Involutive Categories and Monoids, with a GNS-correspondence

In v olutiv e Categorie s and Mon oids, with a GNS- corresp ondence Bart Jacobs Institute for Computing and Information Scienc es (iCIS) R adb oud University Nijme gen, The Netherlands August 2, 2018 Abstract This pa p er dev elops the basics of the theory of in volutiv e categories and shows that such categories provide the natural setting in which to describ e inv olutive monoids. It is shown how categories of Eilenberg- Mo ore algebras o f inv olutiv e monads are in vo lutive, with conjugation for modules and v ector spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS) construction is iden tified as a bijective correspondence b et ween stat es on inv olutiv e monoids and i nner products. This correspondence exists in arbritrary inv olutive (symmetric monoidal) categories. 1 In tro duction In general an in vo lution is a certain en domap i for wh ic h i ◦ i is the id entit y . The in ve rse op eration of a group is a sp ecial example. But there are also mon oids with suc h an inv olutio n, suc h as for ins tance the free monoid of lists, with list rev ersal as in vo lution. An inv olution can also b e defined on a cate gory . It then consists of an endo- functor C → C , w hic h is typically written as X 7→ X . It should satisfy X ∼ = X . In vol utive categories o ccur in the literature, for instance in [ 1 ], but h a v e, as f ar as w e kn o w, not b een stud ied systematically . In vo lutions are of particular imp ortance in th e (categ orical) found ations of quantum mec hanics and compu ting, see [ 2 ]. This pap er will dev elop the b asic elemen ts of such a theory of inv olutiv e categories. W e should note that in vo lutive categories as w e under s tand them h ere are differ- en t fr om dagger categories (whic h hav e an id en tit y-on-ob jects f unctor ( − ) † : C op → C with f †† = f ) and also from ∗ -autonomous categories (w h ic h h av e a dualit y ( − ) ∗ : C op → C giv en by a du alising ob ject D as in X ∗ = X ⊸ D ). In b oth these cases one has con tra v arian t fu nctors, w hereas inv olution ( − ) : C → C is a co v arian t functor. The relation b etw een in vol ution, dagg er and dualit y for Hilb ert spaces is describ ed in [ 2 , §§ 4.1, 4.2]: eac h can b e d efined in terms of th e other t wo . Jacobs In vol utive categories and inv olutiv e monoids are r elated: just lik e the n otion of a monoid is form ulated in a monoidal category , the n otion of inv olutiv e monoid requires an appropriate notion of in vo lutive monoidal categ ory . This is in line with the “micro cosm p r inciple”, formulated by Baez and Dolan [ 4 ], and elab orated in [ 12 , 11 , 10 ]: it in v olv es “outer” structure (lik e monoidal s tr ucture 1 I → C ⊗ ← C × C on a categ ory C ) that enables the definition of “inner” structure (lik e a monoid I 0 → M + ← M ⊗ M in C ). W e br iefly illustrate ho w this connection b et w een inv o- lutiv e mon oids and inv olutiv e categories arises. Consider for instance the additiv e group Z of intege rs with min us − as in v o- lution. In the category Sets of ordinary sets and functions b et w een them w e can describ e min us as an ordinary endomap − : Z → Z . The in tegers form a p artially ordered set, so we ma y wish to consider Z also as inv olutiv e monoid in the cat- egory P oSets of partially ordered set s and monotone fun ctions. Th e problem is that minus reve rses the order: i ≤ j ⇒ − i ≥ − j , and is thus not a m ap Z → Z in P oSets . Ho w ev er, we can describ e it as a map ( Z , ≥ ) → ( Z , ≤ ) in Po Sets , using th e rev ersed order ( ≥ instead of ≤ ) on the int egers. This order r ev ersal f orms an inv olution ( − ) : Po Sets → PoSets on the “outer” category , w h ic h allo ws u s to describ e the inv olution “internally” as − : Z → Z in Po Sets . As said, this pap er in tro du ces th e b asic steps of the theory of inv olutive cate- gories. It introdu ces the category of “self-conjugate” ob jects, and sh o ws ho w inv o- lutions arise on cate gories of Eilen b er g-Mo ore alg ebr as of an “in vol utive” mon ad . This general construction includes the imp ortant example of conjugation on mo dules and vec tor spaces, for th e multiset monad asso ciated w ith an in vo lutive semirin g. It allo ws us to describ e abstractly an in vo lutive monoid in suc h categories of algebras. Pre C ∗ -algebras (without norm) are suc h monoids. Once this setting has b een established we tak e a sp ecial look at the famous Gelfand-Naimark-Segal (GNS) construction [ 3 ]. It relates C ∗ -algebras and Hilb ert spaces, and shows in particular ho w a sta te A → C on a C ∗ -algebra giv es rise an inner pro d uct on A . Using conjugation as inv olution, the latter can b e d escrib ed as a m ap A ⊗ A → C that incorp orates the sesquilinearit y requirements in its t yp e (including conjugate linearit y in its firs t argumen t). The fin al section of th is p ap er giv es the essence of th is construction in the f orm of a bijectiv e corresp on d ence b et wee n suc h states and in ner pro du cts in categorica l terms, using the language of in vo lutive categories and monoids. 2 In v olutiv e categories This section only con tains the most b asic material. Definition 2.1 A category C will b e ca lled involutive if it comes with a fu n ctor C → C , wr itten as X 7→ X , and a natur al isomorph ism ι : X ∼ = − → X satisfying X ι X / / X X ι X / / X (1) Eac h category is trivially in vo lutive via the id en tit y f u nctor. This trivial inv olu- 2 Jacobs tion is certainly us eful. The category PoSets is inv olutive via order reve rsal. T his applies also to categories of, for in s tance, distributive lattice s or Bo olean algebras. The ca tegory Cat of (small) catego ries and fu nctors is also in v olutiv e, b y taking opp osites of categorie s. Next, consider th e category V ect C of v ector spaces ov er the complex num b ers C . It is an in vol utive category via conjugation (see Example 6.5 later on, for a systematic description). F or a v ector s p ace V ∈ V ect C w e d efine V ∈ V ect C with the s ame v ectors as V , but with adapted scalar multiplicat ion: for s ∈ C and v ∈ V , s · V v = s · V v , (2) where s = a − ib is the conju gate of the complex n umb er s = a + ib ∈ C . The follo wing is the first of a series of basic observ ations. Lemma 2.2 The involution functor of an involutive c ate gory is self- adjoint: ( − ) ⊣ ( − ) . As a r esult, involution pr eserves al l limits and c olimts that exist in the c ate gory. Pro of. Obviously there are bijectiv e corresp onden ces: X f / / Y = = = = = = = = X g / / Y One maps f to b f = f ◦ ι X : X ∼ = → X → Y and g to b g = ι − 1 Y ◦ g : X → Y ∼ = → Y .  Definition 2.3 A fun ctor F : C → D b et ween t w o in v olutiv e categ ories is called in vo lutive if it comes with a natural transformation (or distrib utiv e la w) ν with comp onen ts F ( X ) → F ( X ) comm uting appropr iately with th e isomorphism s X ∼ = X , as in: F ( X ) F ( ι X ) ∼ =   F ( X ) ι F ( X ) ∼ =   F ( X ) ν X / / F ( X ) ν X / / F ( X ) (3) A natur al transformation σ : F ⇒ G b etw een tw o in vol utive f unctors F , G : C ⇒ D is called in volutiv e if it commute s with the asso ciated ν ’s, as in: F ( X ) σ X / / ν F   G ( X ) ν G   F ( X ) σ X / / G ( X ) In this w a y we obtain a 2-catego ry ICat of in vol utive catego ries, functors and natural transformations. This 2-categorica l p ersp ectiv e is useful, for instance b ecause it allo ws us to see immediately what an inv olutive adjun ction or monad is, namely one in which the functors and natural transformations inv olved are all in vo lutive. Lemma 2.4 If F is an involutive functor via ν : F ( X ) → F ( X ) , then this ν is automatic al ly an isomorph ism. 3 Jacobs Pro of. W e constru ct an in v erse for ν as comp osite: ν − 1 def =  F ( X ) F ( ι ) / / F ( X ) ν / / F ( X ) ι − 1 / / F ( X )  . W e explicitly c hec k that this is indeed an inv erse to ν , by using the in teraction ( 3 ) b et wee n ν and ι . First we h a v e ν ◦ ν − 1 = id in: F ( X ) F ( ι ) / / H H H H H H H H H H H H H H H H F ( X ) ν / / F ( ι − 1 )   F ( X ) ι − 1 / / ν   F ( X ) ν   F ( X ) F ( X ) ι − 1 / / ι − 1 o o F ( X ) And similarly we get ν − 1 ◦ ν = id in : F ( X ) F ( ι ) / / F ( X ) ν / / F ( X ) ι − 1 / / ι − 1   F ( X ) F ( X ) ν O O F ( ι ) / / F ( X ) ν O O F ( ι − 1 ) / / F ( X ) v v v v v v v v v v v v v v v v  3 Self-conjugates Definition 3.1 F or an in vo lutive cate gory C , let SC ( C ) b e th e category of s elf- conjugates in C . Its ob jects are maps j : X → X making th e triangle b elo w com- m ute. X j / / ι − 1 X $ $ I I I I I I X j   X j X   f / / Y j Y   X X f / / Y It is n ot h ard to see that suc h a map is j is necessarily an isomorphism, with inv erse j ◦ ι X : X → X → X . A m orp hism f : ( X , j X ) → ( Y , j Y ) in S C ( C ) is a map f : X → Y in C making the ab o ve r ectangle commute. Th ere is thus an obvi ous forgetful functor SC ( C ) → C . By the self-adjoin tness of Lemma 2.2 a self-conjugate X → X ma y also b e describ ed as X → X . S ometimes w e call an ob ject X a s elf-conjugate when the map X ∼ = → X inv olv ed is ob vious fr om the conte xt. I n linear algebra, with X giv en b y conju gation (see ( 2 ), or also Example 6.5 ), a map of the f orm X → Y is called an ‘an tilinear’ or ‘conjugate linear’ m ap. Before d escrib ing examples we fir st note the f ollo win g. A more s ystematic d e- scription follo ws in Lemma 3.6 . Lemma 3.2 F or an involutive c ate gory C , the c ate gory S C ( C ) of self- c onjugates is again involutive, via:  X X  j / / def  X j / / X  . (4) 4 Jacobs and the for getfu l functor SC ( C ) → C is an involutive functor, via the identity natur al tr ansformation (as ‘ ν ’ in D efinition 2.3 ). Pro of. Th e map ι X : X ∼ = − → X in C is also a map in SC ( C ) in:  X X  j / / ι X ∼ = / /  X X  j / / since the follo wing diagram commutes by naturalit y . X j   ι X = ι X / / X j   X ι X / / X  Example 3.3 Recall that the category PoSets of p osets and monotone functions is in v olutiv e via th e rev ersed (opp osite) order: ( X, ≤ ) = ( X, ≥ ). The in tegers Z are then s elf-conjugate, v ia min us − : Z ∼ = → Z . Also the p ositiv e rational and real n umb ers Q > 0 and R > 0 are s elf-conjugates in P oSets , v ia x 7→ 1 x . Similarly , for a Bo olean algebra B , n egatio n ¬ yields a self-conjugate ¬ : B ∼ = → B in the categ ory of Bo olean algebras. There are similar self-conjugates via orthosupp lemen ts ( − ) ⊥ in orthomo dular lattices [ 13 ] an d effect algebras [ 9 ]. In C at a self-conjugate is giv en by a self-dual category C op ∼ = C . Recall the conjugation ( 2 ) on v ector sp aces. Supp ose V ∈ V ect C has a basis ( v i ) i ∈ I . T hen we can define a s elf-conjugate V ∼ = → V by: x =  P i x i v i  7− →  P i x i v i  . Finally , if a category C is considered with trivial inv olution X = X , then SC ( C ) con tains the self-inv erse endomaps j : X → X , with j ◦ j = id X . W e first tak e a closer lo ok at these trivial in vo lutions. Lemma 3.4 L et C b e an or dinary c ate gory, c onsider e d as involutive with trivi al involution X = X . Assuming binary c opr o ducts + and pr o ducts × exist in C , ther e ar e left and right adjoints to the f or getful functor: SC ( C ) ⊣ ⊣   C X 7→ 2 × X = X + X @ @ X 7→ X 2 = X × X ^ ^ using the swap maps [ κ 2 , κ 1 ] : X + X ∼ = → X + X and h π 2 , π 1 i : X × X ∼ = → X × X as self-c onjugates. Pro of. Recall that for the trivial inv olution on C , an ob ject ( Y , j ) ∈ SC ( C ) consists of an isomorphism j : Y ∼ = → Y with j − 1 = j . F or the left adjoint the required bijectiv e corresp ondence: ( X + X, [ κ 2 , κ 1 ]) f / / ( Y , j ) in SC ( C ) = = = = = = = = = = = = = = = = = = = = = = X g / / Y in C 5 Jacobs exists b ecause the requiremen t f ◦ [ κ 2 , κ 1 ] = j ◦ f means f ◦ κ 2 = j ◦ f ◦ κ 1 . Hence f is determined by f ◦ κ 1 : X → Y . The argumen t works similarly for the righ t adjoints, giv en by pro d ucts.  Lemma 3.5 L et C b e an involutive c ate gory; SC ( C ) inherits al l limits and c olimits that exist in C , and the f or getful functor SC ( C ) → C pr eserves them. Pro of. W e giv e an exemplaric ske tc h f or binary pr o ducts × . The pr o duct of t w o ob jects ( X , j X ) , ( Y , j Y ) ∈ SC ( C ) is giv en by: X × Y h π 1 ,π 2 i ∼ = / / X × Y j X × j Y / / X × Y , where the (canonical) isomorph ism exists since ( − ) preserves p ro ducts, b y Lemma 2.2 . It is not hard to see that this is a self-conjugate, forming a pro du ct in SC ( C ).  F or the record we note the follo wing (see [ 18 , 6 ] for bac kground ). Lemma 3.6 The mapping C 7→ SC ( C ) i s a 2-functor ICat → ICat , and even a 2-c omonad. Pro of. Essentiall y th is s ays that we can lift inv olutive functors and n atural trans - formations as in: SC ( C )   SC ( F ) , , SC ( G ) 2 2       SC ( σ ) SC ( D )   C F + + G 3 3       σ D (5) Using Lemma 2.4 the lifted functor SC ( F ) is defined as:  X j / / X   / /  F ( X ) ν − 1 / / F ( X ) F ( j ) / / F ( X )  . (6) It is not hard to see that the right- hand -side is a again a self-conjugate. Th e natural transformation SC ( σ ) on X → X is simply σ X . The counit of S C as 2-c omonad is the forgetful fun ctor SC ( C ) → C , w hic h is natural, see ( 5 ). The comultiplic ation SC ( C ) → SC ( SC ( C )) is giv en by:  X X  j / /  / /   X X  j / / j / /  X X  j / /  .  4 In v olutiv e monoidal categories Definition 4.1 An involutive monoidal c ate gory or an involutive symmetric monoi- dal c ate gory , abbreviated as IMC or ISMC, is a catego ry C whic h is b oth inv olu- tiv e and (sym m etric) mono d ial in whic h in vo lution ( − ) : C → C is a (symmetric) monoidal functor and ι : id ⇒ ( − ) is a monoidal natural transformation. 6 Jacobs The fact that inv olution is a (symmetric) monoidal functor means that there are (natural) maps ζ : I → I and ξ : X ⊗ Y → X ⊗ Y comm uting with the monoidal isomorphisms α : X ⊗ ( Y ⊗ Z ) ∼ = → ( X ⊗ Y ) ⊗ Z , λ : I ⊗ X ∼ = → X , ρ : X ⊗ I → X , and also with the swap map γ : X ⊗ Y ∼ = → Y ⊗ X in the sym metric case. That the isomorphism ι is monoidal means that we ha ve commuting diagrams: I I ι   X ⊗ Y ι ⊗ ι   X ⊗ Y ι   I ζ / / I ζ / / I X ⊗ Y ξ / / X ⊗ Y ξ / / X ⊗ Y (7) Lik e in Lemm a 2.4 w e get isomorphy for free. Lemma 4.2 In an IMC the involution functor ( − ) is automatic al ly str ong monoida l: the maps ζ : I → I and ξ : X ⊗ Y → X ⊗ Y ar e ne c essarily i somorphisms. Pro of. All this follo ws from the r equiremen t ι = ι : X → X in ( 1 ) in Definition 2.1 and the monoidal requirements ( 7 ). F or ins tance, the obvious candidate as inv erse for ζ : I → I is ι − 1 ◦ ζ : I → I ∼ = → I . Because ι is a monoidal natural tr ansformation, w e immediately get ι − 1 ◦ ζ ◦ ζ = ι − 1 ◦ ι = id . By p ost-comp osing with the isomorphism ι = ι : I → I w e get by ( 7 ): ι ◦ ζ ◦ ι − 1 ◦ ζ = ζ ◦ ι ◦ ι − 1 ◦ ζ = ζ ◦ ζ = ι = ι. Similarly , the (candidate) in ve rs e for ξ : X ⊗ Y → X ⊗ Y is: X ⊗ Y ι ⊗ ι / / X ⊗ Y ξ / / X ⊗ Y ι − 1 / / X ⊗ Y .  In ord er to b e complete we also hav e to define the follo wing. Definition 4.3 A f unctor F : C → D b et ween IMC’s is called involutive monoidal if it is b oth in v olutiv e, via ν : F ( X ) → F ( X ), and monoidal, via ζ F : I → F ( I ) and ξ F : F ( X ) ⊗ F ( Y ) → F ( X ⊗ Y ), and th ese natural transformations ν, ζ F , ξ F in teract appr opriately with ζ , ξ from( 7 ), as in: I ζ F / / F ( I ) F ( ζ ) / / F ( I ) ν   F ( X ) ⊗ F ( Y ) ξ F / / ν ⊗ ν   F ( X ⊗ Y ) F ( ξ ) / / F ( X ⊗ Y ) ν   I ζ / / I ζ F / / F ( I ) F ( X ) ⊗ F ( Y ) ξ / / F ( X ) ⊗ F ( Y ) ξ F / / F ( X ⊗ Y ) It should then b e ob vious w hat an inv olutiv e symmetric monoidal fun ctor is. An inv olutiv e monoidal n atural transform ation σ : F ⇒ G b etw een tw o inv olu- tiv e monoidal fu nctors is b oth inv olutive and monoidal. Hence also in this case w e ha v e tw o categories IMCat and IMSCat of inv olutiv e (symmetric) monoidal categories. No w we come to the main result of this section. Prop osition 4.4 A c ate gory SC ( C ) inherits (symmetric) monoidal structur e fr om C . As a r esult, the for getful functor SC ( C ) → C i s an involutive (symmetric) monoidal functor. 7 Jacobs In c ase C is monoidal close d, then so is S C ( C ) and SC ( C ) → C pr eserves the exp onent ⊸ . Pro of. Th e tensor un it I ∈ C is a self-conjugate via ζ − 1 : I ∼ = → I . If we hav e self- conjugates j X : X ∼ = → X and j Y : Y ∼ = → Y w e obtain a tensored self-conjugate using Lemma 4.2 : X ⊗ Y ξ − 1 ∼ = / / X ⊗ Y j X ⊗ j Y ∼ = / / X ⊗ Y . It is not hard to see th at, with this tensor pro d uct, the monoidal isomorphisms α, λ, ρ, γ from C are also maps in SC ( C ). S im ilarly , for the requ ir ed maps making the inv olution ( − ) : SC ( C ) → SC ( C ) fr om Lemma 3.2 in to a mon oidal functor, we can tak e the ones from C , in:  I I ζ − 1 / /  ζ / /  I I ζ − 1 / /   X X j X / /  ⊗  Y Y j Y / /  ξ / /  X ⊗ Y X ⊗ Y j X ⊗ j Y ◦ ξ − 1 / /  , The exp onen t of ( X , j X ) , ( Y , j Y ) ∈ SC ( C ) is X ⊸ Y with self-conjugate X ⊸ Y → X ⊸ Y obtained by abstraction from: X ⊗ ( X ⊸ Y ) j − 1 X ⊗ id / / X ⊗ ( X ⊸ Y ) ξ / / X ⊗ ( X ⊸ Y ) ev / / Y j Y / / Y .  5 In v olutiv e Monoids No w that w e ha v e the notion of inv olutiv e cat egory as am bient ca tegory , we can define the notion of in volutiv e monoid in this setting, in the st yle of [ 12 , 11 , 10 ]. W e start with some preliminary obs er v ations. Let M = ( M , · , 1) b e an arbitrary monoid (in Set s ), not necessarily comm u tativ e. An inv olution on M is a sp ecial endofunction M → M whic h we shall write as sup ers cript negation x − , for x ∈ M . It satisfies x −− = x and 1 − = 1. The inte raction of in volutio n and m ultiplication ma y happ en in t w o wa ys: either in a “reversing” manner, as in ( x · y ) − = y − · x − , or in a “non-rev ersing” mann er: ( x · y ) − = x − · y − . Obviously , in a commutat ive monoid there is no difference b etw een a reversing or non-rev ersing inv olution. Eac h grou p is a reversing inv olutive monoid with x − = x − 1 . One adv antag e of in vo lutive m on oids o v er groups is that they inv olve only “linear” equ ations, with axioms con taining v ariables exactly once on b oth sides of the equation sign. Groups ho we ver are non-linear, v ia the axiom x · x − 1 = 1. Hence this equation cannot b e form ulated in a monoidal category , since it r equires diagonals and pr o j ections. Instead, one commonly uses Hopf algebras. As w e ha ve argued in the first section via th e example of in tegers in PoSets , a prop er form ulation of the notion of in vo lutive monoid requ ir es an in vo lutive cate- gory , so that the monoid inv olution can b e describ ed as a map M → M . Definition 5.1 Let C b e an in v olutive symmetric monoidal category . An inv olu- tiv e monoid in C consists of a monoid I u → M m ← M ⊗ M in C tog ether with an 8 Jacobs in vo lution map M j → M satisfying: I u / / ζ ∼ =   M M j / / ι − 1 ∼ = " " E E E E E E E M j   I u / / M j O O M and, one of the follo wing diagrams: “rev ersing” “non-rev ersing” M ⊗ M j ⊗ j   ξ / / M ⊗ M m / / M j   M ⊗ M γ ∼ = / / M ⊗ M m / / M M ⊗ M j ⊗ j   ξ / / M ⊗ M m / / M j   M ⊗ M m / / M One ma y call M a simple in vol utive monoid if C ’s in vol ution ( − ) is the iden tit y . A morph ism of in volutiv e monoids M → M ′ is a m orphism of monoids f : M → M ′ satisfying f ◦ j = j ′ ◦ f . This yields t w o su b categories rIMon ( C ) ֒ → Mon ( C ) and IMon ( C ) ֒ → Mon ( C ) of rev ersing and non-rev er s ing in vo lutive monoids. There is also a comm utativ e ve rsion, f orming a (full) sub cate gory . I C Mon ( C ) ֒ → IMon ( C ). The in v olution map j : M → M of an inv olutiv e monoid is of course a self- conjugate—see Definition 3.1 —and th us an isomorphism . In fact, we ha ve the follo wing resu lt. Lemma 5.2 Involutive monoids (of the non-r eversing kind) ar e or dinary monoids in the c ate gory of self-c onjugates: the c ate g ories IMon ( C ) and Mon ( SC ( C )) ar e the same. Similarly in the c ommutative c ase, I CMon ( C ) = CMon ( SC ( C )) . Pro of. Sin ce the tensors of C and SC ( C ) coincide—see Prop ositi on 4.4 —w e only need to c hec k that the ab o ve defin ition p r ecisely sa ys that the un it u and m ultipli- cation m of an in vo lutive monoid are maps in SC ( C ) of the form:  I I ζ − 1 / /  u / /  M M j / /   M M j / /  ⊗  M M j / /  . m o o The unit u is a m ap as indicated on the left if and only if j ◦ u = u ◦ ζ − 1 . T h is is precisely the first square in Definition 5.1 . Similarly , m is map on the righ t if and on ly if m ◦ ( j ⊗ j ) ◦ ξ − 1 = j ◦ m . Again, this is exactly the (non-reversing) requirement in Definition 5.1 .  This lemma suggests a pattern for defining an inv olutiv e v arian t of certain catego rical structure, namely by d efiniting this structure in the category of self- conjugates. Acti ons form an example, see Definition 5.5 b elo w . Example 5.3 As w e h av e obs er ved b efore, the category P oSets of p osets and monotone functions is inv olutiv e, via ord er-rev ersal ( X, ≤ ) = ( X, ≥ ). The p oset Z of in tegers forms an inv olutiv e monoid in Po Sets , with min us − : Z → Z as 9 Jacobs in vo lution. Also, the p ositive r ationals Q > 0 or reals R > 0 with multiplicat ion · , unit 1, and in ve rse ( − ) − 1 form in vo lutive monoids in P oSets . In the category C a t of categories, with fi nite pro d ucts as m onoidal structure, a monoid is a strictly monoidal category . If suc h a category C has a dagger † : C op → C that comm utes with these tensors (in the sense that ( f ⊗ g ) † = f † ⊗ g † , see e.g . [ 2 ]) then C is an in vo lutive monoid in Cat . Inside such a dagger symmetric (not necessarily strict) monoidal categ ory C with dagger ( − ) † : C op → C the homset of scalars I → I is a comm utativ e in v olutiv e monoid, with inv olution s − = s † . The tensor un it I ∈ C in an arbitrary in vo lutive category C is a commutat ive in vo lutive monoid ob ject, with in vo lution ζ − 1 : I → I . W e briefly describ e free inv olutiv e monoids in the category Sets (with trivial in vo lution), b oth of the rev ersin g and non-reve rsin g k in d. W e recall th at the set V ⋆ of finite lists h v 1 , . . . , v n i of ele ments v i ∈ V , is the fr ee monoid on a set V , with empt y list hi as unit and concatenation of lists as comp osition. W e shall w rite 2 for the tw o-ele ment set 2 = {− , + } of signs with negation (or inv olution) − : 2 → 2 giv en by −− = + and − + = − . Prop osition 5.4 The fr e e non-rev ersing involutive monoid on V ∈ Sets is the set (2 × V ) ⋆ of “signe d” lists, with involution: h ( b 1 , v 1 ) , . . . , ( b n , v n ) i − = h ( − b 1 , v 1 ) , . . . , ( − b n , v n ) i , wher e b i ∈ 2 and v i ∈ V . The fr e e reve rsin g involutive monoid als has (2 × V ) ⋆ as c arrier, but now with involution involving list r eversal: h ( b 1 , v 1 ) , . . . , ( b n , v n ) i − = h ( − b n , v n ) , . . . , ( − b 1 , v 1 ) i . In b oth c ases we use η ( v ) = h (+ , v ) i as insertion η : V → (2 × V ) ⋆ . Pro of. Give n an inv olutive monoid M = ( M , 1 , · , ( − ) − ) in Sets , a map f : V → M can b e extended in a unique w a y to a map of non-reve rsin g in v olutiv e monoids b f : (2 × V ) ⋆ → M , via b f  h ( b 1 , v 1 ) , . . . , ( b n , v n ) i  = f ( v 1 ) b 1 · . . . · f ( v n ) b n , where for x ∈ M w e wr ite x + = x and x − for the result of app lying M ’s inv olution ( − ) − to x . Clearly , b f preserves the un it and comp osition, and satisfies b f ◦ η = f . In the non-rev ersing case it p r eserv es the in volutio n: b f  h ( b 1 , v 1 ) , . . . , ( b n , v n ) i −  = b f h ( − b 1 , v 1 ) , . . . , ( − b n , v n ) i  = f ( v 1 ) − b 1 · . . . · f ( v n ) − b n =  f ( v 1 ) b 1  − · . . . ·  f ( v n ) b n  − =  f ( v 1 ) b 1 · . . . · f ( v n ) b n  − =  b f ( h ( b 1 , v 1 ) , . . . , ( b n , v n ) i )  − . 10 Jacobs Similarly in the reve rs in g case in vo lution is preserve d, b eca use: b f  h ( b 1 , v 1 ) , . . . , ( b n , v n ) i −  = b f h ( − b n , v n ) , . . . , ( − b 1 , v 1 ) i  = f ( v n ) − b n · . . . · f ( v 1 ) − b 1 =  f ( v n ) b n  − · . . . ·  f ( v 1 ) b 1  − =  f ( v 1 ) b 1 · . . . · f ( v n ) b n  − =  b f ( h ( b 1 , v 1 ) , . . . , ( b n , v n ) i )  − .  F or a non-reve rsin g inv olutiv e monoid M ∈ IMon ( C ) = Mon ( SC ( C )) we can consider actions either in C or in SC ( C ). The latter w ill b e called ‘inv olutiv e’ actions. Definition 5.5 F or an inv olutive monoid M ∈ IMon ( C ) = Mon ( SC ( C )) w e write IAct M ( C ) = Act M ( SC ( C )) for the category of in v olutive actions. Its ob jects are actions in SC ( C ) of the form:  M M j / /  ⊗  X X j X / /  a / /  X X j X / /  i.e. M ⊗ X ξ − 1   a / / X j X   M ⊗ X j ⊗ j X   M ⊗ X a / / X together with the usual action requirements inv olving app ropriate in teraction with the unit and multiplic ation of th e monoid M . A morphism f : ( X , j X ) → ( Y , j Y ) in IAct M ( C ) is a morph ism f : X → Y in C that is b oth a map of dualities, in SC ( C ), and of actions, in Act M ( C ). 6 In v olutions and algebras This section introdu ces inv olutions on monads, and will fo cus on algebras of su c h monads. F amiliarit y with the basics of the theory of monads will b e assumed, see e.g. [ 5 , 7 , 17 , 16 ]. Essenti ally , in v olutiv e monads are monads in the 2-category I Cat of in vol utive categ ories. W e describ e them explicitly . Definition 6.1 Let T = ( T , η , µ ) b e a monad on an in vol utive category C . W e shall call T an in vo lutive monad if T : C → C is an inv olutive functor, sa y via ν X : T ( X ) → T ( X ), and the un it η and m ultiplication µ are in vol utive natural transformations. As a r esult, ν forms a distributive law of the monad T ov er C ’s in vo lution ( − ). T h is amoun ts to: X η   η " " E E E E E E E E T 2 ( X ) µ   T ( ν ) / / T ( T ( X ) ) ν / / T 2 ( X ) µ   T ( X ) ι / / T ( ι )   T ( X ) T ( X ) ν / / T ( X ) T ( X ) ν / / T ( X ) T ( X ) ν / / T ( X ) ν O O This monad is called in vo lutive (symmetric) monoidal if T and η , µ are inv olutive (symmetric) monoidal. 11 Jacobs With r esp ect to the iden tit y in volutio n on a (symmetric monoidal) catego ry C , an y monad is in vol utive via the id en tit y distr ibutiv e la w. But the iden tit y inv olution on a category ma y still giv e rise to meaningful inv olutiv e monads, as the semir ing example b elo w sh o ws. Example 6.2 (i) Let M = ( M , m, u, j ) b e an in v olutiv e (non-rev ersing) monoid in an in vol utive catego ry C . As is well- kn own th e fu nctor M ⊗ ( − ) : C → C is a monad; its unit and m ultiplication are: X λ − 1 ∼ = / / I ⊗ X u ⊗ id / / M ⊗ X M ⊗ ( M ⊗ X ) α ∼ = / / ( M ⊗ M ) ⊗ X m ⊗ id / / M ⊗ X. Unsurp r isingly , M ’s inv olution j make s this an in vol utive monad via: ν X =  M ⊗ X j − 1 ⊗ id ∼ = / / M ⊗ X ξ ∼ = / / M ⊗ X  . (ii) Let S b e an in volutiv e comm utativ e semiring, i . e. a commutat ive semirin g with an endomap ( − ) − : S → S that is a semir ing homomorphism with s −− = s . An ob vious example is the set C of complex num b ers w ith conjugation a + ib = a − ib . Similarly , the Gaussian r ational num b ers (with a, b ∈ Q in a + ib ) form an inv olutive semiring, alb eit not a complete one. Consider the multiset m onad M S : Sets → Sets asso ciated w ith S , where we use Sets as trivial inv olutiv e catego ry , with th e ident ity as inv olution. This monad is defined on a set X as: M S ( X ) = { ϕ : X → S | sup p ( ϕ ) is fin ite } . Suc h a multiset ϕ ∈ M S ( X ) may b e written as formal sum s 1 x 1 + · · · + s k x k where supp ( ϕ ) = { x 1 , . . . , x k } and s i = ϕ ( x i ) ∈ S describ es the “multiplicit y” of the elemen t x i ∈ X . F or more in f ormation, see e.g. [ 8 ]. Th e category of algebras of this monad is the catego ry Mo d S of mo dules o v er S . This monad is monoidal / comm utativ e, because S is comm utativ e. It is inv o- lutiv e, with inv olution ν : M S ( X ) → M S ( X ) giv en by ν ( P i s i x i ) = P i s − i x i . F or an inv olutive monad T on an inv olutive category C w e can consider tw o liftings, namely of the monad T to self-dualities S C ( C ) follo wing Lemma 3.6 , or of C ’s in vo lution ( − ) to algebras Alg ( T ), as in th e follo wing t wo d iagrams. SC ( C )   SC ( T ) / / SC ( C )   Alg ( T )   ( − ) / / Alg ( T )   C T / / C C ( − ) / / C (8) The lifting on the left yields a new monad S C ( T ) b ecause lifting in Lemma 3.6 is 2- functorial. The lifting on the right arises b eca use ν in Defin ition 6.1 is a distrib utiv e la w comm uting with unit and m ultiplication. Explicitly , it is giv en by:  T ( X ) X  a / / def  T ( X ) ν X / / T ( X ) a / / X  . (9) 12 Jacobs W e shall s ometime r efer to it as the ‘conjugate’ algebra, b ecause conjugation of mo dules is an imp ortan t in s tance, see Examp le 6.5 b elo w . Prop osition 6.3 Supp ose T is an involutive mona d on an involutive c ate gory C . The c ate gory Alg ( T ) i s then also i nvolutive via ( 9 ) , and: (i) Alg ( SC ( T )) = SC ( Alg ( T )) , f or which we sometimes write IAlg ( T ) ; (ii) the c anonic al adjunction Alg ( T ) ⇆ C is an involutive one. Pro of. Definition ( 9 ) yields a new algebra b ecause ν is a distributiv e law. Th e in vo lution id Alg ( T ) ⇒ ( − ) on algebras is giv en by C ’s inv olution ι , in:  T ( X ) a / / X  ι ∼ = / /  T ( X ) X  a / / It is not hard to see that ι is a m ap of algebras. The inv olution on a morphism f of algebras is just f . F or p oin t (i) notice that on the one h and an SC ( T )-algebra is a map SC ( T )  X X j X / /  a / /  X X j X / /  whic h is a T -algebra a : T ( X ) → X that is a map of self-conjugates, usin g ( 6 ) on the left in: T ( X ) ν − 1   a / / X j X   T ( X ) T ( j X )   T ( X ) a / / X On the other h and a s elf-conjugate in Alg ( T ) consists of an algebra a with a map of the form:  T ( X ) X  a / / j X / /  T ( X ) X  a / / whic h means that j X is a map of algebras: T ( X ) ν   T ( j X ) / / T ( X ) a   T ( X ) a   X j X / / X This is clearly the same as the previous rectangle. As to the second p oin t, th e forgetful fu n ctor Alg ( T ) → C clearly commutes with in v olution. The free fun ctor F : C → Alg ( T ), mapping X to the alge br a µ : T 2 ( X ) → T ( X ), is inv olutive via the map F ( X ) → F ( X ) that is simply ν it self 13 Jacobs b y the (second) diagram in Definition 6.1 , in: T 2 ( X ) µ X   T ( ν ) / / T ( T ( X ) ) ν   T 2 ( X ) µ   T ( X ) ν / / T ( X )  In a next step we wo uld lik e to sho w that these cate gories of algebras of an in vo lutive monoidal monad are also in vo lutive monoidal categories. The monoidal structure is given b y the standard constr u ction of And ers Ko c k [ 15 , 14 ]. T ensors of algebras exist in case certain colimits exist. This is alw ays the case with m onads on sets, due to a result of Lin ton’s, see [ 5 , § 9.3, Prop. 4]. This tensor p ro duct a ⊠ b = ( T X a → X ) ⊠ ( T Y b → Y ) of a lgebras is suc h that algebra morphisms a ⊠ b → c corresp ond to b im orp hisms [ 15 , 14 ]. T h e latter can b e defined abs tr actly . Th is tensor a ⊠ b arises as co equaliser in the category Alg ( T ), of the form: T 2 ( T X ⊗ T Y ) T ( T X ⊗ T Y ) µ   ! T ( a ⊗ b ) / / µ ◦ T ( ξ ) / / T 2 ( X ⊗ Y ) T ( X ⊗ Y ) µ   ! t / / T ( X ⊠ Y ) X ⊠ Y a ⊠ b   ! (10) W e only giv e a sk etc h of the follo wing r esult. Theorem 6.4 Supp ose T is an involutive monoidal monad on an involutive monoidal c ate gory C ; assume the c ate gory Alg ( T ) of algebr as has enough c o e qualisers to make it mono idal, via ( 10 ) . The c ate gory Alg ( T ) is then also involutive monoida l, and the c anonic al adjunction Alg ( T ) ⇆ C is an involutive monoidal one. This r esult extends to symmetric monoidal structur e, an d also to clo sur e (with exp onents ⊸ ). Pro of. F or algebras T ( X ) a → X and T ( Y ) b → Y we need obtain a map of algebras ξ Alg ( T ) : a ⊠ b → a ⊠ b u sing the universal prop ert y d escrib ed ab o ve . The map ⊗ ◦ ξ C : X ⊗ Y → X ⊗ Y → X ⊠ Y is bilinear map, where ⊗ = t ◦ η : X ⊗ Y → X ⊠ Y is the un iv ersal bilinear map. Hence we obtain ξ Alg ( T ) with ξ Alg ( T ) ◦ ⊗ = ⊗ ◦ ξ C . The free algebra F ( I ) is unit for the tensor ⊠ on Alg ( T ) and comes with a map of algebras ζ Alg ( T ) = ν ◦ T ( ζ C ) : F ( I ) → F ( I ).  Example 6.5 In the conte xt of Examp le 6.2 the construction ( 9 ) giv es for an inv o- lutiv e comm utativ e semiring S an in vo lution on the category Mo d S of S -mo du les, whic h m aps a mo dule X to its conjugate space X , with the same v ectors b ut with scalar m ultiplication in X giv en by: s · X x = s − · X x , as we h a v e already seen in ( 2 ). Conjugate mo du les often o ccur in the con text of Hilb ert spaces. The category Hilb is indeed an in vol utive category , via this conju gation. Hence one can consider for instance in vo lutive monoids in Hilb . They are sometimes cal led (unital) H ∗ - algebras. W e tak e a closer lo ok at inv olutiv e monoids in ca tegories of m o dules o ver an in vo lutive semiring. Th ey come close to the notion of C ∗ -algebra. Let S b e thus 14 Jacobs b e an inv olutiv e semirin g with the asso ciated inv olutiv e category Mo d S of mo du les o v er S , lik e ab o ve. W e shall write IMo d S for the asso ciated category of involutive mo dules, whic h can b e describ ed in v arious wa ys: IMo d S = S C ( Mod S ) = SC ( Alg ( M S )) = Alg ( S C ( M S )) = IAlg ( M S ) . An inv olutive mo dule M ∈ S C ( Mod S ) thus consists of a mo du le M = ( M , + , 0 , · ) ∈ Mo d S together with an in vo lution ( − ) − : M ∼ = → M in Mo d S . This in vol ution on M p reserv es th e m on oid structur e ( x + y ) − = x − + y − and 0 − = 0, so that M is an inv olutiv e monoid (in Sets ). I ts inte raction w ith scalar m ultiplication is sp ecia l, b ecause of the conjugation M in its domain. It means that: ( s · M x ) − = s · M x − i.e. ( s · M x ) − = s − · M x − . (11) A morphism f : M → N in IMo d S is a morphism of mo dules satisfying additionally f ( x − ) = f ( x ) − . W e add that the multiset monad M S is ‘additiv e’, and so the p ro ducts × in its catego ry of algebras Mo d S are actually bip ro ducts ⊕ , see [ 8 ]. This add itivity also holds for SC ( T ), using Lemma 3.5 , so th at also IMod S has bipro d ucts ⊕ . They are preserve d by conjugation, essentiall y by Lemma 3.5 . 7 The core of the GNS- construction In this fi nal section we wish to ap p ly the th eory dev elop ed so f ar to obtain wh at can b e considered as the core of the (unital ve rsion of the) Gelfand-Naimark-Segal (GNS) construction [ 3 ], giving a bijectiv e corresp ondence b et w een states on C ∗ - algebras and certain sesqu ilinear maps . Roughly , for an inv olutive monoid A in the catego ry IMo d S , as in Example 6.5 , a state f : A → S gives rise to an inner pro du ct h− | −i : A ⊗ A → S by h a | b i = f ( a − · b ), wher e · is the multiplicat ion of the mon oid A . Notice that using the inv olution ( − ) in the d omain A ⊗ A of the inn er p ro duct giv es a neat w a y of handling conjugatio n in th e condition h s · a | b i = s − · h a | b i , where th is last · is the (scalar) m ultiplication of the semiring S (whic h is the tensor unit in Mo d S ). This ind uced inner pr o duct h a | b i = f ( a − · b ) satisfies t wo sp ecia l p r op erties that w e capture abstractly b elo w, namely: h u | −i = h− | u i and h a · b | c i = h a | b − · c i . These t wo prop erties app ear as cond itions (a) and (b) in the follo wing result. Most commonly the inner p ro duct is d escrib ed as a map p : M ⊗ M → I with the tensor unit as co domain, but the corresp ondence in the next resu lt holds for an arbitrary self-conjugate X instead of I . Theorem 7.1 L et M = ( M , m, u, j ) b e a r eversing involutive monoid in an involu- tive symmetric monoida l c ate gory (ISM C) C and let j X : X → X b e a self-c onjugate. Consider the fol low ing two pr op erties o f a map p : M ⊗ M → X . 15 Jacobs (a) Sameness when r estricte d to units: M j   ρ − 1 ∼ = / / M ⊗ I id ⊗ u / / M ⊗ M p / / X M λ − 1 ∼ = / / I ⊗ M ζ ⊗ id / / I ⊗ M u ⊗ id / / M ⊗ M p O O (b) Shifting of multiplic ations: ( M ⊗ M ) ⊗ M γ ⊗ id ∼ =   ξ ⊗ id / / ( M ⊗ M ) ⊗ M m ⊗ id / / M ⊗ M p / / X ( M ⊗ M ) ⊗ M α − 1 ∼ = / / M ⊗ ( M ⊗ M ) id ⊗ ( j ⊗ id ) / / M ⊗ ( M ⊗ M ) id ⊗ m / / M ⊗ M p O O Then ther e is a bije ctive c orr esp ondenc e b etwe en maps in SC ( C ) , M f / / X = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = M ⊗ M p / / X satisfying (a) and (b) (12) wher e M ⊗ M is pr ovide d with the “twist” c onjugate t define d as: t def =  M ⊗ M id ⊗ ι M / / M ⊗ M ξ / / M ⊗ M ι − 1 / / M ⊗ M γ / / M ⊗ M  . Pro of. V erifi cation of this corresp ond ence in v olv es man y d etails, of whic h we pr esent the essen tials. Giv en f : M → X in SC ( C ), w e define b f def =  M ⊗ M j ⊗ id / / M ⊗ M m / / M f / / X  . This b f is a map in SC ( C ) since we hav e b f ◦ t = j X ◦ b f in: M ⊗ M id ⊗ ι   j ⊗ id * * GF @A t / / ED b f BC o o M ⊗ M ξ   id ⊗ j / / M ⊗ M ξ   j ⊗ j / / γ $ $ J J J J J J J J J J M ⊗ M m   M ⊗ M ι − 1   id ⊗ j / / M ⊗ M γ % % J J J J J J J J J J ι − 1   γ % % J J J J J J J J J J M ⊗ M j ⊗ j / / ξ   M ⊗ M γ : : t t t t t t t t t t t M f   j q q M ⊗ M γ   id ⊗ j / / M ⊗ M γ   M ⊗ M ι − 1 y y s s s s s s s s s s m / / M j 9 9 t t t t t t t t t t t t ι − 1 y y s s s s s s s s s s s s X j X   M ⊗ M j ⊗ id / / @A BC b f O O M ⊗ M m / / M f / / X 16 Jacobs It is not hard to sho w that b f satisfies th e ab o ve t w o prop erties (a) and (b). Con ve rsely , giv en p : M ⊗ M → X in SC ( C ) we tak e: b p =  M λ − 1 / / I ⊗ M ζ ⊗ id / / I ⊗ M e ⊗ id / / M ⊗ M p / / X  . Using pr op ert y (a) one sho ws that b p is a map of self-dualities. Next we chec k that w e get a bijectiv e corr esp ondence ( 12 ). S tarting from f : M → X w e get b b f = f in: M λ − 1 / / V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V I ⊗ M ζ ⊗ id / / u ⊗ id , , X X X X X X X X X X X X X X X X X I ⊗ M u ⊗ id / / M ⊗ M j ⊗ id   M ⊗ M m   M f   X The v erification that b b p = p for p : M ⊗ M → X is more in v olv ed and requires prop erty ( b ), see: M ⊗ M ρ − 1 ⊗ id   M ⊗ M j ⊗ id / / λ − 1   λ − 1 ⊗ id v v l l l l l l l l l l l l l M ⊗ M m / / M λ − 1   ( I ⊗ M ) ⊗ M γ ⊗ id v v m m m m m m m m m m m m m α − 1 / / ( ζ ⊗ id ) ⊗ id   I ⊗ ( M ⊗ M ) id ⊗ ( j ⊗ id ) / / ζ ⊗ id   I ⊗ ( M ⊗ M ) id ⊗ m / / I ⊗ M ζ ⊗ id   ( M ⊗ I ) ⊗ M ρ ⊗ id   (id ⊗ ζ ) ⊗ id % % I ⊗ ( M ⊗ M ) u ⊗ id   I ⊗ M u ⊗ id   M ⊗ ( M ⊗ M ) α   id ⊗ ( j ⊗ id ) / / M ⊗ ( M ⊗ M ) id ⊗ m / / M ⊗ M p   ( I ⊗ M ) ⊗ M γ ⊗ id   ( u ⊗ id ) ⊗ id / / ( M ⊗ M ) ⊗ M γ ⊗ id   ( M ⊗ I ) ⊗ M ξ ⊗ id   (id ⊗ u ) ⊗ id / / ( M ⊗ M ) ⊗ M ξ ⊗ id   ( b ) M ⊗ M ρ − 1 ⊗ id / / ( M ⊗ I ) ⊗ M (id ⊗ u ) ⊗ id / / ( M ⊗ M ) ⊗ M m ⊗ id   M ⊗ M p / / X Remaining details are left to the reader.  As said, this result only captures the heart of th e GNS construction [ 3 ]. 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