has binary products i a binary product exists or all pairs o objects Proposition I < P, outl, outr > and < P, outl, outr > are binary product or the objects A, B o a category C, then there exists a unique isomorphism h : P P such that outl = outl h and outr = outr h. Proo As < P, outl, outr > is a binary product, we know that there exists some unique < outl, outr >: P P such that and outl = outl < outl, outr > outr = outr < outl, outr > Similarly, as < P, outl, outr > is a binary product, we know that there exists some unique < outl, outr >: P P such that outl = outl < outl, outr > and outr = outr < outl, outr >

16 16 We will show that the ollowing diagram commutes: P outl outr <outl,outr> A outl P outr B <outl,outr> outl outr P As per above, rom outl = outl < outl, outr > and outl = outl < outl, outr >, we conclude that outl < outl, outr > < outl, outr >= outl and so < outl, outr > < outl, outr >= id P Using similar logic, but in the other direction, we can show that < outl, outr > < outl, outr >= id P also. Thus, < outl, outr > and < outl, outr > must be isomorphisms Proposition: Relection Law < outl, outr >= id A B Proposition: Fusion Law <, g > h =< h, g h > 4.5 Examples o Binary Products Set Let A and B be sets. A B := {< a, b > a A, b B} outl < a, b >= a outr < a, b >= b Mon Let < M,, u > and < M,, u > be monoids. M M :=< M M,, < u, u >>, where < m, m > < n, n >=< m n, m n >. outl < m, n >= m outr < m, n >= n

17 Cat Let C and D be monoids. C D is the product category. outl(<, g >:< A, B > < C, D >) = ( : A C) outl(<, g >:< A, B > < C, D >) = (g : B D) Pos I C, P, > is a category deined by a poset, then a binary product exists i the poset has a greatest lower bound or all pairs o elements p, q P p q := {p, q} := p q outl : p q p i p q = p outl : p q q i p q = q Proposition I C is a category with a speciied binary product, then C C C given by (A, B) A B (,g) g=< outl,g outr > (A, B ) A B is a biunctor. Proo We need to show the ollowing or all, g Mor(C), dom( g) =dom() dom(g) and cod( g) =cod() cod(g) or all A, B Obj(C), id A id B = id A B or all A, A, A, B, B, B C and C[A, A ], C[A, A ], g C[B, B ], g C[B B ], we have ( ) (g g) = ( g ) ( g) The irst part ollows directly rom the deinitions. The second part is true by noting that id A id B =< id A outl, id B outr >=< outl, outr >= id A B the last part o which ollows by the relection law. To show the inal part, we note the ollowing proposition Proposition: The Absorbtion Law ( g) < p, q >=< p, g q >

18 18 Proo ( g) < p, q >=< outl, g outr > < p, q >=< outl p, g outr q >=< p, g q > Thus, our previous proposition ollows thus: ( g ) ( g) = ( g ) < outl, g outr >= ( outl, g g outr) = ( ) (g g) Proposition I C is a category with binary products and with a terminal object, then the ollowing natural isomorphisms exist or all A, B, C C: Proposition unit A : A 1 A swap A,B : A B B A assoc A,B,C : (A B) C A (B C) A binary product o objects A, B in a category C is a terminal object in the span[a, B](C) o spans over A and B: Objects: pairs o morphisms rom C with a common source, i.e. (, g) where A T g B. m : (, g) (, g ) where m : T T is a morphism on C such that m = and g m = g. 4.6 Binary Sum A Binary Sum o objects A, B in a category C is speciied by an object A + B o C with two injective morphisms inl : A A + B and inr : B A + B such that the ollowing diagram commutes: A inl A + B [,g] T inr g Note that this is the dualized idea o binary products. This gives us the ollowing propositions Proposition: Relection Law [inl, inr] = id A+B B

19 Proposition: Fusion Law h [, g] = [h, h g] 4.7 Examples o Binary Sums Set Let A and B be sets. A + B := A B := ({l} A) ({r} B) inl(a) = (l, a) inr(b) = (r, b) Mon Let M =< M,, u > and M =< M,, u > be monoids, and deine (A + B) to be the set o inite sequences o elements rom the set A + B M + M :=< (M + M ) /,, [ɛ] >, where is the concatenation operation i.e. [(x,..., y)] [(x,..., y )] = [(x,..., y, x,..., y )], [ɛ] is the empty word and is the least equivalence relation such that u ɛ and ɛ u (..., a, a,... ) (..., a a,... ) or all a, a M, (..., b, b,... ) (..., b b,... ) or all b, b B inl(a) = [(a)] inr(b) = [(b)] in the above, what we means by the least equivalence relation is the equivalence relation given by: l A < A,, ɛ > r [ ] A < A / A,, [ɛ] > = < A,, u > inl inl l < (A + B),, ɛ > [ ] < (A + B) / A,, [ɛ] > r inr inr B l r < B,, ɛ > [ ] B < B / B,, [ɛ] > = < B,, u >

20 20 Example: Graph I G is a directed graph, G is the underlying graph o the path category over G, H Obj(Cat) and H is the underlying graph o the category H, then we have the ollowing: G ηg G! H G! H I we are given : (A,, u) (M,, u ) and g : (B,, u ) (M,, u ), then we want to construct a unique h : ((A+B) /,, [ɛ]) (M,, u ) such that h = h inl and g = h inr. Step 1 In Set, we have inl inl A A + B B![,g] g M Step 2 In Set and then in Mon, we have (A + B) η A+B (A + B) [,g]! [,g] M ((A + B),, ɛ)! [,g] (M,, u ) Step 3 x u l ((A + r B), [], ɛ) ((A + B) /,, [ɛ])!h [,g] (M,, u )

21 Chapter 5 More on Functors 5.1 Deinition: Covariant Hom Functor Given a ixed object A o a catagory C the Covariant Hom Functor is a mapping C[A, +] = H A + : C Set such that B h C[A, B] g C[A,g]=H A g B C[A, B ] where C[A, g] = H A g is given by H A g (h) = g h. 5.2 Deinition: Covariant Hom Functor Given a ixed object B o a catagory C the Contravariant Hom Functor is a mapping C[, B] = H B : Cop Set such that A h C[A, B] F op C[F op,b]=h Fop B A C[A, B] where C[F op, B] = H Fop B is given by H B (h) = h F op. 5.3 Deinition: Subcategory C is called a Subcategory o D i there exists an inclusion unctor I : C D such that I( : A B) = ( : A B) Deinition: Faithul A unctor F : C D is called Faithul i the morphism mapping o the unctor is injective.

22 Deinition: Full A unctor F : C D is called Full i the morphism mapping o the unctor is surjective Deinition: Isomorphic Two categories C and D are said to be isomorphic, denoted C = D i there exist mappings F : C D and G : D C such that F G = id C and G F = id D. 5.4 Universal Morphism I G : X A is a unctor and A A is an object, then a universal morphism is a pair < A, η : A G(A ) > consisting o an object A X and a morphism η : A G(A ) o A such that to every pair < X, : A G(X) > with X X an object and a morphism o A, there exist a unique mapping [] : A X with G([]) η =, i.e. X G A A η A G(A )![]!G([]) X G(X) Example: U :Mon Set Let A be a set, deine η(a) =< a > and (a) = 1 or all a A, we have η A U(< A,, u >) < G,, ɛ > where L is the length mapping.!u(l) U(< N, +, 0 >) L < N, +, 0 > Example: U :Graph Cat Let G be a graph and H be a small category. Then, we have η G G U(G ) G U(H)!U([h]) L H

23 Example: U :Mon Set Let A be a set, deine η(a) =< a > and (a) = 1 or all a A, we have η A U(< A,, u >) < G,, ɛ > where L is the length mapping.!u(l) U(< N, +, 0 >) L < N, +, 0 > Example: Diagonal Functor Let : C C C be the unctor given by (A, A) = A, called the diagonal unctor. Then we have: (A, B) (inl,inr) (A + B)! ([,g]) (,g) (T ) A + B [,g] T 5.5 Natural Transormations Given categories C and D and two unctors F, G : C D, we deine a Natural Transormation denoted ψ : F G : C D as a collection φ = {φ A : F (A) G(A) A obj(c)} o morphisms o D indexed by objects o C such that or all (: A B) Mor(C), we have F (A) φ A G(A) C F D F () G() φ F (B) φ B G(B) C G D The morphism φ A is called the component o φ. A natural transormation φ : F G : C D is called an Isomorphism i φ A is an isomorphism or all A Obj(C). 5.6 Equivalence We say that two categories C and D are Equivalent i there exists two unctors F, G : C D together with two natural isomorphisms ɛ : F G = id D and η : id C = GF. 5.7 The Functor Category Let C and D be two categories. The Functor Category usually denoted [C, ] or D C is given by:

24 24 The objects are unctors rom C to D The morphisms are natural transormations between unctors We deine dom(α : F G : C D) = F : C D cod(α : F G : C D) = G : C D We deine the identity transormation on a unctor F as id F = {id F (A) : F (A) F (A) A Obj(C)} We deine the composition o two natural transormations α : F G and β : G H as the mapping β α : F H, where β α = {(β α) A = β A α A : F (A) H(A) A Obj(C)}

25 Chapter 6 Yoneda Embeddings Let C be a locally small category, and deine H : C op [C, Set ] as ollows: A C[A, +] = H A + op H op A C[A, +] = H A + where H op : H A + H A + : C Set is deined or all (g : B B ) Mor(C) as H A B H op B H A B H A g H A g H A B H H op B A B 6.1 The Yoneda Lemma Let C be a locally small category, F : C Set a unctor and A Obj(C). The collection nat[h A +, F ] o natural transormations α : H A + F is a set, so we may deine a unctor Nat[H, +] : C [C, Set ] Set given by < A, F > Nat[H A +, F ] <,µ> Nat[H,µ] < A, F > Nat[H+ A, F ]

26 26 where we deine Nat[H, µ] = (Nat[H, +]) <, µ >. We can also deine the eval unctor, namely ev : C [C, Set ] Set as < A, F > F (A) <,µ> ev<,µ> where or all x F (A), we have < A, F > F (A ) (ev <, µ >)(x) = [F () µ A ](x) = [µ A F ()](x) There exists natural ismorphisms Φ : Nat[H, +] ev : Ψ such that or all A Obj(C) and F : C Set, we have Proo Part A Φ <A,F > : Nat[H A, F ] = F (A) : Ψ <A,F > We ll show that or all A Obj(C), F : C Set, we have Nat[H A +, F ] Set by showing that there exists a bijection rom Nat[H A +, F ] F (A). For all α : H A + F, we have Φ <A,F > = α A (id A ) Also, or all a F (A), B Obj(C), HB A, we have Ψ <A,F > (a) = [F ()](a) We need to show that Ψ <A,F > (a) Nat[H+, A F ], namely or all (g : B B ) Mor(C) we have H A B Ψ <A,F > (a) B F (B) H A g F (g) H A B Ψ <A,F > (a) B F (B ) We note the ollowing: (F (g) Ψ <A,F > (a) B )() = F (g)(ψ <A,F > (a) B ()) = F (g)(f ((a)) = [F (g) F ()](a) = [F (g )](a) = [F (H A g ())](a) = Ψ <A,F > (a) B (H A g ()) = (Ψ <A,F > (a) B H A g )() Nat[H A +, F ]

27 27 We now wish to show that Ψ <A,F > Φ <A,F > = id Nat[H A +,F ] Φ <A,F > Ψ <A,F > = id F (A) For all α Nat[H+, A F ], B C, HB A, we have the ollowing: Also, note that or all a F (A), we have (Ψ <A,F > Φ <A,F > )(α) = Ψ <A,F > (Φ <A,F > (α)) = Ψ <A,F > (α A (id A )) = F ()(α A (id A )) = (F () α A )(id A ) = (α B H A )(id A) = α B (H A (id A)) = α B ( id A )) = α B () = α (Φ < A, F > Ψ <A,F > )(a) = Φ < A, F > (Ψ <A,F > (a)) = Ψ < A, F > (a) A (id A ) = F (id A )(a) = id F (A) (a) = a And so, we have that there exists an isomorphism between F (A) and Nat[H A +] or all A Obj(C) Part B We now wish to show that above-deined morphism actually deined a natural morphism, i.e. we want the ollowing diagram to commute: < A, F > Nat[H A +, F ] Φ <A,> ev < A, F >= F (A) <,µ> Nat[H op,µ] ev<,µ> < A, F > Nat[H+ A, F ] Φ <A,> ev < A, F >= F (A)

28 28 For all α Nat[H A +, F ], we have the ollowing: (Ψ <A,F > Nat[H op +, µ])(α) = Ψ <A,F >(Nat[H op +, µ](α)) = Ψ <A,F >(µ α H op + ) = (µ α H op + ) A (id A ) = (µ A α A H op A )(id A ) = (µ A α A )(H op A (id A )) = (µ A α A )(id A ) = (µ A α A )() = (µ A α A )( id A ) = (µ A α A )(H A(id A)) = (µ A α A H A)(id A) = (µ A F () α A )(id A) = (F () µ A α A )(id A) = (F () µ A )(α A (id A)) = ev <, µ > (α A (id A)) = ev <, µ > (Φ <A,F > (α)) = (ev <, µ > Φ <A,F > )(α) Thus, we have shown that there exists a natural transormation rom Nat[H, +] to ev whose components are isomorphisms. Hence, result. We note that the Yoneda embedding y : C op [C, Set ] is both ull and aithul, by noting that or all A, A Obj(C), we have as required. [C, Set][y(A), y(a )] = Nat[H+, A H A = ev < A, H A = H A + (A) = H A A = C[A, A ] = C op [A, A ] + ] + >

29 Appendix A Supplementary Deinitions A.1 Function and Classes A.1.1 Single Valued Given a unction : A B we let R be the set o all pairs (a, b) such that (a) = b or a A, b B. We say that is Single Valued, i (a, b), (a, b ) R and a = a, then b = b. A.1.2 Totally Deined Given a unction : A B we let R be the set o all pairs (a, b) such that (a) = b or a A, b B. W say that is Totally Deined, i {a A b Bs.t.(a, b) R} = A. A.1.3 Mapping A Mapping is a single-valued, totally deined unction. A.1.4 Partial Mapping A Partial Mapping is a single-valued unction. A.1.5 Relation A Relation is a unction between sets. A.2 Structures A.2.1 Directed Graph A Graph G is a quadruple < N, E, s, t > where E is a set o edges o the graph, N is a set o nodes o the graph and s, t : E N are the source and target mappings. A Graph Morphism h : G G is a quadruple < G, h n, h e, G > where h n : N N and h e : E E are mappings such that h n s = s h e and h n t = t h e. In other

30 30 words, the ollowing diagrams commute: E he E s s N hn N E he E s s N hn N A.2.2 Monoid A Monoid is a triple < M,, u > where M is a set, : M M M is an associative binary operation and u is a unit (identity) or the operation. Examples o monoids are: < N, +, 0 > < N 0,, 1 > where N 0 = N \ {0} < P(A),, > < P(A),, A > < Set[A, A],, id A > Note that a morphism between monoids h : M M is an operation preserving mapping which maps the unit to the unit. In other words, the ollowing diagrams commute: M M h h M M M M h e 1 M e M h A.2.3 Preorder A Preorder is a pair (P, ) where P is a set and is a relexive and transitive binary relation. A.2.4 Poset A Poset (partially ordered set) is a preorder whose binary relation is also anti-symmetric. A.2.5 Bounds Let < P, > be a poset and S P. Then an element z P is called a Lower Bound o S i or all s S, z s. Greatest Lower Bound o S i or all lower bounds y o S, y z. sometimes denoted S This is Upper Bound o S i or all s S, s z. Least Upper Bound o S i or all lower bounds y o S, z y. This is sometimes denoted S

31 31 A.2.6 M-Action An M-action on a ixed monoid M =< M,, u > is a pair < S, δ > where S is a set o states and δ : M S S is a mapping such that or all x, y M and s S, we have δ(x y, s) = δ(x, δ(y, s)) and δ(u, s) = s. In other words, the ollowing diagrams commute: (M M) S = M (M S) id M δ M S S = 1 S u id S M S id S M S S S δ id S δ S