Integral Categories and Calculus Categories

Transcription

1 Intgral Catgoris and Calculus Catgoris Robin Cocktt 1 and Jan-Simon Lmay 2 1 Dpt. o Computr Scinc, Univrsity o Calgary, Calgary, AB, Canada robin@ucalgary.ca 2 Dpt. o Mathmatics and Statistics, Univrsity o Calgary, Calgary, AB, Canada jansimon.lmay@ucalgary.ca Abstract Dirntial catgoris ar now an stablishd abstract stting or dirntiation. Th papr prsnts th paralll dvlopmnt or intgration by axiomatizing an intgral transormation, s A : A, in a symmtric monoidal catgory with a coalgbra modality. Whn intgration is combind with dirntiation, th two undamntal thorms o calculus ar xpctd to hold (in a suitabl sns): a dirntial catgory with intgration which satisis ths two thorm is calld a calculus catgory. Modiying an approach to antidrivativs by T. Ehrhard, it is shown how xampls o calculus catgoris aris as dirntial catgoris with antidrivativs in this nw sns. Having antidrivativs amounts to dmanding that a crtain natural transormation, K :, is invrtibl. W obsrv that a dirntial catgory having antidrivativs, in this sns, is always a calculus catgory and w provid xampls o such catgoris ACM Subjct Classiication F.3.3 Studis o Program Constructs, F.4.1 Mathmatical Logic Kywords and phrass Dirntial Catgoris, Intgral Catgoris, Calculus Catgoris Digital Objct Idntiir /LIPIcs.CSL Introduction Th two undamntal thorms o calculus rlat th two most important oprations o calculus: dirntiation and intgration. Th irst thorm stats that th drivativ o th intgral o a ral unction is th original unction: d( t (x) dx) a dt (x) (x). Whil th scond stats that th intgral o th drivativ o a ral unction on a closd intrval [a, b] is qual to th dirnc o valuatd at th nd points: b d(t) a dt (x) dt (b) (a). Thy ar calld undamntal" thorms bcaus thy ar absolutly undamntal to th dvlopmnt o classical calculus. Sinc th turn o th 21 st cntury, thr has bn signiicant progrss in th abstract undrstanding o dirntiation with th study o dirntial catgoris. Th abstract ormulation o intgration, on th othr hand, has not rcivd th sam lvl o attntion. Nonthlss, on might xpct that, whn suitably adjoind to th ormulation o dirntiation, a commnsurat abstract orm or intgration should ncompass ths undamntal thorms. Th purpos o this xtndd abstract is to xplor th xtnt to which this xpctation is ralizd. This work was partially supportd by NSERC, Canada. Robin Cocktt and Jan-Simon Lmay; licnsd undr Crativ Commons Licns CC-BY 26th EACSL Annual Conrnc on Computr Scinc Logic (CSL 2017). Editors: Valntin Goranko and Mads Dam; Articl No. 20; pp. 20:1 20:17 Libniz Intrnational Procdings in Inormatics Schloss Dagstuhl Libniz-Zntrum ür Inormatik, Dagstuhl Publishing, Grmany

2 20:2 Intgral Catgoris and Calculus Catgoris In th arly 2000 s, T. Ehrhard and L. Rgnir introducd th dirntial λ-calculus [12] and dirntial proo nts [13], which ormalizd dirntiation in linar logic. A w yars latr, R. Blut, R. Cocktt and R. Sly introducd dirntial catgoris [7], which wr th appropriat catgorical structur or modlling Ehrhard and Rgnir s dirntial linar logic. Dirntial catgoris now hav a rich litratur o thir own [4, 3, 2, 6, 14, 10, 9] and thr ar many xampls which hav bn xtnsivly studid [7, 11, 5]. Howvr, as mntiond abov, littl attntion has bn givn to abstracting intgration. In 2014, T. Ehrhard obsrvd that in crtain -autonomous catgoris which had th appropriat structur to b a dirntial catgory, it was possibl with on additional assumption to produc antidrivativs [11]. Th additional assumption was that a crtain natural transormation which h calld J constructd rom th driving transormation was a natural isomorphism. With this assumption, Ehrhard constructd an intgral transormation with an invrs bhaviour to th driving transormation, in th sns that h gav ncssary and suicint conditions or a map to satisy th irst undamntal thorm o calculus by proving Poincaré s Lmma. Furthrmor, whn th driving transormation satisid an xtra non-quational condition, which h calld th Taylor Proprty", h thn showd that vry dirntiabl unction satisid th scond undamntal thorm o calculus. Whil much o th inspiration or our approach to intgration drivs rom ths obsrvations, Ehrhard mad no attmpt to axiomatiz intgration sparatly rom dirntiation. Hr w introduc (tnsor) intgral catgory as a notion which stands on its own (i.. in th absnc o dirntiation). Th inspiration or this indpndnt axiomatization o intgral catgoris coms rom th much oldr notion o a Rota-Baxtr algbra [1, 20, 15], th classical algbraic abstraction o intgration. Brily, or a commutativ ring R and λ R, a Rota-Baxtr algbra o wight λ is an R-algbra A with an R-linar morphism P : A A which satisis th Rota-Baxtr rul: P (a)p (b) P (ap (b)) + P (P (a)b) + λp (ab) or all a, b A. Th map P is calld a Rota-Baxtr oprator o wight λ. A particular xampl o a Rota-Baxtr algbra o wight zro is th R-algbra o ral continuous unctions Cont(R), whr th Rota-Baxtr oprator P : Cont(R) Cont(R) is dind as th intgral o th unction cntrd at zro: P ()(x) x (t) dt. Th Rota-Baxtr rul or 0 this xampl is th xprssion o th intgration by parts rul without th us o drivativs: x 0 (t) dt x 0 g(t) dt x 0 (t) ( t 0 g(u) du) dt + x 0 ( t (u) du) g(t) dt (s [15] or mor 0 dtails). This motivats th Rota-Baxtr rul as an axiom o intgration. Whn dirntiation and intgration ar combind into what w call hr a calculus catgory, w dmand that th two undamntal thorms hold. Th scond undamntal thorm is assumd to hold vrbatim. Howvr, th irst undamntal thorm, as abov, has to b intrprtd as bing on maps rathr than objcts and, undr this intrprtation, bcoms th Poincaré proprty, a conditional proprty which provids ncssary and suicint conditions or a map to b th dirntial o its intgral. Th nam o th condition coms rom th Poincaré Lmma rom cohomology [22] and dirntial topology [8], which stats an analogous rsult o giving critria or a map to b an antidrivativ. To obtain th notion o intgration as an antidrivativ, w insist that a slightly dirnt natural transormation, which w call K, should b invrtibl. W show this is quivalnt to rquiring both that Ehrhard s transormation J is invrtibl and that th Taylor Proprty which Ehrhard had suggstd was dsirabl holds. This improvmnt is asily undrstimatd: th Taylor Proprty is a conditional rquirmnt, rplacing a conditional rquirmnt by a purly quational rquirmnt is always, mathmatically, a signiicant stp. Dmanding that K is invrtibl not only producs an intgral transormation, but also scurs th irst and scond undamntal thorm o calculus. Invrting only Ehrhard s transormation, J,

3 J. R. B. Cocktt and J.-S. Lmay 20:3 dos not by itsl vn produc an intgral transormation; th Taylor Proprty is rquird, in addition to th invrtibility o J, to scur an intgral transormation. Th act that, whn K is invrtibl, J is invrtibl is usul particularly in th proo o th Poincaré s lmma. Thus, it is important to obsrv that, th antidrivativ producd by th invrs o K is prcisly th sam as th antidrivativ producd by th invrs o J whn K is alrady invrtibl. Finally, th notion o a dirntial catgory with anti-drivativs, givn by rquiring K to b invrtibl, provids a plntiul supply o calculus catgoris as w xplain. Bor bginning, w should mntion th convntions that w us in th papr. First o, w will us diagrammatic ordr or composition. Explicitly, this mans that th composit map g is th map which irst dos thn g. Scondly, to simpliy working in symmtric monoidal catgoris, w will allow ourslvs to work in strict symmtric monoidal catgoris and so will gnrally supprss th associator and unitor isomorphisms. For a symmtric monoidal catgory w will us or th tnsor product, I or th monoidal unit, and σ : A B B A or th symmtry isomorphism. Full dtaild proos o all th rsults in this xtndd abstract can b ound in th scond author s mastrs thsis [18]. 2 Coalgbra Modalitis Tnsor intgral and dirntial catgoris ar structurs ovr additiv symmtric monoidal catgoris with a coalgbra modality. W bgin by rcalling th componnts o this structur starting with th notion o an additiv catgory. Hr w man additiv in th sns o bing commutativ monoid nrichd. Thus, w do not assum ngativs nor do w assum biproducts (this dirs rom th usag in [19] or xampl). This allows many important xampls such as th catgory o sts and rlation or th catgory o moduls o a commutativ rig 1. Dinition 1. An additiv catgory is a commutativ monoid nrichd catgory, that is a catgory in which ach hom-st is a commutativ monoid with addition opration + and zro 0 and in which composition prsrvs addition that is: [Add.1] k( +g) k +kg and 0 0; [Add.2] ( +g)hh+gh and 0 0. An additiv symmtric monoidal catgory is an additiv catgory with a tnsor product which is compatibl with th additiv structur in th sns that: [Add.1] ( +g) h h+g h and 0 h0; [Add.2] k ( +g)k +k g and h 00. In any additiv catgory thir is a notion o scalar multiplication o maps by th natural numbrs N. Th scalar multiplication o a map : A B by n N, is th map n : A B dind by summing n copis o togthr. I n 0, thn 0 is simply th zro map rom A to B. Furthrmor, or additiv symmtric monoidal catgoris, on thn has that (n ) g n ( g) (n g). Dinition 2. A coalgbra modality [7, 3] on a symmtric monoidal catgory is a quintupl (!, δ,,, ) consisting o a comonad (!, δ, ), a natural transormation with 1 Rigs ar also known as a smirings: thy ar rings without ngativs. C S L

4 20:4 Intgral Catgoris and Calculus Catgoris componnts A :, and a natural transormation with componnts A : I such that or ach objct A: (i) (, A, A ) is a cocommutativ comonoid, that is, th ollowing diagrams commut: σ (ii) δ A prsrvs th comultiplication, that is, th ollowing diagram commuts: δ! δ δ!! Whn combind with th additiv structur, this nsurs that is a coalgbra in th classical algbraic sns. Furthrmor, on can prov that δ so δ is actually a comonoid homomorphism. Th coklisli maps or th comonad ar important: ths maps ar o th orm : B: amongst ths ar th linar maps g : B whr g : A B. Not that w do not assum that th coalgbra modality,!, is a monoidal unctor: to do so would put us in th ralm o Sly catgoris [3, 14] which is mor than w rquir or this basic thory. 3 Intgral Catgoris Intgral catgoris ar th intgral analogu o dirntial catgoris, thus, th main ingrdint o an intgral catgory is an intgral transormation, s A : A, a natural transormation opposit in orintation to a driving transormation which must satisy just thr quations: Dinition 3. An additiv symmtric monoidal catgory with a coalgbra modality is an intgral catgory i thr is a natural transormation s A : A, calld th intgral transormation, satisying th ollowing quations: [s.1] Constants Rul: s( 1) [s.2] Rota-Baxtr Rul: (s s) s( 1)(s 1 1) + s( 1)(1 σ)(1 1 s) [s.3] Intrchang Rul: s(s 1) s(s 1)(1 σ) Th intgral o a map : A B is dind as th composition S[] : s A : B. This should b thought o as th classical intgral o valuatd rom 0 to x as a unction o x: S[](x) : x (t) dt. To intrprt this as S[] on must rgard as bing a unction o 0 two variabls t and dt, which is linar in dt. Classically, is rgardd as a unction o on (on dimnsional) variabl, t, and to obtain th intrprtation as a unction o two argumnts on simply multiplis by th variabl dt. This allows a simpl intrprtation o th intgral notation or on dimnsional unctions: it lavs opn th intrprtation or multidimnsional unctions an issu to which w shall rturn. Th additiv structur o th catgory nsurs th intgral o a sum o maps is qual to th sum o th intgral o ach map, that is, S[ + g] S[] + S[g] and S[0] 0. Th irst axiom [s.1] stats that th intgral o a constant map is a linar map (in th sns discussd abov). Th scond axiom [s.2] is th Rota-Baxtr rul [15], which is an xprssion

5 J. R. B. Cocktt and J.-S. Lmay 20:5 o intgration by parts using only intgrals. In classical calculus notation, th Rota-Baxtr rul is xprssd as: x 0 (t) dt x 0 g(t) dt x 0 (t) ( t 0 g(u) du) dt + x 0 ( t (u) du) g(t) dt. 0 Th third axiom [s.3] nsurs th indpndnc o th ordr o intgration th intrchang law that is intgrating with rspct to u thn t is th sam as intgrating with rspct to t thn u. It may b tmpting to think this is rlatd to Fubini s thorm. In act, it is not closly rlatd at all: w discuss this at th nd o this sction. [s.3] can b xprssd in classical notation as: x 0 ( t 0 (u) du dt) x 0 ( u (t) dt du). 0 Just lik dirntial catgoris, intgral catgoris hav a graphical calculus (s [21] or an introduction to th graphical calculus in monoidal catgoris and its variations). W rprsnt th intgral transormation in string diagrams as ollows (which should b rad rom top to bottom): s : A Th intgral axioms [s.1], [s.2] and [s.3] ar thn rprsntd in th graphical calculus as ollows (w omit writing th objcts at th nd o th wirs). [s.1] Constants Rul: [s.2] Rota-Baxtr Rul: + [s.3] Intrchang Rul: With th graphical calculus, w ar now in a position to xplor polynomial intgration. Prhaps th irst ormula larnt in irst yar calculus is x 0 xn dx 1 n+1 xn+1. Howvr this ormula cannot b xprssd in a gnral additiv catgory simply bcaus thr may not b ractions. That said, w will soon s that in vry intgral catgory thr is a notion o scalar multiplication by positiv rationals, that is, crtain hom-sts ar Q 0 -moduls, whr Q 0 is th rig o non-ngativ rationals. Th intgral o monomials idntity can b r-xprss as th rquirmnt that (n + 1) x 0 xn dx x n+1 and this idntity dos hold in any intgral catgory! To xprss this idntity in an intgral catgory, w will nd th n-old comultiplication n : n which is dind as n ( 1)( 1 1)...( 1 n 2 ). By convntion w st 0, 1 1 and 2. C S L

6 20:6 Intgral Catgoris and Calculus Catgoris Thorm 4. For vry n N, th intgral transormation satisis th polynomial idntity: [s. Poly] (n + 1) s( n 1)( n 1) n+1 ( n+1 ) n+1 (n + 1) n Proo. Th bauty o this proo is that it uss vry intgral transormation axiom. Th proo is much smoothr using th graphical calculus, which is quivalnt to proos don algbraically as shown in [16]. W will prov th quality or th intgral transormation by induction on n. For th bas cas o n 0, this quality holds dirctly by th constant rul [s.1]. Assum th induction hypothsis [s. Poly] holds or n, w now show it or n + 1: n+2 n+1 (n + 1) n ( ) ( ) (n + 1) n + n (n + 1) n + n n+1 + (n + 1) n+1 n+1 + (n + 1) 2 n n+1 + (n + 1) 2 n+1 + (n + 1) n+1 n An important consqunc o polynomial intgration is that crtain hom-sts ar Q 0 - moduls. In an additiv catgory, or vry objct A and or vry natural numbr n N, din th map n A : A A by summing n copis o th idntitis: n A n 1 A. W will now prov that in any intgral catgory that or vry objct A and n 2, th map n is invrtibl. Thorm 5. In an intgral catgory, or vry natural numbr n N, n 2, and vry objct A X, th map n : is an isomorphism. Rmark. Notic that th cas n 1 is also tru sinc th idntity map is an isomorphism.

7 J. R. B. Cocktt and J.-S. Lmay 20:7 Proo. W will simply din th invrs o n. For ach objct A and n 2, din n 1 :, as: n 1 δ As ( n 1 1)( n 1 1)(1 n 1 ), writtn in th graphical calculus as: δ n 1 n 1 This implis, in an intgral catgory, hom-sts with domain ar Q 0 -moduls. Th scalar multiplication o a map : B with a non-ngativ rational p q Q 0 is th map p q : B dind as p q q 1 (p ). Finally, w discuss th intrprtation o Fubini s thorm. Th thorm rquirs that th coalgbra modality is monoidal and, thus, that thr is a Sly isomorphism [2, 14]: χ :!B!(A B). Fubini s thorm concrns th doubl intgration o a unction o th orm :!(A B) A B C whos typ nsurs it is bilinar in th scond two occurrncs o A and B. Functions o this orm can b intgratd with rspct to ithr A or B, or both A and B: th lattr, th doubl intgral o, is obtaind as ollows:!(a B) χ 1!B s s A!B B 1 σ 1!B A B χ 1 1!(A B) A B C Fubini s thorm assrts that th ordr o intgration in this doubl intgral dos not mattr. At this lvl o gnrality this ordr indpndnc is an immdiat consqunc o th biunctoriality o _ _. 4 Calculus Catgoris In this sction w wish to put intgration togthr with dirntiation and to discuss how thy should intract. W start by brily rcalling th dinition o a dirntial catgory [7] bor introducing calculus catgoris whos structur is inducd by th undamntal thorms o calculus. Dinition 6. An additiv symmtric monoidal catgory with a coalgbra modality is a dirntial catgory i th coalgbra modality coms quippd with a driving transormation [7], that is, a natural transormation d with componnts d A : A, satisying th ollowing quations: [d.1] Constant Rul: d 0 [d.2] Libniz Rul: d ( 1)(1 σ)(d 1) + ( 1)(1 d) [d.3] Linar Rul: d ( 1)λ [d.4] Chain Rul: dδ ( 1)(δ 1 1)(1 d)d [d.5] Intrchang Rul: (d 1)d (1 σ)(d 1)d Th drivativ o a map : B is th composition D[] : d A : A B. Th irst axiom, [d.1], stats that th drivativ o a constant map is zro. Th scond axiom [d.2] is th Libniz rul or dirntiation also calld th product rul. Th third axiom [d.3] says that th drivativ o a linar map is a constant. Th ourth axiom [d.4] is th chain rul. Th last axiom [d.5] is th indpndnc o dirntiation or th intrchang law, which naivly stats that dirntiating with rspct to x thn y is th sam as dirntiation with rspct to y thn x. It should b notd that [d.5] was not a rquirmnt in [7] but C S L

8 20:8 Intgral Catgoris and Calculus Catgoris was latr addd to th dinition [3, 3] to nsur that th coklisli catgory o a dirntial catgory was a Cartsian dirntial catgory. As prviously statd, dirntial catgoris hav a graphical calculus. Th driving transormation is rprsntd as ollows: A d Th string diagram rprsntations o [d.1] to [d.5] ar as ollows: [d.1] Constant Rul: 0 [d.2] Libniz Rul: + [d.3] Linar Rul: [d.4] Chain Rul: δ δ [d.5] Intrchang Rul: W ar now rady to tackl th intraction btwn intgration and dirntiation. W start with th scond undamntal thorm o calculus and rturn to discuss th irst undamntal thorm o calculus: Dinition 7. Lt X b a dirntial catgory and an intgral catgory with driving transormation d and intgrating transormation s on th sam coalgbra modality (!, δ,,, ).

9 J. R. B. Cocktt and J.-S. Lmay 20:9 (i) d and s ar said to satisy th Scond Fundamntal Thorm o Calculus i: sd +!0 1, writtn in th graphical calculus as: +!0 (ii) d and s ar said to b compatibl i: dsd d, writtn in th graphical calculus as: (iii) d is said to b Taylor i or vry pair o maps, g : C B, such that (1 d) (1 d)g thn + (1!(0))g g + (1!(0)). Th irst part o th dinition xprsss th scond undamntal thorm o calculus. Compatibility is a wakr vrsion o th scond undamntal thorm. Th Taylor proprty (s [11]) is th proprty that i two maps hav th sam drivativ thn thy dir by constants. Thorm 8. For a driving transormation d and an intgral transormation s on th sam coalgbra modality, th ollowing ar quivalnt: (i) d and s satisy th Scond Fundamntal Thorm o Calculus; (ii) d and s ar compatibl and d is Taylor. Rmark. This is an xtnsion o Proposition 14 o [11] which provd (ii) (i) or Ehrhard s original intgral using J 1, howvr th notion o compatibility was not idntiid although it was usd in th proo. Proo. (i) (ii): Suppos d and s satisy th Scond Fundamntal Thorm o Calculus. For Taylor, suppos that (1 d) (1 d)g. Thn w hav th ollowing quality: +(1!0)g (1 s)(1 d) +(1!0) +(1!0)g (1 s)(1 d)g+(1!0) +(1!0)g g+(1!0) For compatibility, by naturality, w hav th ollowing quality: d dsd + d!(0) dsd + (!(0) 0)d dsd + 0 dsd. (i) (ii): Suppos d and s ar Compatibl and d is Taylor. Notic by Compatibility w hav: dsd d, and thn by Taylor (whr sd and g 1) w hav th ollowing quality: sd +!(0) 1 +!(0)sd. Howvr, using naturality, w hav: sd +!(0) 1 +!(0)sd 1 + s(!(0) 0)d Th intrprtation o th irst undamntal thorm o calculus, unlik th scond undamntal thorm, is as a proprty o a map: C S L

10 20:10 Intgral Catgoris and Calculus Catgoris Dinition 9. A map : C A B satisis th First Fundamntal Thorm (in th last two argumnts) i (1 (d A s A )), writtn in th graphical calculus as: Thus, i satisis th First Fundamntal thorm, it may b viwd as th dirntial o a map namly th dirntial o its intgral. Clarly not all maps will satisy th First Fundamntal thorm calculus, a ncssary condition is: Lmma 10. I a map, : C A B, satisis th First Fundamntal Thorm, thn: (1 1 σ)(1 d 1) (1 d 1). Proo. As (1 ds), th intrchang rul or th driving transormation [d.5] givs: (1 1 σ)(1 d 1) (1 1 σ)(1 d 1)(1 (ds)) (1 d 1)(1 (ds)) (1 d 1) W shall us th convrs o this lmma as an axiom and call it th Poincaré condition: Dinition 11. A dirntial catgory with an intgral transormation satisis th Poincaré condition i any map : C A B or which: (1 1 σ)(1 d 1) (1 d 1), satisis th First Fundamntal Thorm that is: (1 ds). Th Poincaré condition and Lmma 10 imply th ollowing quivalnc: Th Poincaré condition also implis compatibility o th driving transormation and intgrating transormation. Thorm 12. Th intgral and driving transormation ar compatibl in any dirntial catgory with an intgral transormation which satisis th Poincaré condition. Proo. By [d.5], th driving transormation d satisis th Poincaré pr-condition that (1 σ)(d 1)d (d 1)d. Thror, d satisis th First undamntal thorm o Calculus, which is simply th statmnt o compatibility: dsd d. Corollary 13. A driving and intgral transormation which satisy th Poincaré condition such that d is Taylor, satisis th Scond Fundamntal Thorm o Calculus. This suggsts th ollowing basic dinition: Dinition 14. A calculus catgory is a dirntial catgory and an intgral catgory on th sam coalgbra modality such that th driving transormation and th intgral transormation satisy th Scond Fundamntal Thorm o Calculus and th Poincaré condition.

11 J. R. B. Cocktt and J.-S. Lmay 20:11 5 Antidrivativs In this last sction, w xplor how on obtains a calculus catgory rom a dirntial catgory with antidrivativs. A dirntial catgory has antidrivativs whn a crtain natural transormation, K which is prsnt in all dirntial catgoris is a natural isomorphism. This is a strngthning o Ehrhard s original ida in [11], which rquird a dirnt natural transormation, J, to b a natural isomorphism. Invrting J by itsl dos not appar to giv vn an intgral catgory: to obtain an intgral catgory and th scond undamntal thorm o calculus Ehrhard also dmandd th Taylor proprty. Invrting K, as w shall s, gts all ths proprtis and, thus, a calculus catgory in on stp. In an additiv symmtric monoidal catgory with a coalgbra modality, th codriving transormation is th natural transormation d A : A(1 A ) : A (this calld th annihilation oprator in [14]). W rprsnt th codriving transormation as an upsid down driving transormation in th graphical calculus: d : Th codriving transormation would probably b on s irst attmpt at constructing an intgrating transormation in dirntial catgory. Th ollowing thorm indicats how clos th codriving transormation is to bing an intgrating transormation: Thorm 15. Th codriving transormation d satisis th ollowing proprtis: [cd.1] d ( 1) [cd.2] d ( 1) ( ) [cd.3] d ( 1) (1 d ) [cd.4] d ( 1)(1 σ) (d 1) [cd.5] 2 (d d ) d ( 1)(d 1 1) + d ( 1)(1 σ)(1 1 d ) [cd.6] d (δ 1) δd (1 ) [cd.7] d (d 1) d (d 1)(1 σ) Notic in particular [cd.1], [cd.5] and [cd.7]. I w lt s d, thn [cd.1] and [cd.7] ar prcisly th sam as [s.1] and [s.3]. Howvr, th codriving transormation ails to satisy [s.2], th Rota-Baxtr rul, sinc [cd.5] has an xtra actor o 2. O cours, i th dirntial catgory is in act nrichd ovr idmpotnt commutativ monoids, so that , th codriving transormation would b an intgral transormation: this happns, or xampl, in th catgory o sts and rlations. Anothr important proprty th codriving transormation satisis is its rlation with th driving transormation. Thorm 16. Th driving and codriving transormations satisy th ollowing quality: d A d A W A + 1 A whr W is th natural transormation with componnts W A (d A 1 A)(1 σ)(d A 1 A ). C S L

12 20:12 Intgral Catgoris and Calculus Catgoris Th notation W was introducd by Ehrhard s in [11] whr a proo can b ound. In th graphical calculus, th abov idntity is xprssd as ollows: + In a dirntial catgory thr ar two important natural transormations K and J dind by K A : d A d A +!0 : and J A : d A d A + 1 :, writtn in th graphical calculus as: K +!0 J + K and J satisy a long list o vry similar proprtis which dscrib thir intraction with th dirntial structur. W giv som o th mor important ons in th ollowing thorm: Thorm 17. K and J satisy th ollowing proprtis: [K.1] K!(0)!(0)!(0)K; [K.2] K ; [K.3] K ; [K.4] K ((dd ) 1) + (1 (dd )) + (!(0)!(0)); [K.5] Kδ δd (1 (dd ))d + δ!(!(0)); [K.6] (K 1)W W(K 1); [K.7] (K 1)dd dd (K 1). [J.1] J!(0)!(0)!(0)J; [J.2] J ; [J.3] J 2 ; [J.4] J (J 1) + (1 (dd )) ((dd ) 1) + (1 J) ; [J.5] Jδ δd (1 (dd ))d + δ; [J.6] (J 1)d dk; [J.7] d (J 1) Kd ; [J.8] (J 1)W W(J 1); [J.9] (J 1)dd dd (J 1). Rcall that Ehrhard s original ida was to obtain intgration by rquiring that J b a natural isomorphism. Howvr, Ehrhard s intgral transormation, using only that J is invrtibl, appars to ail th Rota-Baxtr rul [s.2]. This is why w hav strngthnd Ehrhard s approach by rquiring instad that K b a natural isomorphism. W obsrv: Thorm 18. For a dirntial catgory, K is a natural isomorphism i and only i J is a natural isomorphism and th driving transormation is Taylor.

13 J. R. B. Cocktt and J.-S. Lmay 20:13 Proo. W giv th dinitions o and J 1 : (i) (ii) I K is a natural isomorphism, thn: J 1 A th graphical calculus as: : δ A d (!( A) A )ρ, writtn in δ J 1 (ii) (i) I J is a natural isomorphism and th driving transormation is Taylor, thn A : d A (J 1 A 1 A)(J 1 A 1 A)d A +!0, whr 0 : A A, and writtn in th graphical calculus as: J 1 J 1 +!0 Dinition 19. A dirntial catgory has antidrivativs i K is a natural isomorphism. Equivalntly, o cours, a dirntial catgory has antidrivativs i J is a natural isomorphism and th driving transormation is Taylor. Whil our dinition o antidrivativs dirs only slightly rom Ehrhard s, [11], our dinition dos imply Ehrhard s dinition and, at th sam tim, scurs th proprty o bing an intgral catgory or which, as ar as w can s, invrting J is insuicint. Thorm 20. In a dirntial catgory with antidrivativs, and J 1 satisy th ollowing proprtis: [.1]!(0)!(0)!(0) ; [.2] ; [.3] ; [.4] ( ) + (!(0)) + (!(0) ) ( 1) + (1 ) + (!(0)!(0)); [.5] ( 1)W W( 1); [.6] ( 1)dd dd ( 1); [J 1.1] J 1!(0)!(0)!(0)J 1 ; [J 1.2] J 1 ; [J 1.3] 2 J 1 ; [J 1.4] (J 1 1)d d ; [J 1.5] d (J 1 1) d ; [J 1.6] (J 1 1)W W(J 1 1); [J 1.7] (J 1 1)dd dd (J 1 1); In particular, [J 1.5] will imply that th intgral transormation constructd using ithr or J 1 ar qual to on anothr. Finally, with ths proprtis o and J 1, w obtain th main rsult o this sction, namly that a dirntial catgory with antidrivativs is a calculus catgory: C S L

14 20:14 Intgral Catgoris and Calculus Catgoris Thorm 21. A dirntial catgory with antidrivativs is a calculus catgory with th intgral transormation dind by s A : A d A d A (J 1 A 1 A), xprssd in th graphical calculus as: s J 1 Proo. To prov th intgral transormation axioms and th scond undamntal thorm w us th orm o th intgrating transormation. Whil to prov th Poincaré condition w us Ehrhard s J 1. W will us th graphical calculus to hlp us. W irst show that our intgral transormation satisis [s.1] to [s.3]. [s.1]: Hr w us [cd.1] and [.2]: [s.2]: Hr w us [.4], [cd.3], [cd.4] and naturality o th codriving transormation: +!0 +!0 } {{ } 0 } {{ } 0 + +!0!0 + } {{ } 0 [s.3]: Hr w us [J 1.5] and [cd.7]: J 1 J 1 J 1 J 1 Nxt w show th scond undamntal thorm o calculus. Hr w us [.1]: +!0 +!0 K

15 J. R. B. Cocktt and J.-S. Lmay 20:15 Finally w prov th Poincaré Condition. Lt : C A B satisy th Poincaré pr-condition, that is, (1 1 σ)(1 d 1) (1 d 1). First notic that by Thorm 16 and th Poincaré pr-condition, satisis th ollowing idntity: + + J Thn using [J 1.4] and th abov idntity w gt th ollowing quality: J 1 J 1 J Which complts th proo that antidrivativs giv a calculus catgory. Th convrs o Thorm 21 is tru i th coalgbra is monoidal (in th sns xplaind at th nd o Sction 3 whn discussing Fubini s thorm) and th intgral transormation is compatibl with monoidal strngth, that is, a calculus catgory with a monoidal coalgbra modality and a monoidal intgral transormation is a dirntial catgory with antidrivativs. Mor dtails and a proo o this will b givn in a subsqunt papr. W ar now ar in a position to giv two xampls o dirntial catgoris which hav antidrivativs, and thror, two xampls o calculus catgoris: Exampl 22. Th catgory o sts and rlations, REL, is a dirntial catgory [7] with antidrivativs. Th symmtric monoidal strucur is givn by th Cartsian product o sts whil th additiv structur is givn by th union o sts. Th coalgbra modality is givn by th init bag/multist comonad (s [7] or mor dtails), whr or a st X,!X is th st o bags/multists o X. Th driving transormation d X :!X X!X is th rlation which adds an xtra lmnt to th bag: d X {((B, x), B x) x X, B!X} Th additiv idmpotncy o REL maks both K and J th idntity and thus trivially isomorphisms. Thror, th intgral transormation is th codriving transormation :!X!X X, which is th rlation which rmovs an lmnts rom th bag: d X d X {(B, (B {x}, x)) x X, B!X} Exampl 23. Th catgory o vctor spacs ovr a ild K o charactristic o 0, VEC K, is a co-dirntial catgory [7] with antidrivativs, so that, VEC op K is a calculus catgory. Whil having a ild o charactristic zro is not rquird to obtain dirntial structur, it is rquird or antidrivativs. Th additiv symmtric monoidal structur is givn by th standard tnsor product and additiv nrichmnt o vctor spacs. Th algbra modality is givn by th r symmtric algbra monad whr or a vctor spac V,!V is th r commutativ algbra ovr V (s [17] or mor dtails). Equivalntly, i X {x 1, x 2,...} is C S L

16 20:16 Intgral Catgoris and Calculus Catgoris a basis o V, thn!v is isomorphic to th polynomial ring ovr X:!V K[X] [17]. Thn th driving transormation d V :!V!V V (rcall VEC op K is th calculus catgory) on monomials is givn by th sum o partial drivativs o th monomial: d V (x r1 1...xrn n ) n i1 (x r1 1...xri 1...x rn ) x i i On monomials, K multiplis th non-constant monomials by thir dgr and multiplis th constants by on, whil J multiplis monomials by thir dgr plus on. As th rationals ar mbddd in our ild, both ar isomorphisms, and th rsulting intgral transormation s V :!V V!V is dind on monomials by: 1 s V ((x r1 1...xrn n ) x i ) 1 + n j1 r x r1 1...xri+1 i...x rn j At irst glanc this may sm bizarr. On might xpct th intgrating transormation 1 to intgrat a monomial with rspct to th variabl x i and thus only multiply by 1+r i. Howvr, this classical ida o intgration ails th Rota-Baxtr rul [s.2] or any vctor spac o dimnsion gratr than on. 6 Conclusion and Futur Work Th thory o dirntial catgoris was dvlopd in stags: (tnsor) dirntial catgoris [7], cartsian dirntial catgoris [3], dirntial rstriction catgoris [10], and tangnt catgoris [9]. Th dvlopmnt o intgral catgoris, bing vry closly rlatd, has paralll stags. Hr w hav brily introducd th irst stag o this dvlopmnt: tnsor intgral catgoris. Th nxt stag, Cartsian intgral catgoris, is actually wll in hand. Th coklisli catgory o an intgral catgory is a Cartsian intgral catgory. Furthrmor, Cartsian intgral catgoris hav a trm logic which has a mor classic l: w borrowd parts o this trm logic to hlp motivat this papr. Th study o intgration in rstriction catgoris and tangnt catgoris is, by comparison, in its arlist stags. Acknowldgmnts. Th authors would lik to thank Rick Blut or drawing both authors attntion to Rota-Baxtr algbras. Intgral catgoris simply would not hav dvlopd so rapidly without this basic inspiration. Jonathan Gallaghr rmindd us o Ehrhard s work at xactly th right momnt, whil Kristin Baur providd continual constructiv criticism during th volution o our thoughts. Rrncs 1 Gln Baxtr t al. An analytic problm whos solution ollows rom a simpl algbraic idntity. Paciic J. Math, 10(3): , R. Blut, J. R. B. Cocktt, and R. A. G. Sly. Cartsian dirntial storag catgoris. Thory and Applications o Catgoris, 30(18): , R. F. Blut, J. Robin B. Cocktt, and R. A. G. Sly. Cartsian dirntial catgoris. Thory and Applications o Catgoris, 22(23): , Richard Blut, J. R. B. Cocktt, Timothy Portr, and R. A. G. Sly. Kählr catgoris. Cahirs d Topologi t Géométri Diérntill Catégoriqus, 52(4): , Richard Blut, Thomas Ehrhard, and Christin Tasson. A convnint dirntial catgory. arxiv prprint arxiv: , 2010.

17 J. R. B. Cocktt and J.-S. Lmay 20:17 6 Richard Blut, Rory B. B. Lucyshyn-Wright, and Kith O Nill. Drivations in codirntial catgoris. arxiv prprint arxiv: , Richard F. Blut, J. Robin B. Cocktt, and Robrt A. G. Sly. Dirntial catgoris. Mathmatical structurs in computr scinc, 16(06): , Raoul Bott and Loring W. Tu. Dirntial orms in algbraic topology, volum 82. Springr Scinc & Businss Mdia, J. Robin B. Cocktt and Go S. H. Cruttwll. Dirntial Structur, Tangnt Structur, and SDG. Applid Catgorical Structurs, 22(2): , J. R. B. Cocktt, G. S. H. Cruttwll, and J. D. Gallaghr. Dirntial rstriction catgoris. Thory and Applications o Catgoris, 25(21): , Thomas Ehrhard. An introduction to dirntial linar logic: proo-nts, modls and antidrivativs. Mathmatical Structurs in Computr Scinc, pags 1 66, Thomas Ehrhard and Laurnt Rgnir. Th dirntial lambda-calculus. Thortical Computr Scinc, 309(1):1 41, Thomas Ehrhard and Laurnt Rgnir. Dirntial intraction nts. Thortical Computr Scinc, 364(2): , Marclo P. Fior. Dirntial structur in modls o multiplicativ biadditiv intuitionistic linar logic. In Intrnational Conrnc on Typd Lambda Calculi and Applications, pags Springr, Li Guo. An introduction to Rota-Baxtr algbra, volum 2. Intrnational Prss Somrvill, André Joyal and Ross Strt. Th gomtry o tnsor calculus, I. Advancs in Mathmatics, 88(1):55 112, Srg Lang. Algbra rvisd third dition. Graduat Txts in Mathmatics, 1(211):ALL ALL, J.-S. P. Lmay. Intgral Catgoris and Calculus Catgoris. Univrsity o Calgary, Saundrs Mac Lan. Catgoris or th working mathmatician, volum 5. Springr Scinc & Businss Mdia, Gian-Carlo Rota. Baxtr algbras and combinatorial idntitis. I. Bulltin o th Amrican Mathmatical Socity, 75(2): , Ptr Slingr. A survy o graphical languags or monoidal catgoris. In Nw structurs or physics, pags Springr, Charls A. Wibl. An introduction to homological algbra. Cambridg univrsity prss, C S L