A skeleton for the proof of the Yoneda Lemma (working draft)

Transcription

1 skeleton or te proo o te Yoneda Lemma (workin drat) Eduardo Ocs February 2, 20 bstract ese notes consists o ive parts. e irst part explains ow to draw te internal view o a diaram (or o a unction, unctor, natural transormation, etc). e second part sows tat a certain diaram, tat we call diaram Y0, is te skeleton o te proo o te Yoneda Lemma in te ollowin sense. In order to interpret tat diaram ormally we ave to iner te types o all its entities, and ten iner (by term inerence, as in [Oc3], obtainin untyped λ-terms) te actions on morpisms o te our unctors in Y0, and also te actions o te our natural transormations and te actions o tree bijections. e bijections are called, 2 and 3, were is easy to construct, 2 is obtained rom by substituin a eneric unctor and a eneric object tat appear in by speciic ones, and 3 is 2 composed wit two trivial bijections, one at eac side. e statement o te Yoneda Lemma is essentially just 3 is a bijection. In ateory eory texts above a certain level most term inerences are treated as obvious, so a (skeleton o a) proo o te Yoneda Lemma is just diaram Y0 plus do te obvious type inerences and term inerences. e tird part discusses a ap in te second part. e bijection 3 converts a map Hom (, ) into a natural transormation Nat((, ), (, )) and a into an, but wat we ot in te second part is just a pair o λ-terms o te rit types, ( 3, 3 ), witout te proos tat 3 (3()) = and tat 3( 3 ( )) =. In te lanuae o [Oc3] wat we did was to drop, or erase, a lot o inormation (mainly te equational parts ) and ten work in te syntactical world ; we obtained a skeleton o a proo tat must now must be lited to te real world by completin some missin parts. It turns out tat 3 (3()) = is trivial, but 3( 3 ( )) = only olds i obeys te naturality condition tat comes rom it bein a natural transormation. e moral o te story so ar is tat 90% o te proo o te Yoneda Lemma can be extracted rom diaram Y0 i we do te obvious type and term inerences on it ( or some value o 90%, o course); only a tiny part o te proo needs tins tat et erased in te passae to te skeleton. e ourt part uses tese tools to state and prove tree oter Yoneda Lemmas and to deine universal arrows, universal elements, representable

2 2 unctors, and to sow ow some o tese ideas are motivated by adjunctions. e it part uses all tis to build brides between several notations. e less trivial case is ow to translate between our notation and te one in Reyes, Reyes and Zolaari s Generic Fiures and eir Glueins; te translations between our notation and MacLane s, Riel s and wodey s are easy (but only RRZ as been written in details at te moment). Index o sections: Internal views anin sape, canin notation Interpretin diarams Y0, Y, and Y is really a bijection Makin te bijections more explicit stroner Yoneda Lemma Representable unctors Universal elements and universal arrows djunctions wo contravariant Yoneda Lemmas e Yoneda Embeddins Readin Generic Fiures and ter Glueins Morpisms o -sets e Yoneda Lemma in RRZ Readin MacLane s WM Readin Emily Riel s in ontext Readin wodey s ateory eory Related projects Internal views Note: tis section is an introduction to te idea o internal views o cateorical diarams. I irst saw tis idea in [LS97], p.3, but it is used in oter places too or example in p.7 o [Rie6]. I used it a lot in [Oc3], but tere I insisted on a notion o downcasin tat I ve since abandoned. Wen I was a kid my irst exposure to unctions was trou diarams like tis: yoneda February 2, 20 7:26

3 3 ater a wile actually years te blob-sets ot names, like,, N, R, te unctions ot names like,,, and several conventions were establised: we didn t ave to draw all elements in te blob-sets; we could draw a eneric element, n, and indicate tat it oes to n; and we could draw an external view o te unction above or below te internal view iven by te blobs: n n N R en te internal view radually disappeared rom our matematical practice, and we started to write unctions like tis, : N R n n, : a (a), : D E (c, d) c d wic makes a clear distinction between te tailless arrow,, and te arrow wit tail, : : is a unction tat takes elements (plural!) rom to elements o, and n (n) is an element (in te sinular) bein taken to anoter. Rewritin our diaram or te internal and te external views o witout blobs, it becomes: 4 2 n n, or simply: n n N R N R We will oten use te convention tat : is a unction rom to, but is te set o all unctions rom to i.e., ( ) = and : means ( ) on s tis doesn t old, and te names on s can be omitted. e internal view o a unctor F : is more complex. e cateory as not only points (te objects o ) but also arrows (te morpisms 207yoneda February 2, 20 7:26

4 4 o ). e unctor F takes a morpism : in to a morpism F : F F in ; and sometimes we will denote te action o F on objects by F 0 and its action on morpisms by F, so a diaram wit te internal and te external views o F may be drawn, or example, as: F 0 F 0 F F 0 F 0 F F or as: F F F F e action o a natural transormation : F G, were F, G : are unctors, consists o a sinle operation not two as in unctors tat expects an object and returns a morpism : F G in. We can represent tat action as ( : F G) or (F G), or as a diaram: F G F G F G F e naturality condition o a natural transormation : F G is te assurance tat or every arrow α : in tis square commutes: G F G α F α Gα F G F G Diarams like te one above will be our avorite ways to draw internal views o natural transormations. Note tat te arrows or te unctors F and G are let implicit. We will sometimes use diarams like tis to sow te internal view o a commutative diaram, especially wen it is in Set: a (a) k D (a) k ((a)) k((a)) 207yoneda February 2, 20 7:26

5 5 te internal view sows tat ((a)) = k((a)) or every a. Our avorite way to coose names or te components o an adjunction and to draw its internal view is tis: L L R R e adjunction L R is a bijection Hom (L, ) Hom (, R) or eac in and eac in. Note tat te unctor L appears at te let o te and o te,, and it oes let; te unctor R appears at te rit o te and o te,, and it oes rit; te direction o te bijection oes let in te diaram, and it pulls te unctor in : R to te let o te ; te direction o te bijection oes rit in te diaram and puses te unctor in : L to te rit o te. Wen is a inite cateory tat can be drawn explicitly, like tis, we can represent unctors rom to oter cateories very compactly usin a positional notation similar to te ones in sec. o [Oc7]. For example, tis diaram {3, 4} {, 2} {5, 6} can be intepreted as a unctor F : Set wit F () = {, 2}, F () = {(, 5), (2, 6)} and so on we deine F by te internal view o its imae. 2 anin sape, canin notation e translation between two lanuaes or diarams can be done in two steps canin sape and canin notation tat are independent and can be applied in any order. In te example below we start wit a diaram rom [RRZ04] (see sec.2.2) at te top let; movin rit canes its sape to sow 207yoneda February 2, 20 7:26

6 6 te internal view o its natural transormation, and movin down canes its notation to te one in sec.4: F Φ F X F F F F Φ F F F X(F ) F Φ F X() F F X(F ) Φ F X F Φ( F ) Φ( F ). Φ() (, ) R id (, ) R (,) R (, ) R (, ) R id id id ; R 3 Interpretin diarams Y0, Y, and Y2 My avorite diaram or rememberin te proo o (one o te orms o) te Yoneda Lemma is tis one ( diaram Y0 ): R (, ) (, ) (, R ) (, ) (, (, )) (, ) It is made o objects in dierent cateories, 6 morpisms, two unctors, two bijections, and a middle arrow tat perorms some substitutions on te irst 207yoneda February 2, 20 7:26

7 7 bijection to obtain te second one. Let s name (or number ) all o tem: O O F 2 O 3 O 7 O m 2 m S O 4 O F 2 5 O 6 2 m 3 O 9 m 4 O 0 m 6 m 5 O m e existence o a morpism O O3 tells us tat O and O 3 belon to te same cateory; as O = let s call tat cateory. Similarly, O 2 = O 5 =, so O 2 and O 5 belon to a cateory tat we will call. O 4 and O 6 belon to te same cateory, and O 4 =, wic is an object o Set, so O 4 and O 6 are objects o Set. Similarly, O 7 = O 9 = (, ), so O 7, O, O 9, O 0, O all belon to te same cateory. e unctor F = R oes rom to and te unctor F 2, tat will turn out to be (, ), oes rom to Set. (, ) is a sortand or Hom (, ); te two objects and ave to belon to te same cateory so is an object o. (, ) is a sortand or te unctor Hom (, ), wic oes rom to Set (obs: as to be locally small). (, ) is an object o te cateory o unctors rom to Set, and O 7 to O, so: : : : Set (, ) : Set : R : (, ) : Set (, ) : Set R : (, R ) : Set (, (, )) : Set (, R ) is a sortand or Hom (, R ) : Set, (, ) or Hom (, ) : Set, and (, (, )) or Hom Set (, Hom((, ))). O 7 to O are all unctors rom to Set and so objects o te cateory Set, and te morpisms m 3, m 4, m 5, m 6, are natural transormations; m 5 is a natural isomorpism. I we indicate in te diaram tat O 7 to O are unctors and m 3 to m 6 are 207yoneda February 2, 20 7:26

8 Ns, we et: O O F 2 O 3 F 4 F 4 m S O 4 O F 2 5 O 6 2 F 5 F 6 2 m F 7 Warnin: te bijection is between m and, not between F and, even tou we draw it vertically; similarly, te bijection 2 is between m 2 and 2. e reason or drawin te diaram in tis way instead o makin O and O 2 switc places wit one anoter and doin te same wit O 4 and O 5 will be explained later. e arrow S is a substitution tat produces 2 rom. It s better to write it in a notation or simultaneous substitutions, not in λ-calculus notation: R := (, ) S = := Set := Now tat we ave typed most objects in diaram Y0 let s o back to te oriinal notation, and ive names to some arrows. is is diaram Y: R R S (, ) (, ) 2 2 (, ) (, R ) (, ) (, (, )) I I (, ) e next step is to deine precisely ow te our unctors work. We can do tat 207yoneda February 2, 20 7:26

9 9 by drawin internal views: (, ) k (, ) λ.(;k) (, R )) k (, R )) λ.( ;Rk) (, ) Set (,R ) Set (, ) (, ) λ.(;) (, (, )) (, (, )) λ.λe.( (e);) (, ) Set (,(, )) Set e actions o te unctors (, ), (, ), and (, R ) can be inerred by term inerence or by lookin at te diarams below: ;k ;k R ;Rk k k k Rk R and te action o (, (, )) is a variant o (, ). We et: (, ) 0 = λ. Hom (, ) (, ) = λk.λ.(; k) (, ) 0 = λ. Hom (, ) (, ) = λ.λ.(; ) (, R ) 0 = λ. Hom (, R ) (, R ) = λk.λ.( ; Rk) (, (, )) 0 = λ. Hom Set (, Hom (, )) (, (, )) = λk.λ.( ; Rk) We can do te same or te natural transormations. (, ) λ.(;r) (, R ) (, ) (, R ) (, ) λ.λe.(;) (, (, )) (, ) (, (, )) (, λ.(;) ) (, ) (, ) (, ) (, (, λ. )) (e) (, ) (, (, )) (, ) I (, ) λ.λe. (, (, )) (, ) I (, (, )) 207yoneda February 2, 20 7:26

10 0 We et: = λ.λ.(; R) = λ.λ.λe.(; ) = λ.λ.(; ) I = λ.λ. (e) I = λ.λ.λe. nd we can also do te same or te bijections. : R :=λ.λ.(;r) := (id ) : (, ) (, R ) : (, ) :=λ.λ.(;(, )()) (?) := (id ) : (, ) (, (, )) so: = λ.λ.λ.(; R) = λ. (id ) 2 = λ.λ.λ.(; (, )()) (?) 2 = λ. (id ). Note tat we used only type inerence and term inerence wic is not little, but most books and articles on pretend tat simple type inerences and term inerences like tese are obvious and now ave te types and te terms or everytin in diaram Y. Let s call te diaram below diaram 207yoneda February 2, 20 7:26

11 Y2 ; it is Y plus lots o inormation. R R S (, ) (, ) 2 2 (, ) (, R ) (, ) (, (, )) I I (, ) : : : Set (, ) : Set : R : (, ) : Set (, ) : Set R : (, R ) : Set (, (, )) : Set (, ) 0 = λ. Hom (, ) (, ) = λk.λ.(; k) (, ) 0 = λ. Hom (, ) (, ) = λ.λ.(; ) (, R ) 0 = λ. Hom (, R ) (, R ) = λk.λ.( ; Rk) (, (, )) 0 = λ. Hom Set (, Hom (, )) (, (, )) = λk.λ.( ; Rk) = λ.λ.(; R) = λ.λ.λe.(; ) = λ.λ.(; ) I = λ.λ. (e) I = λ.λ.λe. S = [ R:=(, ) ] :=Set := = λ.λ.λ.(; R) = λ. (id ) 2 = λ.λ.λ.(; (, )()) (?) 2 = λ. (id ) 207yoneda February 2, 20 7:26

12 2 4 is really a bijection In tis diaram, tat is just a part o diaram Y wit te bijection made more explicit, (, ) := ():= λ.λ.(;r) it is easy to see tat ( ()) = : R := ( ):= (id ) (, R ) ( ()) = (λ.λ.(; R)) = (λ.λ.(; R))(id ) = (λ.(; R))(id ) = ; R(id ) = ; id R = Let s try to calculate ( ( )): ( ( )) = ( (id )) = λ.λ.( (id ); R) is is not necessarily equal to... but note tat i is a natural transormation ten its naturality condition means tat or every k : tis square commutes, k (, ) (, R ) (, )k (, ) (, R ) (, ) (, R ) (,R )k (λk.λ.( ;Rk))k ( ); Rk ; k (; k) (λk.λ.(;k))k i.e., ( ); Rk = (; k); tis diaram elps understandin te types: r R R R k R Rk (;k) 207yoneda February 2, 20 7:26

13 3 I we replace k : by : and by id we et: ] (( ); Rk = (; k)) [ := := k:= :=id wic lets us continue te calculation o ( ( )): = ( id ; R = (id ; )) ( ( )) = ( (id )) = λ.λ.( (id ); R) = λ.λ.( (id ; )) = λ.λ. tis means tat or all and we ave ( ( )) = (λ.λ. ) = (λ. ) = so by η-reduction ( ( )) = and ( ( )) =. Note tat te proo o id ; R = can be represented as a diaram: (, ) (, R) id id (, ) (,R ) (, ) (, R ) ( id ); R (, ) (, R ) 5 Makin te bijections more explicit Let s introduce a new diaram tat stresses te bijections and names a ew bijections tat were unnamed beore. is is diaram Y3: : : R : (, ) (, R ) S 4 4 : (, ) : (, ) (, (, )) : (, ) (, ) 207yoneda February 2, 20 7:26

14 4 e statement o te Yoneda Lemma is just tis: 3 is a bijection. I we build 4 and 5, deine 3 as and simpliy te λ-terms we obtain tat 3 is just tis: : :=λ.(;) :=( )(id ) : (, ) (, ) direct proo tat 3 and 3 are inverses to one anoter requires naturality like we did in section 4 (trust me!), and less direct proo can be structured like tis: is a bijection implies tat 2 is a bijection, tat implies tat 3 is a bijection. 6 stroner Yoneda Lemma I we don t replace te unctor R by (, ) in Y0 and we make := Set and := we can build tis diaram ere ( diaram Y4 ), p R R (, ) (, R ) R tat yields a bijection between points o R and natural transormations rom (, ) to R ( diaram Y5 ): p R p : R : (, ) (, R ) : (, ) R is bijection eels muc more abstract tan te one tat we were lookin at beore. 207yoneda February 2, 20 7:26

15 5 7 Representable unctors Universal elements and universal arrows We say tat an element p R is a universal element wen te natural transormation associated to it by diaram Y4 is a natural isomorpism, i.e., wen or every te map = λ.rp is an iso: :=λ.λ.rp p R p:= id : (, ) R universal arrow is an arrow : R suc tat te associated (= λ.λ.(; R)) is a natural isomorpism: 9 djunctions (, ) := λ.λ.(;r) R := (id ) (, R ) t te end o sec. we presented a convention or namin te components o an adjunction and drawin its internal view, but we didn t include units or counits. For any te unit map η, deined like tis, id L L L L,L,L R L R RL η := (id L ) induces a map ( (η ; R)) : (L, ) (, R) tat is equal to te bijec- 207yoneda February 2, 20 7:26

16 6 tion : (L, ) (, R): η L RL R L R R R η ;R te map ( (η ; R)) is natural in, and we can see (I m omittin te details now) tat it induces a natural transormation : (L, ) (, R ): η L R RL (L, ) (, R ) We are now in a situation similar to diaram Y0 we can see tat any natural transormation : (L, ) (, R ) yields a map η : RL (tat is not necessarily te unit o te adjuction, o course). Now make te cateory be Set, and make := and := L = L. en RL = RL = R, and we et tese diarams: p R R (, ) (, R ) p R p : R : (, ) (, R ) R : (, ) R We ave a bijection between R = RL and te set o natural transormations rom (, ) to R, but we also ave more: wen p : RL = R is a unit map o te adjunction ten te correspondin : (, ) R is a natural isomorpism, so tis unctor R is representable and represented by, te map p : R is a universal arrow, p R is a universal element. Most (or all?) 207yoneda February 2, 20 7:26

17 7 items in Examples 2..5 in pp.5 53 o [Rie6] are applications o tis idea usin adjunctions o te orm F U e.., in item (ii) te unctor F : Set Group takes eac set to te ree roup F avin as its set o enerators. (o do: debu te ideas above!) 0 wo contravariant Yoneda Lemmas Let s introduce some notations or dealin wit opposite cateories. I and are objects o ten op and op are objects o op ; a morpism : in is written as op : op op wen rearded as a morpism in op. Lookin at om-sets, we ave tat Hom (, ) i op Hom op( op, op ), and in te sortand notation tis means tat (, ) and ( op, op ) are equal except or te op s in te elements o ( op, op ). Let G : op D be a contravariant unctor. We will write its action on objects as op GG, and te internal view o G is: op op op G G G op G D I we take Diaram Y0 and replace te cateory by op we et tis; note tat R : op : op R R ( op, ) (, R ) op (, ) op ( op, op ) ( op, ) (, ( op, )) ( op, ) 207yoneda February 2, 20 7:26

18 We can simpliy tis a bit, rewritin it as: op (, ) R R (, R ) op (,) (, ) (, ) (, (, )) (, ) I we replace by Set and by and complete te trianle at te lower let we et a sinle diaram tat states te two contravariant Yoneda Lemmas: op R R (, ) (, R ) R op (,) (, ) (, ) (, (, )) (, ) e diarams tat elp us understand ow te unctors and natural transormations above work are: op op op R R R op (, ) op op op (, ) (, ) (, ) (, R ) R (, ) (, (, )) (, ) 207yoneda February 2, 20 7:26

19 9 e statements o tese contravariant Yoneda Lemmas are: := λ.λ.(r)(p) p : R p:= id : (, ) R := λ.λ.(;) : := id : (, ) (, ) Note tat te action o te (contravariant) unctor (, ) on objects can be written op (, ); sometimes by abuse o lanuae we will denote te wole unctor (, ) by op (, ), and, similarly, denote te covariant unctor (, ) used in sec.3 by (, ). e Yoneda Embeddins Our two less abstract Yoneda lemmas can be drawn like tis: (, ) (, ) (, ) (, ) ey are usually presented at a slitly ier level, as: op (, ) op op op (, ) op (, ) (, ) op y Set y Set op e Yoneda Lemma says tat te unctors op (, ) and (, ) wose sort names are y and y are ull and aitul. ese unctors are usually called te Yoneda Embeddins I we expand te (, ) and te (, ) in op (, ) and (, ) we et op ( (, )) and ( op (, )), and tis makes te actions o y and y on objects very easy to understand. trick to iure out ow y and y act on morpisms is to draw te internal views o te natural transormations and at te rit o te diaram, and rewrite y and y 207yoneda February 2, 20 7:26

20 20 in several equivalent notations: op (, ) op op op (, ) op y:= := (, ):= λd.(;):= λd.λ.(;) (, D) (, D) yd:= D:= (,D):= λ.(;)= (;) ; op y Set (, ) (, ) y := := (,):= λ.(;):= λ.λ.(;) (, ) (, ) y := := (,):= λ.(;)= (;) ; y Set op 2 Readin Generic Fiures and ter Glueins Wen I irst tried to read Reyes, Reyes and Zolaari s [RRZ04] ( RRZ rom ere on) I ot very stuck, as I didn t ave any ood metods to work on its notation bit by bit to make it make sense to me... ake tis diaram rom pae o te book: We can type its entities: F F σ X σ. F σ X σ. F X(F ) X(F ) Set op Set ( ).:= X() σ σ. ( ). is a cateory F, F : F F X : op Set X(F ), X(F ) Set σ X(F ) σ. X(F ) σ. = X()(σ) In sec. we said tat we would sometimes write or or Hom(, ); we can do sometin similar or. In RRZ F σ X means σ X(F ), so we will interpret F X as X(F ) and F σ X as σ : F X. 207yoneda February 2, 20 7:26

21 2 We can make te examples in RRZ more elementary i we work wit inite matematical objects built rom inteers, pairs, and sets, as done in [Oc7] (sec.2 and onwards). Let M be te directed (multi-)rap wit labeled arrows ( DGL ) below: M = {7,, 9}, (7,7,77), (7,,7), (7,,70), (,9,9), (9,9,99) We can set to tis cateory (te identity arrows are omitted), V to deine iures made o vertices and arrows. is unctor M : op Set s t M() M(s) M(t) M(V ) {77, 7, 70, 9, 99} {7,, 9} is te DGL M above. I am not sure wat tis notation means wen it appears in RRZ: V X a, b, c, d, e It may be eiter a, b, c, d, e : V X or (V X) = {a, b, c, d, e}... anyway, in M we ave: V M 7,, 9 M 77, 7, 70, 9, 99 nd tis is a cane o iure: 70 M s V 70.s =7 M() M(s) M(V ) M(s) 70 7 Set op Set 207yoneda February 2, 20 7:26

22 22 2. Morpisms o -sets Morpisms o -sets Φ : X Y (see p. o RZZ) is a natural transormation rom X to Y, were bot X and Y are -sets, i.e., X, Y : op Set. e iure at let below appears in p. o RRZ except or te last line wit te te cateory annotations; at te rit o it is an internal view, in RRZ s notation, o te natural transormation Φ: F σ Φ X Y σ. F F F Φ F X(F ) Y (F ) X() X(F Φ F ) Y (F ) Y () σ σ. Φ(σ) Φ(σ). Φ(σ.) Set op Set op X Φ Y Here is a type inerence or te subexpressions o te naturality condition Φ(σ). = Φ(σ.): Φ }{{} :X Y ( σ }{{} X(F ) ) }{{} Y (). }{{} :F F }{{} :Y (F ) Y (F ) } {{ } Y (F ) = }{{} Φ ( }{{} σ. ) }{{} :X Y X(F ) :F F }{{} X(F ) } {{ } Y (F ) One diiculty in translatin Φ(σ). = Φ(σ.) to standard notation is tat te aruments to te dot operation are in te wron order. I we rewrite σ. as (.)(σ) ten te naturality condition becomes (.)(Φ(σ)) = Φ((.)(σ)), i.e., (.) Φ = Φ (.), and te easiest way (or me) to understand te types is to write irst : F F and Φ : X Y and ten all te rest in te diaram below: Φ F F F (X) F (Y ) σ Φ(σ) F X() F Φ F (X) F (Y ) X Φ Y Y () σ. Φ(σ). Φ(σ.) and (.)(Φ(σ)) = Φ((.)(σ)) becomes Y ()(Φ F (σ)) = Φ F (X()(σ)), and (.) Φ = Φ (.) becomes Y () Φ F = Φ F X(). 2.2 e Yoneda Lemma in RRZ e Yoneda Lemma appears in paes and aain at paes o te book. Let s examine te enlared versions drawn wit internal views o some o te iures used in te proo. Our enlared versions will be called diarams YR, YR2, and YR3. Important: we will make one cane in RRZ s notation were te book writes F we will write (, F ), and were it writes we will write (, ); we 207yoneda February 2, 20 7:26

23 23 saw in sec. tat te action o te natural transormation (, ) (a.k.a. ) is essentially ( ). is ( YR ) is rom p.23: F (, F ) 3 (, F ) F (,) Set op Set op is ( YR2 ) is rom p.29: F F (F,) =( ) (F, F ) (F, F ) (,F ) =( ) (F,) =( ) (,F ) =( ) (F, F ) (F, F ) (, F ) (,) (, F ) ( ) ( ) F F F (, F ) Φ X F F Φ F (F, F ) X(F ) (,F ) =( ) Φ F X() (F, F ) X(F ) F Φ F ( F ) Φ F ( F ). Φ F () Set op Set op (, F ) Φ X is ( YR3 ) is also rom p.29: F Φ (, F ) X F F F Φ F (F, F ) X(F ) (,F ) =( ) Φ F X() (F, F ) X(F ) Φ F () Φ F (). Φ F ( ) Set op Set op (, F ) Φ X (YR is used to prove tat (, ) is morpism o -sets) (YR2 is used to prove tat Φ F () can be calculated as Φ F ( F ). = X()(Φ F ( F ))) (YR3 is used to prove tat (???)) (o do: debu tis, compare wit sections 4 and 0...) 3 Readin MacLane s WM MacLane (in [Mac9], section III.2, paes 59 62) starts by ixin a unctor S : D and sowin tat or any pair r, u : c Sr, tat we draw like tis, u u r S Sr D S 207yoneda February 2, 20 7:26

24 24 any coice o an object d D induces a map ϕ d : D(r, d) (c, Sd), c u r S Sr d S Sd S ϕ d S u D(r, d) ϕ d (c, Sd) D(r, d) ϕ d (c, Sd) It turns out tat D(r, ) and (c, S ) are unctors, d D(r, d) d d D(r, d ) (D(r, ))():= λ.( ) d d (c, Sd) d (c, Sd ) ((s,s ))():= λk.(s k) D D(r, ) Set D (c,s ) Set and ϕ : D(r, ) (c, S ) is a natural transormation between tem: d d D(r, d) ϕ d (c, Sd) D(r,) ϕ d D(r, d ) (c, Sd ) (c,s) S u S (S u) S( ) u D(r, ) ϕ (c, S ) However, MacLane introduces, rit in te beinnin, a concept tat I eel tat sould better be let to a second moment... (o be continued!!!) 4 Readin Emily Riel s in ontext e Yoneda Lemma is proved in [Rie6] in section 2.2, paes Here s Riel s ormula rom paes 57 5 in te sape o our diaram 207yoneda February 2, 20 7:26

25 25 Y4: x c F F c (c, ) Set(, F ) α:= Ψ(x):= λd.λ.f (x) x F c a:= Φ(α):= α c( c) α Hom((c, ), F ) α F e Yoneda embeddins appear as orollary 2.2. in [Rie6], in p.60. Se makes tis diaram: y Set op c (, c) d (, d) op y Set c (c, ) d (d, ) Se uses te y wit two dierent meanins, one in te let al and anoter in te rit al o er diaram. I we modiy er diaram to add some o te inormation rom our diarams in sec. and cane er letters just a little bit, we et tis: y Set op b (, b) c (, c) := y := (,):= λa.λ.( )= λa.( ) op y Set b (b, ) c (c, ) := y:= (, ):= λd.λ.( )= λd.( ) 5 Readin wodey s ateory eory (o do: sow internal views etc o sections.2.4 o [wo06] (pp.60 67)) 6 Related projects ese notes are related to tree, aem, tins: a worksop called Loic or ildren, a series o papers on Planar Heytin lebras or ildren (tese 207yoneda February 2, 20 7:26

26 REFERENES 26 notes prepare te round or te material on preseaves, seaves and eometric morpisms tat will be presented in te tird paper), and a very introductory course on λ-calculus, loics and ateories tat I am ivin every semester in my university since 206. Links: ttp://an.twu.net/loic-or-cildren-20.tml ttp://an.twu.net/mat-b.tml#zas-or-cildren-2 ttp://an.twu.net/mat-b.tml#lclt Reerences [wo06] S. wodey. ateory eory. Oxord University Press, [LS97] [Mac9] [Oc3] [Oc7] [Rie6] [RRZ04] W. Lawvere and S. Scanuel. onceptual Matematics: irst introduction to cateories. ambride, 997. S. MacLane. ateories or te Workin Matematician, 2nd ed. Spriner, 99. E. Ocs. Internal Diarams and rcetypal Reasonin in ateory eory. In: Loica Universalis 7.3 (Sept. 203). ttp://an.twu. net/mat-b.tml#idarct, pp E. Ocs. Planar Heytin lebras or ildren. vailable at ttp: //an.twu.net/mat-b.tml#zas-or-cildren E. Riel. ateory eory in ontext. ttp:// ~eriel/context.pd. Dover, 206. M. P. Reyes, G. E. Reyes, and H. Zolaari. Generic Fiures and eir Glueins. ttps://marieetonzalo.iles.wordpress.com/ 2004/06/eneric-iures.pd. Polimetrica, yoneda February 2, 20 7:26