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Category Theory and Its Applications - ITI9200 Category theory and its applications - ITI9200 https://compose.ioc.ee Yoneda lemma In set theory, we have a rule to decide when two sets are equal A=B X(X A X B) ⇐⇒ ∀ ∈ ⇐⇒ ∈ Yoneda lemma In set theory, we have a rule to decide when two sets are equal A=B X(X A X B) ⇐⇒ ∀ ∈ ⇐⇒ ∈ On the other hand, in category theory there can’t be such thing, because objects are impenetrable ‘dots’ and one can’t look inside them. Yoneda lemma In set theory, we have a rule to decide when two sets are equal A=B X(X A X B) ⇐⇒ ∀ ∈ ⇐⇒ ∈ On the other hand, in category theory there can’t be such thing, because objects are impenetrable ‘dots’ and one can’t look inside them. Idea Use morphisms to decide whether two objects are ‘the same’. Idea If for every morphismX A the result of looking insideA equals the → result of looking insideB, thenA ∼= B. (Not equal: isomorphic!) Prelude Yoneda lemma or How does a particle accelerator work? Prelude Prelude Prelude • physicists throw stuff into other stuff (using particle accelerators) to ‘see’ what the second stuff’s shape is, measuring the amplitude of an angle; • category theorists throw object into other objects (using morphisms) to ‘see’ what the second object looks like, measuring the amplitude (cardinality) of an hom-set. • Yoneda lemma: if you throw every object atA, and the result is the same of throwing every object atB, then A ∼= B. Representable functors Let Set be the category of sets and functions; a special class of functors (A, ): Set ( ,A): op Set C − C→ C − C → is defined as follows: • on objects, (A,X) is the set of all morphismsA X in ; C → C • on morphisms, (A,f) post-composes withf:X Y. C → Definition F: Set( G: op Set) is called representable if there existsA C→ C → and an isomorphism of functorsF = (A, )( G = ( ,A) ) ∼ C − ∼ C − Examples Let = Set be the category of sets and functions; let be a terminal C • object (a singleton). The identity functor is representable by : • A = Set( ,A). ∼ • Examples Let = Set be the category of sets and functions; let be a terminal C • object (a singleton). The identity functor is representable by : • A = Set( ,A). ∼ • Let = Cat be the category of categories and functors; let be a C • terminal category (one object, one identity). The functor( ) that − o sends a category to its set of objects is representable by : A • = Cat( , ). Ao ∼ • A Examples Let = Set be the category of sets and functions; let be a terminal C • object (a singleton). The identity functor is representable by : • A = Set( ,A). ∼ • Let = Cat be the category of categories and functors; let be a C • terminal category (one object, one identity). The functor( ) that − o sends a category to its set of objects is representable by : A • = Cat( , ). Ao ∼ • A Again, let = Cat; the functor( ) that sends a category to its C − a A set of arrows is representable by the category 0 1 : { → } = Cat( 0 1 , ). Aa ∼ { → } A Examples Let = Top be the category of topological spaces; letS= 0,1 be C { } the Sierpinski space, with topology ∅, 0 ,S ; the functor { { } } Op: op Set that sends a topological space to the set of its open C → subsets Op(X) is representable byS: Op(X) = Top(X,S) continuous mapsf:X S ∼ → Examples Let = Top be the category of topological spaces; letS= 0,1 be C { } the Sierpinski space, with topology ∅, 0 ,S ; the functor { { } } Op: op Set that sends a topological space to the set of its open C → subsets Op(X) is representable byS: Op(X) = Top(X,S) continuous mapsf:X S ∼ → Let again = Top; letD= 0,1 d be the discrete topological space C { } on two points: the functorD: Top op Set that sends a topological → spaceX into its set of disconnections, i.e. the set of pairs(U,V) such thatU V=X andU V=∅, is representable byD. ∪ ∩ Representable preserve limits LetD: be a diagram such that limD exists in ; let (A, ) J→C C C − be a representable functor. Representable preserve limits LetD: be a diagram such that limD exists in ; let (A, ) J→C C C − be a representable functor. Then, the limit of (A,D ): Set C − J→ exists, and it is canonically isomorphic to (A, limD). C Representable preserve limits LetD: be a diagram such that limD exists in ; let (A, ) J→C C C − be a representable functor. Then, the limit of (A,D ): Set C − J→ exists, and it is canonically isomorphic to (A, limD). C The universal property of the limit limD yields r (A, limD) i � (A, Di) C C and thus a morphism (A, limD) � lim (A, Di) C C Representable preserve limits Now, if ζ :X (A, Di) i →C is a cone for (A,D ), it sends an elementx:X to a morphism C − ζ (x):A Di. i → Representable preserve limits Now, if ζ :X (A, Di) i →C is a cone for (A,D ), it sends an elementx:X to a morphism C − ζ (x):A Di. i → The family ζ (x):A Di i now forms a cone forD with { i → | ∈J} domainA, and by the UP of limD this yields a unique ζ¯(x):A limD. → Representable preserve limits Now, if ζ :X (A, Di) i →C is a cone for (A,D ), it sends an elementx:X to a morphism C − ζ (x):A Di. i → The family ζ (x):A Di i now forms a cone forD with { i → | ∈J} domainA, and by the UP of limD this yields a unique ζ¯(x):A limD. → Sendingx ζ¯(x) is a functionX (A, limD), and this is the only �→ →C one with the property thatr ζ¯=ζ . I ◦ i X ζ¯ � � (A, limD) � (A, Di) C ri C (F) has terminal F representable E ⇐⇒ Definition (category of elements) LetF: Set( G: op Set) be a functor. We define the category C→ C → of elements (F) ofF as follows: E • objects: the pairsA: ,a:FA; C (F) has terminal F representable E ⇐⇒ Definition (category of elements) LetF: Set( G: op Set) be a functor. We define the category C→ C → of elements (F) ofF as follows: E • objects: the pairsA: ,a:FA; C • morphisms(A,a) (B,b) the arrowsh:A B in such that → → C Fh(a) =b. (F) has terminal F representable E ⇐⇒ Definition (category of elements) LetF: Set( G: op Set) be a functor. We define the category C→ C → of elements (F) ofF as follows: E • objects: the pairsA: ,a:FA; C • morphisms(A,a) (B,b) the arrowsh:A B in such that → → C Fh(a) =b. • morphisms(A,a) (B,b) the arrowsh:A B in such that → → C Gh(b) =a. (F) has terminal F representable E ⇐⇒ LetF: Set( G: op Set) be a functor; then,F is C→ C → representable if and only if El(F) has an initial (terminal) object. (F) has terminal F representable E ⇐⇒ LetF: Set( G: op Set) be a functor; then,F is C→ C → representable if and only if El(F) has an initial (terminal) object. Proof. IfF is representable, sayF= (A, ), then El(F) = A/ , and 1 is the C − ∼ C A initial object; conversely, letF be such that El(F) has an initial object (V,t:FV). (F) has terminal F representable E ⇐⇒ LetF: Set( G: op Set) be a functor; then,F is C→ C → representable if and only if El(F) has an initial (terminal) object. Proof. IfF is representable, sayF= (A, ), then El(F) = A/ , and 1 is the C − ∼ C A initial object; conversely, letF be such that El(F) has an initial object (V,t:FV). Then for all(A,a): El(F) there is a unique h:V A with the → property that Fh:t a. This gives a bijection �→ FA (V,A) →C that is natural inA. Yoneda lemma: statement Theorem Let F: Set be a functor, and let (A, ) be the representable C→ C − functor associated to an object A; then there is a bijection :=Nat( (A, ),F)=[A,F] C − natural transformationsα: (A, ) F = FA { C − ⇒ } ∼ natural both in A: and in F. C Yoneda lemma: statement Theorem Let F: Set be a functor, and let (A, ) be the representable C→ C − functor associated to an object A; then there is a bijection :=Nat( (A, ),F)=[A,F] C − natural transformationsα: (A, ) F = FA { C − ⇒ } ∼ natural both in A: and in F. C α (A,X) X � FX C f _ Ff � � a:FA ◦ { } � � (A,Y) � FY C αY Yoneda lemma: proof Given a n.t.α: (A, ) F, we can look at a special component C − ⇒ α : (A,A) FA A C → because the set (A,A) is always nonempty, C Yoneda lemma: proof Given a n.t.α: (A, ) F, we can look at a special component C − ⇒ α : (A,A) FA A C → because the set (A,A) is always nonempty, and thusα (id ) is an C A A element ofFA. Yoneda lemma: proof Given a n.t.α: (A, ) F, we can look at a special component C − ⇒ α : (A,A) FA A C → because the set (A,A) is always nonempty, and thusα (id ) is an C A A element ofFA. id A is a universal element in the sense that it uniquely determinesα. Yoneda lemma: proof Given a n.t.α: (A, ) F, we can look at a special component C − ⇒ α : (A,A) FA A C → because the set (A,A) is always nonempty, and thusα (id ) is an C A A element ofFA.
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