The Frobenius relations meet linear distributivity

Transcription

1 Th Frobnius rlations mt linar distributivity J.M. Eggr Fbruary 18, 2007 Abstract Th notion of Frobnius algbra originally aros in ring thory, but it is a fairly asy obsrvation that this notion can b xtndd to arbitrary monoidal catgoris. But, is this rally th corrct lvl of gnralisation? For xampl, whn studying Frobnius algbras in th -autonomous catgory Sup, th standard concpt using only th usual tnsor product is lss intrsting than a similar on in which both tnsor and its d Morgan dual ( par ) ar usd. Thus w maintain that th notion of linar-distributiv catgory (which has both a tnsor and a par, but is nvrthlss mor gnral than th notion of monoidal catgory) provids th corrct framwork in which to intrprt th concpt of Frobnius algbra. 1 Introduction W rcall that a linarly distributiv catgory is dfind to b a catgory quippd with two tnsor products, hr dnotd and, which ar rlatd by a pair of (gnrally noninvrtibl) natural transformations x (y z) () z (x y) z κ (rr) x (y z) which ar rquird to satisfy a larg numbr of cohrnc conditions. Th radr is rfrrd to [2] for th full dfinition, as wll as for proofs of all statmnts in this sction. Th dfinition of linar functor btwn linarly distributiv catgoris can b found in [3]. Exampl 1.1 Evry (non-commutativ) -autonomous catgory (K,,,,, d) has an undrlying linarly distributiv catgory, in which th scond tnsor product is dfind as th d Morgan dual of th first. x y := (y x ) Rsarch partially supportd by NSERC. 1

2 Hr, as usual, x is an abbrviation for x d, and x is an abbrviation for d x. [Not that (y x ) = ( y x) holds in any (non-commutativ) -autonomous catgory, so th asymmtry prsnt in th dfinition abov is only on of apparanc.] In particular, a boolan algbra may b viwd as a linarly distributiv catgory with = and =. On of th profoundst and most usful obsrvations containd in [2] is that what on might call duality thorms in a (non-commutativ) -autonomous catgory can b charactrisd in trms of its undrlying linarly distributiv structur. Thorm 1.2 Lt (K,,,,, d) b a (non-commutativ) -autonomous catgory, and suppos that w ar givn a pair of arrows x ϑ y. ϕ Lt γ d dnot th transpos of ϑ, and τ th composit ϕ y x y x. Thn ϑ and ϕ ar invrs to ach othr if and only if th following diagrams commut. x ι τ x (y x) υ (r) x (lti 1 ) υ (l) 1 () x d x γ ι y τ ι (y x) y κ (rr) υ (l) y (lti 2 ) υ (r) 1 y () y d ι γ [Hr w abus notation by using th sam symbol to dnot both of th lft-unit isomorphisms; similarly, th right-unit isomorphisms.] 2

3 Dfinitions A pair of arrows in a linarly distributiv catgory (K,,,, d) γ d τ y x which satisfis (lti 1 ) and (lti 2 ) is calld a linar adjoint. 2. A pair of linar adjoints γ 1 d τ 1 y x y x γ 2 d τ 2 x y is calld a cyclic linar adjoint. Rmark 1.4 It is to b mphasisd that any monoidal catgory (K,, i) can b rgardd as a linarly distributiv catgory by choosing = =, = i = d, κ (rr) = α, and = α 1. In this sns, linarly distributiv catgoris ar actually mor gnral than monoidal catgoris and whn, as hr, duality thory is at th cntr of on s attntion, this is oftn a usful point of viw: that monoidal catgoris ar mrly dgnrat linarly distributiv catgoris. Not that in th strict dgnrat cas, th diagrams dfining linar adjoint rduc to th mor usual triangl idntitis. Lmma 1.5 In an arbitrary linarly distributiv catgory, ach half of a linar adjoint dtrmins th othr in th sns that, if γ 1 d τ 1 y x γ 2 d τ 2 y x ar linar adjoints, thn (γ 1 = γ 2 ) (τ 1 = τ 2 ). 2 Linar points and Frobnius monoids Throughout th nxt two sctions (K,,,, d) shall dnot an arbitrary linarly distributiv catgory. In particular, w shall not b considring symmtris, braidings, or vn non-planar linar distributions on (K,,,, d), until sction 5. Also, it shall b our convntion that: th trm monoid rfrs to a monoid in (K,, ); th trm comonoid rfrs to a comonoid in (K,, d); and, th trm monoid/comonoid pair rfrs to a monoid and a comonoid (in th snss abov) which hav th sam undrlying objct. 3

4 Lmma 2.1 Lt M = (m,, ) b a monoid, (x, σ (l) ) a lft M-objct, and (y, σ (r) ) a right M-objct; thn th maps m (x y) (m x) y σ (l) ι x y (x y) m κ (rr) x (y m) ι σ (r) x y dfin a two-sidd action of M on x y. Proof W shall prov only th lft/right-compatibility diagram, somtims also known as middl associativity, as an illustration. (m (x y)) m α m ((x y) m) ι ι κ (rr) ((m x) y) m (coh) m (x (y m)) (σ (l) κ ι) ι (rr) ι (ι σ (r) ) (x y) m (nat) (m x) (y m) (nat) m (x y) κ (rr) σ (l) ι ι σ (r) x (y m) σ (l) σ (r) (m x) y ι σ (r) σ (l) ι Th rmaindr is lft as an xrcis. q..d. Rmark 2.2 In th cas whr (K,,,, d) ariss from a (non-commutativ) -autonomous catgory (K,,,,, d), w hav canonical isomorphisms x y = x y = x y. Now th lft (right) m-action w hav dfind on x y is th sam as th obvious on on x y (rspctivly, x y). But in an arbitrary (lft- and right-) closd monoidal catgory, thr would sm to b no way of xprssing th ida that ths lft- and right- actions combin to form a two-sidd action. For th most part, w shall b intrstd in only th following cass of Lmma 2.1, and thir duals: if M = (m,, ) is a monoid, thn of cours m carris a canonical two-sidd M-action; and so any two-sidd action on x inducs canonical two-sidd actions on both x m and m x. This obsrvation is ncssary to mak th following dfinition typ corrctly. 4

5 Dfinitions A cyclic nuclar monoid in (K,,,, d) consists of: (a) a monoid M = (m,, ); (b) a comonoid G = (g, δ, ); (c) a two-sidd M-action (σ (l), σ (r) ) on g; and (d) a two-sidd G-coaction (ϑ (l), ϑ (r) ) on m; such that: () m ϑ(l) g m and m ϑ(r) m g ar M-quivariant; and (f) m g σ (l) g and g m σ(r) g ar G-coquivariant. 2. A Frobnius monoid in (K,,,, d) is a cyclic nuclar monoid with m = g, σ (l) = = σ (r), and ϑ (l) = δ = ϑ (r). Thorm 2.4 A cyclic nuclar monoid in (K,,,, d) is th sam thing as a linar point of (K,,,, d) i.., a linar functor 1 (K,,,, d). A Frobnius monoid is th sam thing as a monoid/comonoid pair (m,,, δ, ) which satisfy th linar Frobnius rlations dpictd blow. m (m m) ι δ m m δ ι (m m) m (lfr 1 ) (m m) m ι m δ m m (lfr 2 ) κ (rr) ι m (m m) Proof It follows dirctly from th dfinition of linar functor that a linar point consists of: a monoid M = (m,, ), a comonoid G = (g, δ, ), and maps σ (l), σ (r), ϑ (l), ϑ (r) whos sourcs and targts match thos givn in th dfinition of cyclic nuclar monoid. Thrfor all that rmains to show is that th ightn cohrnc conditions rquird of σ (l), σ (r), ϑ (l), ϑ (r) by th dfinition of linar functor ar (collctivly) quivalnt to th conditions placd on thm by th dfinition of cyclic nuclar monoid. In fact, Dfinition 2.3 also rquirs that σ (l), σ (r), ϑ (l) and ϑ (r) satisfy a total of ightn diagrams: 1. fiv diagrams for σ (l), σ (r) bing an M-action; 2. fiv diagrams for ϑ (l), ϑ (r) bing a G-coaction; 5

6 3. four diagrams for ϑ (l), ϑ (r) bing M-quivariant; and 4. four diagrams for σ (l), σ (r) bing G-coquivariant. W claim that th two sts of conditions ar not mrly quivalnt, but actually qual. To dmonstrat this claim, w nd mrly idntify th fiv symmtry-basd groupings which appar in [3, Dfinition 1]. Ths ar as follows: 1. th four (co)unit diagrams; 2. th four (co)associativity diagrams; 3. th two lft/right-compatibility diagrams; 4. th four sam-parity (co)quivarianc diagrams i.., th lft quivarianc of ϑ (l), th right quivarianc of ϑ (r), th lft coquivarianc of σ (l) and th right coquivarianc of σ (r) ; 5. th rmaining four (co)quivarianc diagrams. [What distinguishs th sam-parity (co)quivarianc diagrams is that, in ach cas, thy contain an altrnating trnary string: ithr mgm or gmg for an xampl, s blow.] Turning to th cas of Frobnius monoids, w s that th first tn diagrams (according to ithr grouping) ar trivial; th rmaining ight diagrams collaps to two. Considr, for xampl, th M-quivarianc of ϑ (l). In gnral, this stats that th diagrams m m ι ϑ (l) m (g m) m m ϑ (l) ι (g m) m (m g) m κ (rr) g (m m) m σ (l) ι ϑ (l) g m m ϑ (l) g m ι commut but, if g = m, σ (l) = and ϑ (l) = δ, thn ths rduc to (lfr 1 ) and (lfr 2 ), rspctivly. Each of th thr othr pairs of diagrams th G-coquivarianc of σ (l), that of σ (r), and th M-quivarianc of ϑ (r) do th sam. [Not that ach pair contains on diagram with a and on with a κ (rr) th formr rducs to (lfr 1 ) and th lattr to (lfr 2 ).] q..d. 6

7 Scholium 2.5 For a monoid/comonoid pair (m,,, δ, ) in (K,,,, d), th following ar quivalnt: 1. m δ m m is lft (m,, )-quivariant; 2. m δ m m is right (m,, )-quivariant; 3. m m m is lft (m, δ, )-coquivariant; 4. m m m is right (m, δ, )-coquivariant; 5. (m,,, δ, ) is a Frobnius monoid. In th dgnrat cas, this fact is wll-known and appars, for xampl, in [5]. 3 Duality In th dgnrat cas, Frobnius monoids ar usually studid in connction with duality thory, so thr should b littl surpris that th sam is tru in th gnral cas. Indd, it follows from gnral rsults in [3] and [1] that cyclic nuclar monoids and Frobnius monoids must b slf-dual. Hr w formulat and prov th sam rsult with somwhat mor prcision. Thorm 3.1 If ((m,, ), (g, δ, ), (σ (l), σ (r) ), (ϑ (l), ϑ (r) )) form a cyclic nuclar monoid, thn th maps m ϑ (l) g m m g σ (l) g d m ϑ (r) m g g m σ (r) g d form a cyclic linar adjoint. In particular, if (m,,, δ, ) is a Frobnius monoid, thn m δ m m m m m d form a linar adjoint. Proof Each of th four sam-parity (co)quivarianc diagrams is usd onc, togthr with appropriat (co)unit axioms, to obtain on of th four ncssary diagrams. For instanc, th lft-quivarianc of ϑ (l) (s proof of Thorm 2.4 abov) can b combind with its lft counit axiom and th right unit axiom for to produc (lti 1 ) for ; ϑ (l) 7

8 and σ (l) ;. m ι m m ι ϑ (l) m (g m) (m g) m υ (r) m ϑ (l) g m σ (l) ι υ (l) 1 ι d m q..d. Scholium 3.2 By xamining th proof of Thorm 3.1, w can driv th following formula for ϑ (l) : υ (r) 1 m m ι ( ; ϑ) ϑ (l) m (g m) (m g) m g m σ (l) ι but rcall from Lmma 1.5, that th composit ; ϑ (l) is uniquly dtrmind by σ (l) ;. Thus, th structur of a cyclic nuclar monoid is ovrdtrmind. In particular, ϑ (l) is dtrmind, indirctly, by σ (l) and. In th cas of a Frobnius monoid in a dgnrat linarly distributiv catgory, this is a clbratd rsult: th map δ = ϑ (l) is dtrmind by = σ (l) and s, for xampl, [5]. Rmark 3.3 It is intrsting to not that only four of th ight (co)quivarianc conditions ar actually usd in th proof of Thorm 3.1. Also intrsting is th fact that ach linar adjoint is constructd using only on half of th action/coaction structur. This suggsts that thr xists a mor gnral notion than that of cyclic nuclar monoid, for which non-cyclic linar adjoints can b constructd. Th lattr ida will b mor fully xplord in a subsqunt papr. 4 Frobnius quantals Throughout th nxt two sctions: E shall dnot an arbitrary lmntary topos; 1 2 its subobjct classifir; P th powr-e-objct monad, (2 ( ),, {}); and Ě = Sup(E) th catgory 8

9 of P-algbras quivalntly, th catgory of (intrnally) cocomplt ordrd E-objcts and (intrnal) sup-homomorphisms. Hncforth, w shall spak of th objcts of E as if thy wr sts, and thrfor also drop th modifirs intrnal and intrnally. As dmonstratd in [4], Ě can b givn a symmtric -autonomous structur, and hnc also a (symmtric) linarly distributiv structur. [W shall postpon th dfinition of symmtric linarly distributiv catgory until th nxt sction, as w will not nd this xtra structur until thn.] Following th convntion laid out in Exampl 1.1, x dos not dnot x op but rathr x 2 op ; w writ for th canonical isomorphism x op x = x. W shall continu to writ and for th two tnsor products on Ě, but w shall abus notation, slightly, by dnoting thir units by 2 and 2 op rspctivly. [Each of ths carris a canonical P-algbra structur.] W writ for th forgtful functor Ě Ord(E), also for th canonical map x y hnc (,, ) is th forgtful monoidal functor (Ě,, 2) (Ord(E),, 1). A quantal is a monoid in (Ě,, 2). [W shall only considr unital quantals in th currnt papr.] Givn a quantal (q,, ), w writ ( q,, ) for th corrsponding monoid in (Ord(E),, 1); in particular, dnots th composit q q q q q. W writ and for th lft- and right-closd structurs on ( q,, ), rspctivly. W rcall that a quantal quippd with a cyclic dualising lmnt is commonly calld a Girard quantal, [7]. [By a (cyclic) dualising lmnt for (q,, ), on actually mans a (cyclic) dualising lmnt for ( q,,,, ).] Surprisingly, thr dos not sm to b a standard (short) nam for a quantal quippd with an arbitrary dualising lmnt; but this provs to b fortunat, in light of th following thorm. Thorm 4.1 A Frobnius monoid in (Ě,, 2,, 2 op ) amounts to a quantal quippd with a dualising lmnt; quivalntly, a -autonomous cocomplt post. Proof Suppos that (q,,, δ, ) is a Frobnius monoid in (Ě,, 2,, 2 op ), and lt = kr (so that = ). W will abbrviat ( ) and ( ) to ( ) and ( ) rspctivly. By Thorm q δ q q q q q 2 op form a linar adjoint. Hnc, by Thorm 1.2, th transpos of th lattr composit q ( ; ) r 2 op q = q 9

10 is invrtibl. But it is asy to chck that this transpos quals th map sinc w hav α α α (β) = β α = α (α β) = α β ((α β)) =. [This may not look contructiv, but it is: w r just using th univrsal proprtis of and of 2. Hr, mans 2 op i.., 2.] Morovr, this suffics to dmonstrat that is a dualising lmnt, sinc for posts w always hav ( (( ) )) = ( ). Th prvious argumnts ar rvrsibl in th sns that if w start with a dualising lmnt, thn w can dfin =, and th composit q q q 2 op dfins half of a linar adjoint. According to Scholium 3.2, thr is thn a uniqu δ making (q,,, δ, ) into a Frobnius monoid in (Ě,, 2,, 2 op ). q..d. W thrfor propos that a quantal quippd with a (not ncssarily) dualising lmnt should b calld a Frobnius quantal. W shall also us th trm Frobnius local for a Frobnius quantal whos undrlying quantal is a local; this amounts to a complt boolan algbra. By way of comparison, not that a Frobnius monoid in (Ě,, 2,, 2) whos undrlying quantal is a local amounts to a powr-objct, [6]. Thus, in a boolan topos, E, Frobnius quantals ar mor gnral than Frobnius monoids in (Ě,, 2,, 2); but, in non-boolan toposs, nithr concpt is containd in th othr. 5 Canonical cyclicity W rcall that, by dfinition, a linarly distributiv catgory (K,,,, d) is calld symmtric if both and ar quippd with a symmtry, and if ths symmtris (which shall both b dnotd ) satisfy on xtra cohrnc condition: x (y z) ι x (z y) (z y) x () z z () ι κ (rr) z (y x). Th following lmma should b fairly obvious, but its proof dos utilis this xtra cohrnc condition, dmonstrating th lattr s worth. 10

11 Lmma 5.1 If (K,,,, d) is a symmtric linarly distributiv catgory, and γ d τ y x is a linar adjoint in (K,,,, d), thn y x γ d τ y x x y is also a linar adjoint in (K,,,, d). Brifly, vry linar adjoint in (K,,,, d) givs riss to a cyclic linar adjoint. Proof W can obtain (lti 1 ) as follows: υ (r) y y υ (l) y ι τ τ ι ι y (y x) (y x) y κ (rr) υ (r) 1 y (x y) y (y x) ι y () y d ι γ (y x) y ι () y d y γ ι υ (l) 1 and (lti 2 ) by a symmtric argumnt. q..d. What is, prhaps, not obvious is whthr vry cyclic linar adjoint in (K,,,, d) ariss in this way. As w shall s latr in this sction, th answr turns out to b no. Dfinitions A cyclic linar adjoint in (K,,,, d) is calld canonically cyclic if th diagram y x γ 1 d γ 2 d τ 1 y x τ 2 x y. commuts. 11

12 2. A cyclic nuclar monoid in (K,,,, d) is calld canonically cyclic if th cyclic linar adjoint dscribd in Thorm 3.1 is so i.., if th diagram commuts. m m ϑ (l) g m m g ϑ (r) m g g m 3. A Girard monoid in (K,,,, d) is a Frobnius monoid whos undrlying cyclic nuclar monoid is canonically cyclic i.., if it satisfis m m δ m m m m δ m m m m Rmark 5.3 As a rsult of Lmmata 1.5 and 5.1, it suffics to chck only on of ach pair of diagrams in th dfinitions abov. From an abstract point of viw, th significanc of canonically cyclic linar adjoints has to do with th fact that in a symmtric -autonomous catgory (K,,,,, d), w hav a canonical natural isomorphism ω x x dfind as th transpos of th composit σ (l) σ (r) g g m m d d d d x x x x (r) d. [Not that, for us, th trm -autonomous catgory dnots a (lft- and right-) closd monoidal catgory quippd with a dualising objct; and that th trm symmtric -autonomous catgory dnots a -autonomous catgory quippd with a symmtry. Thus w do not assum in gnral that x y = y x, although this is what frquntly occurs in practic.] Thorm 5.4 Lt (K,,,,, d) b a symmtric -autonomous catgory. Thn a cyclic linar adjoint in (K,,,, d) γ 1 d τ 1 y x y x γ 2 d τ 2 x y is canonically cyclic if and only if th diagram x x γ r 1 γ l 2 12 y y ω

13 commuts. Proof It suffics to show that th composit y x ι γ r 1 y y ι ω y y (l) d quals γ 2. But th dfinition of ω amounts to y y y y ι ω (r) y y d (l) and combining this with a naturality squar, w obtain y x ι γ r 1 y y ι ω y y γ r 1 ι y y (r) d. (l) γ 1 q..d. Rmark 5.5 Th prvious thorm is th main rason w hav concntratd on symmtris rathr than braidings. In a braidd -autonomous catgory, w hav at last two isomorphisms x ω x which nd not b qual; thrfor nithr is gnuinly canonical. Radrs familiar with th classical thory of Frobnius algbras (i.., Frobnius monoids in th dgnrat linarly distributiv catgory (Vc,, k,, k)) will hav alrady rcognisd that not vry Frobnius monoid is a Girard monoid, and hnc that not vry cyclic linar adjoint is canonically cyclic. [Girard monoids in (Vc,, k,, k) normally go by th uninspird, and potntially mislading, nam symmtric Frobnius algbras.] For vryon ls w offr th following class of countr-xampls. Corollary 5.6 A Girard monoid in (Ě,, 2,, 2 op ) is th sam thing as a Girard quantal. 13

14 Proof By th prvious thorm, a Girard monoid in (Ě,, 2,, 2 op ) is th sam thing as a Frobnius quantal (q,,, δ, ) such that th diagram q ( ; ) r q q ( ; ) l q commuts. But w hav alrady shown, in th proof of Thorm 4.1, that th top map quals α α ; by a symmtric argumnt, th bottom map quals α α. q..d. Acknowldgmnts Th author would lik to thank: Rick Blut, Robin Cocktt and Robrt Sly, for much ncouragmnt in dvloping th matrial of sctions 3 and 4; David Kruml, for long and intrsting convrsations which, among othr things, providd th imptus for proving th rsults of sction 5; and Ptr Johnston, for (r-)introducing th wondrful word scholium to th mathmatical vocabulary. Rfrncs [1] J. R. B. Cocktt, J. Koslowski, and R. A. G. Sly. Introduction to linar bicatgoris. Math. Structurs Comput. Sci., 10(2): , Th Lambk Fstschrift: mathmatical structurs in computr scinc (Montral, QC, 1997). [2] J. R. B. Cocktt and R. A. G. Sly. Wakly distributiv catgoris. J. Pur Appl. Algbra, 114(2): , [3] J. R. B. Cocktt and R. A. G. Sly. Linarly distributiv functors. J. Pur Appl. Algbra, 143(1-3): , Spcial volum on th occasion of th 60th birthday of Profssor Michal Barr (Montral, QC, 1997). [4] André Joyal and Myls Tirny. An xtnsion of th Galois thory of Grothndick. Mm. Amr. Math. Soc., 51(309):vii+71, [5] Joachim Kock. Frobnius algbras and 2D topological quantum fild thoris, volum 59 of London Mathmatical Socity Studnt Txts. Cambridg Univrsity Prss, Cambridg, [6] M. C. Pdicchio and R. J. Wood. Groupoidal compltly distributiv lattics. J. Pur Appl. Algbra, 143(1-3): , Spcial volum on th occasion of th 60th birthday of Profssor Michal Barr (Montral, QC, 1997). 14

15 [7] Kimmo I. Rosnthal. Quantals and thir applications, volum 234 of Pitman Rsarch Nots in Mathmatics Sris. Longman Scintific & Tchnical, Harlow,