Quotients of the Multiplihedron as Categorified Associahedra
We describe a new sequence of polytopes which characterize A_infinity maps from a topological monoid to an A_infinity space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Later term(s) in our sequence of polytop…
Authors: Stefan Forcey
QUOTIENTS OF THE MUL TIPLIHEDRON AS CA TE GORIFIED ASSOCIAHEDR A STEF AN F ORCEY Abstract. W e describ e a new sequence of p oly top es which characterize A ∞ maps from a top ologic a l monoid to an A ∞ space. Therefore ea ch of these p olytop es is a quotient of the corresp onding m ultiplihedron. Our sequence of p olytop es is demonstr a ted no t to be co mb inatorially equiv alen t to the asso cia hedra, a s was prev iously assumed in bo th topolo g ical and categorical liter ature. They are giv en the new collective na me comp osihedra. W e po int out how these p olytop es are used to para meterize co mpo sitions in the formulation of the theories o f enriched bica tegories and pseudo monoids in a monoidal bicategory . W e also present a simple algor ithm for determining the e x tremal po int s in Euc lidea n space whos e conv ex h ull is the n th po lytop e in the sequence of comp osihedra, that is, the n th comp osihedron C K ( n ). x x x x q q q q q & & & & M M M M M , , , , z z z z z D D D D D + + + + + + + y y y y y y y y E E E E E E E E L L L L , , , , , y y y y y y y y o o o O O O E E E E E E E E * * * * r r r r / / / O O O O O O o o o o o o O O O O O O o o o o o o e e e e e e Y Y Y Y Y Y & & & & M M M M M , , , , , , , y y y y y y y y E E E E E E E E 9 9 9 9 9 9 y y y y y y y y o o o O O O E E E E E E E E , , , , O O O O O O o o o o o o N N N N N N N N N p p p p p p p p p x x x x q q q q q , , , , z z z z D D D D k k k k k k k k k S S S S S S S S S , , , , , n n n n n n n n n P P P P P P P P P ) ) ) ) ) ) ) O O O O O O O o o o o o o o e e e e e e e Y Y Y Y Y Y Y s s s s s s s s s s K K K K K K K K K K N N N N N N N N N p p p p p p p p p r r r r r r r r r r L L L L L L L L L L r r r r r r r r r r L L L L L L L L L L p p p p p p p p p p N N N N N N N N N N Figure 1: The ca st of character s. Left to right: C K (4), J (4), 3- d cub e, a nd K (5 ). Contents 1. In tro duction 2 2. P olytopes in top ology a nd categories 4 2.1. Asso ciahedron 4 2.2. Multiplihedron 5 2.3. Quotien ts o f the multiplihe dron 7 2.4. Enric hed bicategories and t he comp osihedron 8 3. Recursiv e definition 12 4. V ertex Com binato rics 17 Key wor ds and phr ases. enriched categorie s , n-categories , monoidal categorie s , poly to pe s . Thanks to X Y -pic for the diagr ams. 1 5. Con v ex hull realizations 19 6. Pro of o f Theorem 5.2 23 References 28 1. Intro duction Categorification, as describ ed in [ 1 ], refers to the pro cess o f creating a new mathemati- cal theory by 1 ) c ho osing a demonstrably useful concept that y ou understand fairly w ell, 2) replacing some of the sets in it s definition with catego ries, and 3) replacing some of the inte resting equalities with morphisms. The asso ciahedra are a sequence of p olytopes, in v ented by Stasheff in [ 39 ], whose face p os et is defined to corresp ond to brac ketings of a giv en list of elemen t s from a set. If instead w e ta ke lists of elemen ts from a category , and define a sequence of p olytop e s whose faces corresp ond to either brack etings or to certain maps in that category , t hen we ma y naiv ely describ e our new definition as a categorified v ersion o f the asso ciahedra. Of course there ma y b e man y in teresting w ay s to mak e this idea precise. Here we fo cus o n just one, whic h a rises na t ur a lly in the study of bo th top ological monoids and category theory . Categorification of the concept of category itself can b e ac hiev ed b y replacing sets of morphisms (and/or sets of ob jects) b y categories. This implies considering enric hed (or in ternal) catego r ies of Cat , the category of categories. There is ro om for the comp osition la ws in enric hed or in ternal categories in Cat to b e w eaken ed. T o ac hiev e this in a coheren t wa y the familiar definitions of bi- and tricategory utilize the Stasheff p olytop es, or asso ciahedra, as the underlying shap es of axiomatic commu t ing diagrams. A parallel categorification of enric hed categories creates a definition allowing the hom- ob jects to b e lo cated in a monoidal bicategory W . This is kno wn as the the ory of enric hed bicategories. Rather than the asso ciahedra, it is a new se quence of p olytopes whic h arises in the corresp onding coherence axioms of enric hed bicategories. This new sequence comprises the categorified v ersion of the asso ciahedra whic h w e will b e studying. S ince they app ear in the comp osition laws of enriche d categories w e ha v e c hosen to refer to them collectiv ely as the comp osihedra, or singulary as t he n th comp osihedron, denoted C K ( n ). The capital “ C ” stands for “comp ose”, or for “categorify ,” or for “ cone”– the last since the n th comp osihedron can b e seen a s b eing a sub division of the top o logical cone of the n th asso ciahedron. Indeed the p olytop e C K ( n ) is o f dimension n − 1, while the p olytop e K ( n ) is of dimension n − 2 . D ecomp ositions o f the b oundaries of the earliest terms in our new sequence of p olytop es ha ve b een seen b efore, in t he classic sources on enric hed categories suc h as [ 22 ], the definition of enriched bicategories in [ 10 ], and in the single-ob ject v ersion of the latter: pseudomonoids as defined in [ 35 ]. The other half of our title r efers to the fact that the p olytop es we study here also arise in the study of A n and A ∞ maps. The m ultiplihedra w ere in v ente d by Stasheff, describ ed b y Iw ase and Mim ura, and generalized b y Boardman and V ogt. They represen t the fundamen tal structure of a w eak map b etw een w eak structures, suc h as w eak n -categories or A n spaces. They for m a bimo dule ov er the asso ciahedra, and collapse under a quotien t map to b ecome the asso ciahedra in the sp ecial case of a strictly asso ciative monoid as range. In the case of a strictly asso ciative monoid as do main the multiplihedra collapse to form our new family of polytop es. This is pictured ab ov e in Fig ure 1, in dimension 2 3. The multiplihedron is at the top, comp osihedron at left, associahedron on the righ t, and the cub e which results in the case of b oth strict domain and range structures at the b ottom. In sec t io n 2 w e briefly review the app earance of p olytop e sequen ces in top ology and category theory . In section 3 we pr ovide a complete recursiv e definition of o ur new p oly- top es, as w ell as a description of them as quotien ts of the multiplihedra. In section 4 w e go o v er some basic comb ina torial results ab out the compo sihedra. In se ction 5 w e presen t an alternative definition of the comp osihedra a s a con v ex hull, based up on the conv ex h ull realization of the multiplihedra in [ 13 ] and whic h reflects the quotien ting pro cess. In section 6 we pro ve that the con v ex h ull defined in section 5 is indeed com bina t o rially equiv alen t to the complex defined in section 3 . A w ord of in tro duction is appropriat e in r ega rd to the conv ex h ull algorithm in sec- tions 5 a nd 6 . In the pap er on the multiplihedra [ 13 ] w e describ ed ho w to represen t Boardman and V ogt’s spaces of painted trees with n lea ve s as conv ex p o lytop es whic h are combinatorially equiv alent to the CW-complexes describ ed b y Iw ase and Mim ura. Our algorithm for the v ertices of the p olytop es is flexible in that it allows an initial c hoice of a c onstan t q betw een zero and one. In the limit as q → 1 the conv ex h ull approac hes that of Lo day’s con ve x hull represen tation of the associahedra as describ ed in [ 28 ]. The limit as q → 1 corresponds to the case for whic h the mapping is a homo- morphism in that it strictly respects the m ultiplication. The limit as q → 0 represen ts the case fo r which multiplication in the domain of the morphism in ques tio n is strictly asso ciativ e. In the limit as q → 0 the con ve x hulls of the m ultiplihedra approach our newly disco vered sequence of p olytop es, the comp osihedra. There are t w o pro jects f o r the future that are supp o rted by this w ork. One is to mak e rigorous the implication that enric hed bicategories ma y be exemplified b y certain maps of top ological monoids. It could b e hop ed that if this endea v or is successful that A ∞ categories and their maps might also b e amenable to the same approac h, yielding more in teresting examples. The other pro ject already underw ay is to extend the concept of quotien t multiplihe dra described here to t he graph asso ciahedra introduced by Carr and Dev adoss, in [ 11 ]. There is a lso a philosophical conclusion t o b e argued from the results of this w or k. Historically , w eak 2 a nd 3-categories were defined using the asso ciahedra, whic h form an op erad of top ological spaces. T hen the op erad structure w as tak en as fundamen tal in man y of the functioning definitions of w eak n -category , as describ ed in [ 26 ] and [ 27 ]. Again historically , weak maps of bi- and tri-categories w ere defined using the mu ltipli- hedra, whic h form a 2-sided operad bimo dule ov er the asso ciahedra. M o re recen tly the op erad bimodule structure has b een used to define w eak maps of w eak n -catego ries, in [ 19 ] and [ 5 ]. Th us the facts t hat the comp osihedra are used for defining enric hed cat- egories and bicategories, and that they form an op era d bimo dule (left-mo dule o ver the asso ciahedra a nd righ t-mo dule ov er the a sso ciativ e op erad t ( n ) = ∗ ) lead us to prop ose that enric hing ov er a we a k n -category should in general b e accomplishe d by us e of operad bimo dules as w ell. The philosophy here is that the structure o f a bimo dule will tak e into accoun t the w eakness of the base of enric hmen t, (where a w eakly associative pro duct is used to form the domain for comp osition) as w ell as providing for t he w eakness of the enric hed comp osition itself. 3 2. Pol ytopes in topology and ca tegories Here we review the app eara nce of fundamen tal families of p o lytop es in the axioms of higher dimensional category theory . In b oth top olo gy a nd in category theory , the use of these p olytop es has prov en to b e a source of imp ortan t clues rather than the final solution. The algebraic structure of the p olytop e sequence is more imp o rtan t than its com binatorial structure, although certainly one dep ends on the other. Th us the operad structure on the asso ciahedra can b e seen as foreshadowing the in tuition fo r the use of actions of the operad of little n -cubes t o recognize lo op spaces, as we ll as the use o f n -op erad a ctio ns in Bat anin’s definition of n - cat ego ry [ 5 ]. 2.1. Asso ciahedron. The asso ciahedra are the famous seque nce of p olytop es denoted K ( n ) from [ 39 ] which characterize the structure o f w eakly asso ciativ e pro ducts. K (1) = K (2) = a single p oin t , K (3) is the line segmen t, K (4) is the p en tago n, and K (5) is the follo wing 3d shap e: K (5) = P P P P P P P P P . . . . . . . . . n n n n n n n n n P P P P P P P P P . . . . . . . . . n n n n n n n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? K K K K K K K K K s s s s s s s s s s s s s s s s s s K K K K K K K K K The o riginal examples of w eakly associative pro duct structure are the A n spaces, top o - logical H -spaces with w eakly asso ciative m ultiplication of p oints . Here “w eak” should b e understo o d as “up to homoto p y .” That is, there is a path in the space from ( ab ) c to a ( bc ) . An A ∞ -space X is c haracterized b y its admission of an action K ( n ) × X n → X for all n. Categorical example s b egin with the monoidal categories as defined in [ 29 ], where there is a w eakly asso ciativ e tensor pro duct of ob jects. Here “weak” officially means “naturally isomorphic.” There is a natural isomorphism α : ( U ⊗ V ) ⊗ W → U ⊗ ( V ⊗ W ) . Recall that a mo n oidal c ate go ry is a category V together with a f unctor ⊗ : V × V → V suc h that ⊗ is asso ciative up to the coheren t natura l isomorphism α . The coherence axiom is giv en b y a commuting p en ta g on, which is a cop y of K (4) . (( U ⊗ V ) ⊗ W ) ⊗ X α U V W ⊗ 1 X / / α ( U ⊗ V ) W X | | y y y y y y y y y y y y y y y y y y ( U ⊗ ( V ⊗ W )) ⊗ X α U ( V ⊗ W ) X " " E E E E E E E E E E E E E E E E E E ( U ⊗ V ) ⊗ ( W ⊗ X ) α U V ( W ⊗ X ) R R R R R R R R R R R R R ( ( R R R R R R R R R R R R R U ⊗ (( V ⊗ W ) ⊗ X ) 1 U ⊗ α V W X l l l l l l l l l l l l l v v l l l l l l l l l l l l l U ⊗ ( V ⊗ ( W ⊗ X )) 4 Monoidal categories can b e view ed as single ob ject bicatego r ies. In a bicategory eac h hom( a, b ) is lo cated in Cat rat her than Set . The comp osition o f morphisms is not exactly asso ciativ e: there is a 2-cell σ : ( f ◦ g ) ◦ h → f ◦ ( g ◦ h ) . This asso ciator ob eys t he same p en tago nal commuting diagram as for monoidal categories, as seen in [ 24 ]. Another iteration of categorification results in the t heory of tricategories. The se ob ey an a xiom in which a comm uting pasting diagram has the underlying form of the asso ci- ahedron K (5) , as not iced b y the authors of [ 16 ]. The term “co cycle condition” fo r this axiom w as p opularized b y Ross Street, and its connections to homology ar e describ ed in man y of his pap ers, including [ 42 ]. The pattern con tinues in T rim ble’s definition of tetra- categories, where K ( 6) is found as the underlying structure of the corresp onding co cycle condition. The asso ciahedra are also seen as the classifying spaces of certain categories of trees, as illustrated in [ 24 ], and as the foundation fo r the free strict ω -categories defined in [ 42 ]. The asso ciahedra are also the starting p oin t for defining the one-dimensional analogs of the full n -categorical comparison of delo oping and enrichme nt. One dimensional we a k- ened v ersions of enric hed categories hav e b een w ell-studied in the field of differential graded algebras and A ∞ -categories, the man y ob ject generalizations of Stasheff ’s A ∞ - algebras [ 39 ]. An A ∞ -category category is basically a category “w eakly” enric hed o v er c hain complexes of mo dules, where the w eak ening in this case is accomplished b y sum- ming the comp osition c ha in maps to zero (rather than by requiring commuting diagr a ms). It is also easily described a s an algebra o v er a certain op era d. 2.2. Multiplihedron. The complexes now known as the m ultiplihedra J ( n ) we r e first pictured b y Stasheff, for n ≤ 4 in [ 40 ]. They w ere in tro duced in order to approach a f ull description of the catego ry of A ∞ spaces b y pro viding the underlying structure for morphisms whic h preserv ed the structure of the domain space “up to homotopy” in the ra nge. Recall that an A ∞ space itself is a monoid only “up to homotop y ,” and is recognized by a contin uous action of the a sso ciahedra as describ ed in [ 39 ]. Thus the m ultiplihedra a re used to c haracterize t he A ∞ -maps. A map f : X → Y b etw een A ∞ - spaces is an A ∞ -map if t here exists an action J ( n ) × X n → Y for all n , whic h is equal to t he action of f for n = 1 , and ob eying asso ciativit y constraints as describ ed in [ 40 ]. Stasheff describ ed ho w to construct the 1 - sk eleton of these complexes in [ 40 ], but stopp ed short o f a full combin atorial description. Iw a se and Mim ura in [ 20 ] giv e the first detailed definition of the sequenc e of complexes J ( n ) now know n as the m ultiplihedra, and describe their com binatorial prop erties. Spaces of pain ted trees we r e first introduced b y Boardman and V ogt in [ 9 ] to help describe m ultiplication in (and mor phisms of ) top ological monoids that are not strictly asso ciativ e (and whose morphisms do no t strictly respect that m ultiplication.) T he n th m ultiplihedron is a C W -complex whose v ertices cor r esp o nd to the unam biguous w ays of m ultiplying and applying an A ∞ -map to n ordered elemen ts of an A ∞ -space. Th us the v ertices corresp ond to the binary pain ted trees with n lea ve s. In [ 13 ] a new realization of the multiplih edra ba sed up on a map from these binary painted trees to Euclidean space is used to unite the approac h to A n -maps o f Stasheff, Iw ase and Mim ura to that of Boardman and V ogt. 5 Here are the first few lo w dimensional multiplihe dra . The v ertices are lab eled, all but some of those in the last picture. There the b o ld ve rtex in the large p en tagonal facet has lab el (( f ( a ) f ( b )) f ( c )) f ( d ) and the b old v ertex in the small p en ta g onal facet has lab el f ((( ab ) c ) d ) . The others can b e easily dete rmined based on the f a ct that those t wo p en tago ns are copies o f the asso ciahedron K (4) , that is to say all their edges are asso ciations. J (1 ) = • f ( a ) J (2 ) = f ( a ) f ( b ) f ( ab ) • • J (3 ) = ( f ( a ) f ( b )) f ( c ) • f ( a )( f ( b ) f ( c )) • / / / / / f ( a ) f ( bc ) / / / / / / • f ( a ( bc )) • f (( ab ) c ) • / / / / / f ( ab ) f ( c ) / / / / / / • J (4 ) = • • f ( a )( f ( bc ) f ( d )) ( f ( a ) f ( bc )) f ( d ) f ( a ( bc )) f ( d ) f (( ab ) c ) f ( d ) ( f ( ab ) f ( c )) f ( d ) f ( ab )( f ( c ) f ( d )) f ( ab ) f ( cd ) ( f ( a ) f ( b )) f ( cd ) f ( a )( f ( b ) f ( cd )) f ( a )( f ( b ( cd ))) f ( a ) f (( bc ) d ) , , , , , , , , z z z z z z z z z z z D D D D D D D D D D D + + + + + + + + + + + + + + + y y y y y y y y y y y y y y y y y y E E E E E E E E E E E E E E E E E E L L L L L L L L , , , , , , , , , , , y y y y y y y y y y y y y y y y y y o o o o o o O O O O O O E E E E E E E E E E E E E E E E E E * * * * * * * * * r r r r r r r r / / / / / / O O O O O O O O O O O O o o o o o o o o o o o o O O O O O O O O O O O O O o o o o o o o o o o o o o e e e e e e e e e e e e e Y Y Y Y Y Y Y Y Y Y Y Y Y The multiplihe dra also app ear in higher category theory . The definitions of bicategory and tricategory homomorphisms each include comm uting pasting diagrams as seen in [ 24 ] and [ 16 ] resp ectiv ely . The tw o halve s of the axiom for a bicategory homomorphism together fo rm the b oundary of the multiplihedra J (3) , and the t w o halv es of t he axiom for a tricategory homomorphism together for m the b oundary of J (4) . Sinc e w eak n - categories can b e understo o d as b eing the algebras of higher o p erads, these facts can b e 6 seen as the motiv ation for defining morphisms of op erad (and n -op erad) algebras in terms of their bimo dules. This definition is men tioned in [ 5 ] and dev elop ed in detail in [ 1 9 ]. In the la t t er pap er it is p ointed out that the bimo dules in question m ust b e co-rings, which ha ve a co-multiplication with respect to the bimo dule pro duct o v er the op erad. The m ultiplihedra ha v e also app eared in sev eral areas related to deformation theory and A ∞ category theory . A diagonal map is constructed for these p olytop es in [ 36 ]. This allo ws a functorial monoidal structure for certain categories of A ∞ -algebras and A ∞ - categories. A differen t, p ossibly equiv alen t, v ersion of the diagonal is presen ted in [ 31 ]. The 3 dimens iona l v ersion of the mu ltiplihedron is called b y the name Chinese lan tern diagram in [ 46 ], and used to describ e deformation of functors. There is a forthcoming pap er by W o o dw ar d and Mau in whic h a new realization o f the m ultiplihedra as mo duli spaces of disks with additional structure is presen ted [ 33 ]. This realization allo ws the authors to define A n -functors a s w ell as morphisms of cohomolo gical field theories. 2.3. Quotien ts of the m ultiplihedron. The sp ecial m ultiplihedra in the case for whic h m ultiplication in the range is strictly asso ciative w ere found by Stasheff in [ 40 ] to be precise ly the asso ciahedra. Sp ecifically , the quotient of J ( n ) under the equiv- alence generated b y ( f ( a ) f ( b )) f ( c ) = f ( a )( f ( b ) f ( c )) is com binatorially equiv alent to K ( n + 1) . This pro jection is pictured on the rig ht hand side of Figure 1 , in dimen- sion 3. Recall that the edges o f the m ultiplihedra corresp ond to either an asso ciation ( ab ) c → a ( bc ) o r to a preserv atio n f ( a ) f ( b ) → f ( ab ) . The asso ciations can either be in the range: ( f ( a ) f ( b )) f ( c ) → f ( a )( f ( b ) f ( c )); or the image of a domain asso ciation: f (( ab ) c ) → f ( a ( bc )) . It was long assumed that the case fo r whic h the domain w as asso ciative w o uld lik ewise yield the asso ciahedra, but we will demonstrate otherwise. The n th comp osihedron ma y b e describ ed as the quotient of the n th m ultiplihedron under the eq uiv alence generated b y f (( ab ) c ) = f ( a ( bc )). Of course this is implied b y associativity in the domain, where ( ab ) c = a ( bc ) . W e will t a k e this latter view throughout, but w e note that there ma y b e inte r esting functions for whic h f (( ab ) c ) = f ( a ( bc )) ev en if the domain is not strictly asso ciativ e. Here a re the first few comp o sihedra with ve rt ices lab eled a s in the multiplihe dra , but with the assumption that the do main is asso ciative. (F or these pictures the lab el actually app ears ov er the vertex .) Notice how the n - dimensional comp o sihedron is a sub divided top ological cone on the ( n − 1)- dimensional asso ciahedron. The Sc hlegel diagram is sho wn for C K (4) , view ed with a copy of K (4) as the p erimeter. C K (1) : f ( a ) C K (2) : f ( a ) f ( b ) f ( ab ) C K (3) : ( f ( a ) f ( b )) f ( c ) p p p p p p f ( a )( f ( b ) f ( c )) N N N N N N f ( ab ) f ( c ) V V V V V V V V V V V V V V f ( a ) f ( bc ) h h h h h h h h h h h h h h f ( abc ) 7 C K (4) : (( f ( a ) f ( b )) f ( c )) f ( d ) P P P P P P P P ( f ( a )( f ( b ) f ( c ))) f ( d ) 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 n n n n n n n n ( f ( ab ) f ( c )) f ( d ) P P P P P P P P n n n n n n n n ( f ( a ) f ( bc )) f ( d ) n n n n n n n n @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ f ( ab )( f ( c ) f ( d )) O O O O O O O O O f ( abc ) f ( d ) ( f ( a ) f ( b ))( f ( c ) f ( d )) m m m m m m m m m m N N N N N N N N N N K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K f ( ab ) f ( cd ) f ( abcd ) f ( a )( f ( bc ) f ( d )) q q q q q q q q q q q f ( a )(( f ( b ) f ( c )) f ( d )) u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u ( f ( a ) f ( b )) f ( cd ) q q q q q q q q q V V V V V V V V V V V V V V V V V V V f ( a ) f ( bcd ) f ( a )( f ( b ) f ( cd )) f ( a )( f ( b )( f ( c ) f ( d ))) An y confusion can pr o bably b e traced to the f act that the t wo sequenc es o f p olytop es are iden tical for the first f ew terms. T hey div erge first in dimension three, at whic h dimension the asso ciahedron has 9 fa cets and 14 v ertices, while the comp osihedron has 10 fa cets and 15 v ertices. Another similarit y at this dimension is that b oth p olytop es ha ve exactly 6 p en tag onal f acets; the difference is in the n um b er of quadrilateral facets. The difference is also clear fro m the fact that the ass o ciahedra ar e simple p olytop es, whereas starting at dimension three the comp osihedra are not . Here a re easily compared pictures of the asso ciahedron and comp osihedron in dimen- sion 3. K (5) = n n n n n n n n n n n n n n . . . . . . . . . . . . . . P P P P P P P P P P P P P P n n n n n n n n n n n n n n . . . . . . . . . . . . . . P P P P P P P P P P P P P P ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? X X X X X X X X X X X X X X X X X X X X f f f f f f f f f f f f f f f f f f f f C K (4) = n n n n n n n n n n n n n n . . . . . . . . . . . . . . P P P P P P P P P P P P P P n n n n n n n n n n n n n n . . . . . . . . . . . . . . P P P P P P P P P P P P P P ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? X X X X X X X X X X X X X X X X X X X X & & & & & & & & & & & & & & & & & & & & J. Stasheff points out that w e can obtain the complex K (5) by deleting a single edge of C K (4) . 2.4. Enriche d bicategories and the comp osihedron. Remark ably , the same minor error of recognition b et w een the asso ciahedron and comp osihedron ma y ha ve b een made b y category theorists who wrote do wn the coherence axioms of enriche d bicategory theory . The first few p o lyto p es in our new sequence corresp ond to co cyle coherence conditions in 8 the definition of enric hed bicategories as in [ 10 ]. It is incorrectly implied there that the final co cycle conditio n has the com binat o rial form of t he asso ciahedron K (5). The axiom actually consists of t w o pasting diagrams whic h when glued along their b oundary are seen to form the comp osihedron C K (4) instead. Also note that the first few comp osihedra corresp ond as we ll to the axioms fo r pseudomonoids in a mono idal bicategory as seen in [ 35 ]. This fact is to b e expected, since pseudomonoids are j ust single ob ject enriched bicategories. Little has been published ab out enriche d bicategories, although the theory is used in recen t researc h pap ers suc h as [ 14 ] and [ 3 8 ]. The full definition of enric hed bicategory is w orked out in [ 10 ]. It is rep eated with the simplification of a strict monoidal W in [ 23 ], in whic h case the comm uting diagrams ha v e the form of cub es. (Recall that when b oth the range a nd domain are strictly asso ciative that the m ultiplihedra collapse to b ecome the cub es, as sho wn in [ 9 ].) An ear lier definition of (lax) enric hed bicategory as a lax functor of certain tricategories is found in [ 16 ] and is also review ed in [ 23 ]; in retrosp ect this form ulatio n is to b e exp ected since the comp osihedra are sp ecial cases of m ultiplihedra. Here f or reference is the definition of enric hed bicategories, closely following [ 10 ]. Let ( W , ⊗ , α , π , I ) b e a mono ida l bicategory , as defined in [ 16 ] or in [ 10 ], or comparably in [ 2 ], with I a strict unit (but w e will include the iden t it y cells in our diagrams). Note that item (5) in the following definition corrects an obvious t yp o in the corresp onding item of [ 10 ], and that the 2 - cells in item (6) ha ve b een some what rearranged from t ha t source, for easier comparison to the p o lytop e C K (4). 2.1. Definition. An enric hed bicatego ry A o v er W is: (1) a collection of ob jects Ob A , (2) hom-ob jects A ( A, B ) ∈ Ob W for eac h A, B ∈ O b A , (3) comp osition 1- cells M AB C : A ( B , C ) ⊗ A ( A, B ) → A ( A, C ) in W f o r eac h A, B , C ∈ Ob A , (4) an iden tity 1-cell J A : I → A ( A, A ) fo r eac h ob ject A, (5) 2-cells M 2 in W f o r eac h A, B , C , D ∈ Ob A : ( A ( C, D ) ⊗ A ( B , C )) ⊗ A ( A, B ) α / / M ⊗ 1 t t h h h h h h h h h h h h h h h h A ( C, D ) ⊗ ( A ( B , C ) ⊗ A ( A, B )) 1 ⊗ M * * V V V V V V V V V V V V V V V V A ( B , D ) ⊗ A ( A, B ) M , , Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z M 2 + 3 A ( C, D ) ⊗ A ( A, C ) M r r d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d A ( A, D ) (6) ...whic h ob ey the follo wing co cycle condition fo r eac h A, B , C , D , E . In the fol- lo wing w e abbreviate the hom-ob jects and comp osition b y M : B C , AB → AC . 9 D E , ( C D , ( BC , AB )) ( DE , C D ) , ( B C, AB ) (( DE , C D ) , B C ) , AB ( C E , B C ) , AB B E , AB AE D E , AD D E , ( C D , AC ) D E , (( C D , B C ) , AB ) ( DE , ( C D , B C )) , AB ( DE , B D ) , AB D E , ( B D , AB ) M 2 ⇑ M 2 ⇑ M 2 ⇑ π ⇑ = 6 6 n n n n n n n n n n n n n n n n n n H H . . . . . . . . . . . . . . . . . . . . . ( ( P P P P P P P P P P P P P P P P P P P 6 6 n n n n n n n n n n n n n n n n n n n . . . . . . . . . . . . . . . . . . . . . ( ( P P P P P P P P P P P P P P P P P P ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? O O / / / / = D E , ( C D , ( BC , AB )) ( DE , C D ) , ( B C, AB ) (( DE , C D ) , B C ) , AB ( C E , B C ) , AB B E , AB AE D E , AD D E , ( C D , AC ) ( DE , C D ) , AC C E , A C C E , ( B C , AB ) M 2 ⇑ M 2 ⇑ = = = 6 6 n n n n n n n n n n n n n n n n n n H H . . . . . . . . . . . . . . . . . . . . . ( ( P P P P P P P P P P P P P P P P P P P 6 6 n n n n n n n n n n n n n n n n n n n . . . . . . . . . . . . . . . . . . . . . ( ( P P P P P P P P P P P P P P P P P P ? ? ? ? ? ? ? ? ? ? ? + + X X X X X X X X X X X X X X X X X X X X X X X X X X / / ? ? & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & 10 (7) Unit 2-cells: I ⊗ A ( A, B ) = ) ) T T T T T T T T T T T T T T T T J B ⊗ 1 A ( A, B ) ⊗ I 1 ⊗J A = u u j j j j j j j j j j j j j j j λ ⇑ A ( A, B ) ρ ⇑ A ( B , B ) ⊗ A ( A, B ) M ABB j j j j j 5 5 j j j j j A ( A, B ) ⊗ A ( A, A ) M AAB T T T T T i i T T T T T (8) ... whic h ob ey a pasting condition of their ow n, whic h w e will omit for brevit y . If instead of the existence of 2- cells p ostulated in (5) and (7 ) w e had required tha t the diag rams comm ute, w e would reco ve r the definition of a n enric hed catego ry . Note that the p en tago na l axiom for a monoidal categor y at the b eginning o f this section has the form of the asso ciahedron K (4) but the commuting p en tagon for enric hed categories (here the domain and range of M 2 ), is actually better des crib ed as ha ving the for m of C K (3) . Here is the co cycle condition (6), pasted together and show n as a Sc hlegel diagram for comparison to the similarly displa y ed picture ab o ve of C K (4) . T o sa ve space “ • • → • ” will represen t M 1 : A ( B , C ) ⊗ A ( A, B ) → A ( A, C ). (( • • ) • ) • / / ! ! C C C C C C C ( • ( • • )) • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } } { { { { { { { ( • • ) • ! ! C C C C C C C C } } { { { { { { { M 2 = ⇒ ( • • ) • } } { { { { { { { 2 2 2 2 2 2 2 2 2 2 2 2 2 • ( • • ) " " D D D D D D D M 2 ⇓ • • M 2 ( • • )( • • ) 9 9 r r r r r r r r % % L L L L L L L L B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B • • / / • • ( • • ) v v l l l l l l l l l l l l l • (( • • ) • ) o o ~ ~ } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } ( • • ) • < < z z z z z z z ( ( R R R R R R R R R R R M 2 ⇓ • • O O M 2 • ( • • ) O O • ( • ( • • )) O O F rom this p ersp ectiv e it is easier to visualize how, in the case that W = Cat , an enric hed bicategory is just a bicategory . In that case M 2 is renamed σ , a nd the enric hed co cycle axiom b ecomes the ordinary bicategory co cycle axiom with the underlying form of K (4) . Th us the n th comp osihedron can b e seen to contain in its structure tw o copies o f the n th asso ciahedron: one copy as a particular upp er fa cet and another as a certain collection of fa cets that can function as lab els f o r the facets of t he corresp onding asso ciahedron. Of course the second cop y is only seen up on decategorification! 11 3. Recursive definition In [ 20 ] the authors giv e a geometrically defined C W -complex definition of the multi- plihedra, and then demonstrate the recursiv e com binat orial structure. Here we describ e ho w to collapse that structure for the case of a strictly asso ciativ e domain, a nd ac hiev e a recursiv e definition of t he comp osihedra. Pictures in the fo r m of p ainte d bin a ry tr e es can b e dra wn to represen t the m ultiplicatio n of sev eral o b jects in a monoid, b efore or after their passage to the imag e of tha t monoid under a homomorphism. W e use the t erm “painted ” rather than “colored” to distinguish our trees with t wo edge colorings, “painted” and “unpain ted,” from the other meaning of colored, as in colored op erad or m ulticategory . W e will refer to the exterior v ertices of the tree as the ro ot and the lea v es , and to the in t erio r v ertices as no des. This will b e handy since then w e can reserv e the term “v ertices” for reference to p olytop es. A pain ted binary tree is pain t ed b eginning at the ro ot edge (the leaf edges a r e unpain ted), and alwa ys painted in such a w a y that there are only three t yp es of no des. T hey a r e: • • (1) (2) • (3) / / / / / / / / / / / / / / / / / / / / / / / / This limitation on no des implies that pain ted regions m ust b e connected, that pa inting m ust nev er end precisely at a tr iv alen t no de, and that pain ting m ust pro ceed up b oth branc hes of a triv alen t no de. T o see the pro mised represen tation w e let the left-hand, t yp e (1) triv alen t no de ab ov e stand for m ult iplicatio n in the domain; the middle, painte d, t yp e (2) triv alen t no de ab ov e stand for m ultiplication in the range; and t he right-hand t yp e ( 3 ) biv alen t no de stand f or t he action of the mapping. F or instance, given a, b, c, d elemen ts of a monoid, and f a monoid morphism, the follo wing diagra m represen ts the op eration resulting in the pro duct f ( ab )( f ( c ) f ( d )) . a b c d • • • f ( ab )( f ( c ) f ( d )) • • • / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / T o define the face p oset structures of the multiplihe dr a and comp osihedra w e need pain ted trees that are no longer binary . Here are the three new t yp es of no de allow ed in a general painted tr ee. They correspond to the the no de t yp es (1), (2) a nd (3) in that they are pain ted in similar fashion. They generalize t yp es (1), (2 ) , a nd (3 ) in that each 12 has greater or equal v alence than the corresp onding earlier no de t yp e. . . . . . . • • (4) (5) . . . • (6) ? ? ? ? ? ? ? ? ? ? * * * * * * * ? ? ? ? ? ? ? ? ? ? * * * * * * * / / / / / / / / ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? * * * * * * * * * * * * * * 3.1. Definition. By r efinement of pain ted trees w e refer to the relatio nship: t refines t ′ means that t ′ results from the collapse of some of the internal edges of t . This is a partial order o n n -leav ed pain ted trees, and w e write t < t ′ . Thus the binary pain ted trees are refinemen ts of t he trees ha ving no des of t yp e (4)-(6). Minimal r efinement refers to the follo wing sp ecific case of refinemen t: t minimally refines t ′′ means that t refines t ′′ and also that there is no t ′ suc h that b ot h t refines t ′ and t ′ refines t ′′ . The p oset of pain ted trees with n leav es is precisely the face p oset of the n th m ultipli- hedron. 3.2. Definition. Tw o painted tr ees are said to b e domain e quivalent if tw o requiremen ts are satisfied: 1) they b oth refine the same tree, and 2) the collapses in v o lv ed in b oth refinemen ts are of edges whose t wo no des are of ty p e (1) or type ( 4 ). That is, these collapses will b e of in ternal unpain ted edges with no a dj a cen t pain ted edges. Lo cally the equiv alences will a pp ear as follo ws: ∼ ∼ • • • • • • • • ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? That is, the domain equiv alence is generated by the tw o mov es illustrated a b o ve. There- fore we often c ho o se to represen t a domain equiv a lence class of trees by the unique mem b er with least refinemen t, that is with its unpainted subtrees a ll corolla s. 3.3. Definition. A p ainte d c or ol la is a painted tree with only one no de, of type (6). The n th comp osihedron may b e describ ed as the quotien t of the n th m ultiplihedron under domain equiv alence. This quotien t can b e p erformed b y applying the equiv alence either to the metric trees which define the multiplihe dra in [ 9 ] or to the combinatorial definition of the m ultiplihedra, i.e. to the face p oset of the m ultiplihedra. Here w e will follo w the lat t er sc heme to unpack the definition of the comp o sihedra in to a recursiv e description if facet inclusions. The facets of the n th m ultiplihedron are of t w o t yp es: upp er facets whose v ertices corresp ond to sets of related w ay s of m ultiplying in the ra ng e, and low er facets whose v ertices corresp ond to sets of related w ays of multiply ing in the domain. A lo wer facet is denoted J k ( r , s ) and is a com binatorial cop y of the complex J ( r ) ×K ( s ) . Here r + s − 1 = n. A ve r t ex of a low er facet represen ts a w ay in whic h s o f the p oints are multiplie d in the domain, and then how the images of their pro duct and of the other r − 1 p oin t s are 13 m ultiplied in the range. In the case for whic h the domain is strictly asso ciative , the cop y of K ( s ) here should b e replaced b y a single p oin t, denoted {∗} . Th us in the comp osihedra the lo wer faces will hav e reduced dimension. In fact only certain of them will still b e facets. The recursiv e definition of the new sequence of p olytop es is as fo llo ws: 3.4. Definition. The first comp osihedron denoted C K (1) is defined to b e the single p oint {∗} . It is asso ciated to the painted tr ee with o ne leaf, and thus one ty p e (3) in ternal no de. Assume that the C K ( k ) ha ve b een defined for k = 1 . . . n − 1 . T o C K ( k ) w e a sso ciate the k -leav ed painted corolla. W e define an ( n − 2)-dimensional C W -complex ∂ C K ( n ) as follo ws, and then define C K ( n ) to b e the cone on ∂ C K ( n ). Now the t op-dimensional cells of ∂ C K ( n ) (f acets of C K ( n )) are in bijection with the set o f pain ted trees of tw o ty p es: upp er trees u ( t ; r 1 , . . . , r t ) = 0 r 1 z}|{ . . . r 2 z}|{ . . . r t z}|{ . . . n − 1 • • • . . . • ? ? ? ? ? ? ? ? ? ? : : : : / / / / 4 4 4 4 * * * * * * * ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? * * * * * * * * * * * * * * and minimal lower trees l ( k , 2) = 2 z}|{ 0 n − 1 k − 1 • • . . . . . . ? ? ? ? ? ? ? ? ? ? Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q m m m m m m m m m m m m m m m / / / / There are 2 n − 1 − 1 upp er trees, as coun ted in [ 13 ]. The index t can range from 2 . . . n, and r i ≥ 1 . The upp er facet which w e call C K ( t ; , r 1 , . . . , r t ) asso ciated with the upp er tree of iden t ical indexing is a cop y o f K ( t ) × C K ( r 1 ) × · · · × C K ( r t ) . Here the sum of the r i is n. There are n − 1 minimal low er trees. The lo w er facet whic h w e call C K k ( n − 1 , 2) asso ciated with the low er tree of iden t ical indexing is a cop y of C K ( n − 1 ) × K (2), that is, a cop y of C K ( n − 1) . Here k ranges f rom 1 to n − 1 . B ( n ) is the union of all t hese facets, with in tersections describ ed as fo llows: Since the facets are pro duct p olytop es, their sub-fa cets in turn are pro ducts of fa ces (of smaller asso ciahedra and comp osihedra) whose dimensions sum to n − 3 . Each of these sub-facets thus comes (inductiv ely) with a list of asso ciated trees . There will alw ays b e a unique w ay of grafting these trees to construct a pain ted tree that is a minimal refinemen t of the upp er or minimal lo wer tree asso ciated to the facet in question. F or the sub-facets of an upp er facet the recip e is to pain t entirely the t -leav ed tree asso ciated 14 to a fa ce of K ( t ) and to gr a ft to eac h of its branc hes in turn the t r ees associated to the appropriate fa ces o f C K ( r 1 ) through C K ( r t ) resp ectiv ely . A sub-facet of the lo we r facet C K k ( n − 1 , 2) inductiv ely comes with an ( n − 1)- lea v ed asso ciated upp er or minimal lo we r tree T . The recip e for assigning our sub-fa cet an n -lea ve d minimal refinemen t of the n -leav ed minimal low er tree l ( k , 2) is to graft an unpain ted 2-lea v ed tree to the k th leaf of T . See the follo wing Example 3.5 for this pictured in low dimensions. The in tersection of tw o facets in ∂ C K ( n ) consists of t he sub-facets of eac h whic h ha ve asso ciated trees that are domain equiv alen t. Then C K ( n ) is defined to b e the cone on ∂ C K ( n ) . T o C K ( n ) w e assign the pain ted corolla o f n lea v es. 3.5. Example. C K (1) = • • Here is C K (2) with the upper facet K (2) × C K (1) × C K (1) on the left and the low er facet C K (1 ) × K (2) on the righ t. • • • • • • • • ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? And here is the complex C K (3) . The pro duct structure of the upp er facets is listed, and the t wo low er facets are named. Note that b oth lo wer facets are copies of C K (2) × K (2) . Notice also how t he sub-f a cets (v ertices) are lab eled. F or instance, the upp er r ig h t verte x is lab eled b y a tree tha t could b e constructed b y grafting three copies of the single leaf pain ted corolla onto a completely pain ted binary tree with three leav es, or b y grafting a single leaf pa in ted corolla and a 2-leaf painted corolla onto the leav es of a 2-leaf completely pain ted binary tree. 15 K (3) ×C K (1) ×C K (1) ×C K (1) • / / / / / / / / / / / / / K (2) ×C K (1) ×C K (2) K (2) ×C K (2) ×C K (1) • / / / / / / / / / / / / / • z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z C K 2 (2 , 2) C K 1 (2 , 2) • D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D • • / / / • • • • • / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / • • • • • / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / • • • • / / / / / / / / / / / / / / / / / / / / / / / / / / / / • • • • / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / • • • ∼ / / / / / / / / / / / / / / / / / / / / / / / / • • • ∼ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / • • • • ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? • • • ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? • • • ? ? ? ? ? ? ? ? ? ? ? ? • • J J J J J J J J t t t t • • ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? • • ? ? ? ? ? ? ? ? ? 3.6. R emark. The tot a l num b er of facets of the n th comp osihedron is t hus 2 n − 1 + n − 2 . Therefore the num b er of facets of C K ( n + 1) is 2 n + n − 1 = 0 , 2 , 5 , 10 , 19 , 36 , . . . . This is sequence A052 9 44 in the OEIS. This se quence also giv es t he n um b er of v ertices of a n - dimensional cub e with one truncated corner. Now the relationship b et wee n the p o lyto p es in Fig ur e 1 can b e alg ebraically stated as an equation: (2 n + n − 1) + ( n + 1)( n + 2) 2 − 1 − n ( n + 1) 2 + 2 n − 1 = 2 n That is: |{ facets of C K ( n +1 ) }| + |{ facets o f K ( n +2) }|−|{ fa cets of J ( n +1) } | = |{ facets of C ( n ) }| where C ( n ) is the n -dimensional cub e. 3.7. R ema rk. Consider the existence o f fa cet inclusions: K ( k ) × ( C K ( j 1 ) × · · · × C K ( j k )) → C K ( n ) where n is the sum of the j i . This is the left mo dule structure: the comp osihedra together form a left o p erad mo dule ov er the op erad of spaces formed b y t he a sso ciahedra. Notice that the description of the assignation of trees t o sub-fa cets (via graf ting) in the definition ab o ve guaran tees the mo dule axioms. 16 3.8. R emark. By definition, the n th comp osihedron is seen to b e the quotien t of the n th m ultiplihedron under domain equiv alence. R ecall that the low er facets of J ( n ) are equiv alen t to copies of J ( r ) × K ( s ) . The a ctual quotient is achie ved by iden tif ying any t w o p oin t s ( a, b ) ∼ ( a, c ) in a lo wer fa cet, where a is a p oin t of J ( r ) and b, c are p oin ts in K ( s ). An a lternate definition of the comp osihedra describ es C K ( n ) as the p olytop e ac hiev ed b y taking the n th m ultiplihedron J ( n ) and sequen tially collapsing certain facets. An y cell in J ( n ) whose 1-sk eleton is en tirely made up of edges that corresp ond to images of domain a sso ciat io ns f (( ab ) c ) → f ( a ( bc )) is collapsed to a single v ertex. In the resulting complex t here will b e redundan t cells. First any cell with exactly tw o v ertices will b e collapsed to an edge betw een them. Then an y cell with 1-sk eleton S 1 will be collapsed to a 2-disk spanning that b oundary . This pro cess contin ues inductive ly un t il any cell with ( n − 3)- sk eleton S ( n − 3) is collapsed to a ( n − 2)- disk spanning that bo undar y . F or a picture of this see the upp er left pro jection of Figure 1 , where the small p en tagon and t wo rectangles on the bac k of J (4) collapse t o a v ertex a nd tw o edges resp ectiv ely . 3.9. R em a rk. In [ 36 ] there is describ ed a pro jection π from the n th p erm utohedron P ( n ) to J ( n ) , and so there follow s a compo site pro jection from P ( n ) → C K ( n ) . F or comparison sak e, recall that in the case of a strictly a sso ciativ e ra ng e the m ultiplihedra b ecome the asso ciahedra, in fact the quotient under range equiv alence of J ( n ) is K ( n + 1) . The implied pro jection of this quotien t comp osed with π yields a new pro jection from P ( n ) → K ( n + 1 ) . Saneblidze and Umble describ e a v ery different pro jection from J ( n ) to K ( n + 1 ) . When comp osed with π this yields a (differen t) pro jection fr o m P ( n ) to K ( n + 1 ) that is sho wn in [ 36 ] to b e precisely the pro jection θ describ ed in [ 45 ]. 3.10. R e mark. There is a bij ection b et w een the 0-cells, or ve r tices, of C K ( n ) and the do- main equiv alence classes of n -lea ved pain ted binary tr ees. This follo ws from the recursiv e construction, since the 0-cells m ust be asso ciated to completely refine d pain ted trees. Ho we ver, the domain equiv alen t trees must b e assigned to the same v ertex since domain equiv alence determin es inte r section of sub-facets. Recall that w e can also lab el the ve r- tices of C K ( n ) b y applications of an A ∞ function f to n -fold pro ducts in a top ological monoid. 3.11. R emark. As defined in [ 40 ] an A n map b et w een A n spaces f : X → Y is describ ed b y an action J ( n ) × X n → Y . If the space X is a topolo gical monoid then w e ma y equiv alen tly describe f : X → Y b y simply replacing the action of J ( n ) with an action of C K ( n ) in the definition of A n map. 4. Ver tex Combina torics No w w e men tion results regarding the coun ting of t he ve rt ices of the comp o sihedra. Recall that vertice s corresp ond bijectiv ely to domain equiv a lence classes of pain ted binary trees. 4.1. Theorem. The numb er of vertic es a n of the n th c om p osi h e dr on is given r e cursively by: a n = 1 + n − 1 X i =1 a i a n − i wher e a 0 = 0 . 17 Pr o of. By a w eigh ted tree w e refer t o a tree with a real num b er assigned to eac h of its lea v es. The tot al w eight of the tree is the sum of the w eigh ts of its leav es. The set of domain equiv alence classes of painted binary tr ees with n -leav es is in bijection with t he set of binary w eighted trees of total w eight n where lea ve s ha v e p ositiv e in teger we ig h ts. This is eviden t fro m the represen ta tion o f a domain equiv alence class b y its least refined mem b er. Here is a picture that illustrates the general case of the bijection. • • • • • • • , , , , , , , / / / 4 4 4 * * * * / / / / / / / / / / / / / / / / / / / / / / / / / / / ⇋ 3 • • 1 2 / / / / / / / / / / / / / / There is one w eigh ted 1- leav ed tree with t o tal w eigh t n . Now we coun t the binary trees with tota l we igh t n that ha v e at least one t r iv alen t no de. Eac h o f these consists of a choice of tw o binary subtrees whose ro ot is the initial triv alen t no de, and whose w eights m ust sum to n. Th us w e sum o ver the wa ys that n can b e split in to tw o natural n umbers. 4.2. R emark. This formula gives the sequence whic h b egins: 0 , 1 , 2 , 5 , 15 , 51 , 188 , 73 1 , 2950 , 12235 . . . . It is sequence A007317 of the On-line Encyclop edia of in teger sequences. This form ula for the sequenc e w a s originally stated b y Benoit Cloitre. The recursiv e formula ab ov e yields the equation A ( x ) = x 1 − x + ( A ( x )) 2 where A ( x ) is the ordinary generating function of t he sequence a n ab ov e. Thus b y use of the quadratic form ula w e hav e A ( x ) = x 1 − x c ( x 1 − x ) . where c ( x ) = 1 − √ 1 − 4 x 2 x is the g enerating function of the Catala n num b ers. This form ula w as originally deriv ed b y Emeric D eutsc h from a commen t of Mic ha el Somos. No w if w e wan t to generate the sequence { a n +1 } ∞ n =0 = 1 , 2 , 5 , 1 5 , 51 , . . . , then w e m ust divide our generating function by x , to get 1 1 − x c ( x 1 − x ) . This we recognize as the generating function of the binary tra nsform of the the Catalan num b ers, f r om the definition of binary transform in [ 3 ]. Therefore a direct form ula for the sequence is giv en b y a n +1 = n X k =0 n k C ( k ) where C ( n ) are the Catalan num b ers. 4.3. R emark. This sequence also coun ts the non-commutativ e non-asso ciativ e pa rtitions of n . Tw o other com binatoria l items that this sequence en umerates are: the n um b er of Sc hro eder paths (i.e. consisting of steps U = (1 , 1) , D = (1 , − 1) , H = (2 , 0 ) and nev er going b elo w the x-axis) from (0 , 0) to (2 n − 2 , 0 ), with no p eaks at ev en leve l; and the n um b er of tree-lik e p olyhexes (including the non-planar helic enic p olyhexes) with n or few er cells [ 12 ]. 18 5. Convex hull realiza tions In [ 28 ] Lo da y giv es a n algo rithm for ta king the binary trees with n leav es and finding for each an extremal p o int in R n − 1 ; t o gether whose con v ex h ull is K ( n ) , the ( n − 2)- dimensional asso ciahedron. Note tha t Lo day writes form ulas with the con ven tio n that the num b er of lea v es is n + 1 , where w e instead alw ays use n to refer to the num b er of lea v es. Given a (non-pain ted) binary n - lea v ed tree t, Lo da y arr ives at a p oin t M ( t ) in R n − 1 b y calculating a co ordinate from eac h triv a lent no de. Thes e are ordered left to righ t based up o n the ordering of the leav es from left to r igh t. F o llowing Lo da y w e n um b er the lea v es 0 , 1 , . . . , n − 1 a nd the no des 1 , 2 , . . . , n − 1 . The i th no de is “b etw een” leaf i − 1 and leaf i where “b etw een” might be describ ed to mean that a rain drop falling b et w een those leav es w o uld b e caugh t at that no de. Eac h triv alen t no de has a left and righ t branc h, whic h each suppo rt a subtree. T o find the Lo da y co ordinate for the i th no de w e ta k e the product of the num b er o f lea v es of the left subtree ( l i ) and the n umber of leav es of the righ t subtre e ( r i ) for that no de. Th us M ( t ) = ( x 1 , . . . x n − 1 ) where x i = l i r i . Lo day pro ves that the con vex h ull of t he p oin ts th us calculated for all n -lea ve d binary trees is the n th asso ciahedron. He a lso sho ws that the p o in ts thus calculated all lie in the n − 2 dimensional affine h yp erplane H giv en by the equation x 1 + · · · + x n − 1 = S ( n − 1) = 1 2 n ( n − 1) . In [ 13 ] w e adjust Lo day’s algorit hm to apply to painted binary tr ees as describ ed ab o ve, with only no des of type (1), (2), and (3), b y c ho osing a n um b er q ∈ (0 , 1) . The n giv en a pain ted binary tree t with n lea v es w e calculate a p o in t M q ( t ) in R n − 1 as follows : we b egin b y finding the co ordinate fo r each triv alen t no de from left to righ t given by Lo day’s algorithm, but if the no de is of ty p e (1) (unpainted, or colored b y the do main) then its new co ordinate is found by further multiply ing its Lo day co ordinate by q . Th us M q ( t ) = ( x 1 , . . . x n − 1 ) where x i = ( q l i r i , if no de i is t yp e (1) l i r i , if no de i is t yp e (2). Note that whenev er w e sp eak of the num b ered no des (1 , . . . , n − 1 fro m left to righ t ) of a binary tree, w e are referring only to the triv alent no des, of t yp e (1) or (2). F or an example, let us calculate the p oin t in R 3 whic h corresp onds to the 4 -lea ve d tree: t = • • • • • • / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Then M q ( t ) = ( q , 4 , 1 ) . No w w e turn to consider the case when in the algo rithm describ ed ab o ve w e let q = 0 . M 0 ( t ) = ( x 1 , . . . x n − 1 ) where x i = ( 0 , if no de i is t yp e (1) l i r i , if no de i is type (2). F or the same tr ee t in the ab o ve example w e ha v e M 0 ( t ) = (0 , 4 , 1 ) . 19 5.1. Lemm a. If (p ainte d binary) tr e e t is domai n e quivalent to t ′ then M 0 ( t ) = M 0 ( t ′ ) . That is, e ach domain e quivalenc e class of binary p ainte d tr e es c ontributes exactly one p oin t. Pr o of. This is cle a r from the fact tha t all the unpain t ed no des of a tree contribute a 0 co ordinate. 5.2. Theorem. The c onvex hul l of al l the r esulting p oints M 0 ( t ) for t in the se t of n -le ave d binary p ain te d tr e es is the n th c om p osi h e dr on. That is , our c onvex hul l is c ombin atorial ly e quival e nt to the CW-c omplex C K ( n ) . The pro of will follow in section 6 . Here are all the pain ted binary t r ees with 3 leav es, t o gether with their p oin ts M 0 ( t ) ∈ R 2 . M 0 • • • • • / / / / / / / / / / / / / / / / / / / / / / / / / = (1 , 2) , M 0 • • • • • / / / / / / / / / / / / / / / / / / / / / / / / / = (2 , 1) M 0 • • • • / / / / / / / / / / / / / / / / / = (0 , 2) M 0 • • • • / / / / / / / / / / / / / / / / / / / / / = (2 , 0) M 0 • • • / / / / / / / / / = (0 , 0) , M 0 • • • / / / / / / / / / / / / / = (0 , 0) Note tha t the b o ttom t w o p oin ts are b oth the origin. The con v ex h ull of the five total distinct p oin ts app ears as follows: O O / / ? ? ? ? ? ? ? ? ? ? ? • • ? ? ? ? ? ? ? ? ? ? ? • • • • • The (redundan t) list of v ertices for C K (4) based on pain ted binary trees with 4 lea v es is: 20 (1, 2 ,3) (0 ,2 ,3) (0 ,0 ,3) (0, 0 ,0) (2, 1, 3) (2 ,0 ,3) (0 ,0 ,3) (0, 0 ,0) (3 ,1 ,2) (3, 0, 2) (3 ,0 ,0) (0, 0 ,0) (3, 2, 1) (3 ,2, 0) (3 ,0, 0) (0 ,0 ,0) (1 ,4 ,1) (0, 4, 1) (1, 4, 0) (0, 4, 0) (0, 0 ,0) These are suggestiv ely listed as a table where the first column is made up of the co ordi- nates calculated b y Lo da y for K (4 ) , whic h here corr esp o nd to trees with ev ery triv alen t no de en tirely painted. The ro ws ma y b e found b y applying the factor 0 to eac h co or- dinate in turn, in order of increasing size of those co ordinates. Here is the con v ex hu ll of the fifteen total distinc t p oints , where w e see that eac h ro w of the t a ble corresp onds to shortest paths from the big p en tag on to the origin. O f course sometimes there a re m ultiple suc h paths. O O | | / / • • • • • • (3 , 1 , 2) • (3 , 2 , 1) • (1 , 4 , 1) • (1 , 2 , 3) • (2 , 1 , 3) • • • • • • • • • • • z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z y y y y y y y y y y y y y y y y y y y y y y y y z z z z z z z z z z z z z z z z z z y y y y y y y y y y y y y y y y y y y y y y y y O O O O O O O O O O O O O O O O O O O O O O O O O O O q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H k k k k k k k k k k k k k k k k k k k k k k k k k k k k k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H t t t t t t t t t t t t t t t t t t t t t t t t t t t t t T o see the picture of C K (4 ) t ha t is in Figure 1 of this pap er, just rota te this view of the conv ex h ull b y 90 degrees clo ckw ise. T o compare to other pictures of C K (4) in this pap er note the smallest quadrilateral facet in this picture containing the unique v ertex with four inciden t edges. 21 T o see a rotatable v ersion of the con vex hull which is the fourth comp osihedron, ente r the following homog eneous co ordinates into the W eb Demo of p olymake (with option vi- sual), at http://www. math.tu- berlin.de/polymake /index.html#apps/polytope . In- deed p olymak e w as instrumen tal in t he exp erimen tal phase of this researc h [ 15 ]. P O I N T S 1 1 2 3 1 0 2 3 1 0 0 3 1 0 0 0 1 2 1 3 1 2 0 3 1 3 1 2 1 3 0 2 1 3 0 0 1 3 2 1 1 3 2 0 1 1 4 1 1 0 4 1 1 1 4 0 1 0 4 0 5.3. R em ark. It is also fairly simple to devise a mapping fr om n -lea v ed pain ted bina r y trees to Euclidean space whic h r eflects the quotien t of the m ult iplihedron b y range equiv alence, where f ( a )( f ( b ) f ( c )) = ( f ( a ) f ( b )) f ( c ) . It may be done b y reflecting suc h equiv alences as: • • • • • / / / / / / / / / / / / / / / / / / / / / / / / / ∼ • • • • • / / / / / / / / / / / / / / / / / / / / / / / / / . . . b y mapping them b oth to a single v ertex in R 2 . One p ossible map is M ′ ( t ) = ( x 1 , . . . x n − 1 ) where x i = ( q l i r i , if no de i is t yp e (1) i ( n − i ) , if no de i is t yp e (2). This will yield a new realization of K ( n + 1) in R ( n − 1) , with v ertices corresp onding to range equiv alence classes of n - leav ed pa inted trees. Here is the (redundan t) list of v ertices for K (5) based on painted binary trees with 4 lea ves : The list of v ertices fo r J (4) based on pain ted binary trees with 4 lea v es, for q = 1 2 , is: 22 (3, 4 ,3) (1/2 ,4 ,3) (1/ 2 ,2/2 ,3) (1/2, 2/2 ,3/2 ) (3, 4, 3) (3 ,1/2 ,3) (2/ 2 ,1/2 ,3) (2/2, 1/2 ,3/2 ) (3 ,4 ,3) (3, 1/2, 3) (3 ,1/2 ,2/2 ) (3/ 2, 1/2 ,2/2) (3, 4, 3) (3 ,4, 1/2) (3 ,2/2, 1/2 ) (3/ 2 ,2/2 ,1/2) (3 ,4 ,3) (1/2, 4, 3) (3, 4, 1/2) (1/2, 4, 1/2) (1/2, 4/2 ,1 /2) These are suggestiv ely listed as a table whe r e the first column is the single image of all the trees with ev ery triv alen t no de en tirely pain ted. Indeed there are 14 total differen t v ertices. Here is the con v ex hull of those v ertices. O O { { / / • • • • • • • • • • • • • • (3 , 4 , 3) y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y O O O O O O O O O O O O O O O O O O O O O O t t t t t t t t t t t t t t t t t t L L L L L L L L L L L L j j j j j j j j j j j j j O O O O O O O O O O O x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x h h h h h h h h h h h h h y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u 6. Proof of Theorem 5.2 Our algorithm fo r g enerating extremal p oin ts whose conv ex hull giv es the comp osihedra can b e describ ed as the q → 0 limit of the algo r it hm f o r the m ultiplihedra, which w as giv en in [ 13 ]. What w e need to demonstrate is that the limiting pro cess do es indeed deliv er the combinatorial equiv alen t of the CW-complex C K ( n ) . T o demonstrate that our con vex h ulls a re eac h com binato rially equiv alen t to the cor- resp onding C W -complexes of Definition 3.4 w e need only che ck that they b oth hav e the same v ertex-facet incidence. W e will show tha t for eac h n there is an isomorphism f b et we en the v ertex sets (0 -cells) of our con v ex h ull and C K ( n ) whic h preserv es the sets of v ertices corresp onding to facets; i.e. if S is the set of v ertices of a facet of our con v ex h ull then f ( S ) is a v ertex set of a facet of C K ( n ) . T o demonstrate the existenc e of the isomorphism, noting tha t the v ertices of C K ( n ) corresp ond to the doma in equiv a lence classes of binary painted trees, w e only need to 23 c hec k that the p o ints w e calculate from those classes are indeed the v ertices of t heir con ve x h ull. Recall that the calculation o f a p oin t from a class is we ll defined b y Lemma 5.1 . The isomorphism f is the one that takes the v ertex calculated from a certain class to the 0-cell asso ciated to the same class. Now a giv en f a cet o f C K ( n ) corresp onds to a tree T whic h is one of the tw o sorts o f trees pictured in Definition 3.4 . T o sho w tha t our implied isomorphism o f v ertices preserv es v ertex sets o f facets w e need to sho w that that our f acet is the con v ex h ull of the p oin ts corresponding to the classes of binary trees represen ted b y r efinements of T . By refinemen t of pa in ted tr ees w e refer to the relationship: t refines t ′ if t ′ results fro m the collapse of some of the in ternal edges of t . The pro ofs of b oth k ey p oin ts will pro ceed in tandem, and will b e inductiv e. That is, w e will sho w that for each facet tree T , the po in ts M 0 ( t ) for t < T are precisely those p oints that lie in a a b ounding h yp erplane of our conv ex hull. Then we will c hec k that those p o in ts M 0 ( t ) are the extremal p oints of their con v ex h ull, whic h is indeed an ( n − 2)-dimensional p olytop e of the t yp e decreed in Definition 3.4 . W e will use the fact that if P ( q ) and R ( q ) a re tw o p olytop es with some v ertices parameterized contin uously b y q that lim q → a ( P × Q ) ≡ lim q → a P × lim q → a Q 6.1. Definition. The low er facets C K k ( n − 1 , 2) corresp ond to minimal lo wer trees suc h as: l ( k , 2) = 2 z}|{ 0 n − 1 k − 1 • • . . . . . . . . . ? ? ? ? ? ? ? ? ? ? Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q m m m m m m m m m m m m m m m / / / / These are assigned a h yp erplane H 0 ( l ( k , 2)) determined by the equation x k = 0 . Recall t hat n − 1 is the n um b er of branche s extending from the lo w est no de. Th us 1 ≤ k ≤ n − 1 . Not ice that if q w ere not zero as in definition 5.1 of [ 13 ] then the t he equation would app ear: x k = q . W e conjecture that these latter hy p erplanes w o uld function as replacemen ts fo r the ones giv en b y x k = 0 in that after the replacemen t the resulting p olytop e w ould still be the comp osihedron. 24 6.2. Definition. The upp er facets C K ( t ; r 1 , . . . , r t ) corresp ond to upp er tr ees suc h as: u ( t ; r 1 , . . . , r t ) = 0 r 1 z}|{ . . . r 2 z}|{ . . . r t z}|{ . . . n − 1 • • • . . . • ? ? ? ? ? ? ? ? ? ? : : : : / / / / 4 4 4 4 * * * * * * * ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? * * * * * * * * * * * * * * These are assigned a h yp erplane H 0 ( u ( t ; r 1 , . . . , r t )) determined b y the equation x r 1 + x ( r 1 + r 2 ) + · · · + x ( r 1 + r 2 + ··· + r t − 1 ) = 1 2 n ( n − 1) − t X i =1 r i ( r i − 1) ! or equiv alently: x r 1 + x ( r 1 + r 2 ) + · · · + x ( r 1 + r 2 + ··· + r t − 1 ) = X 1 ≤ i 0 , since the k th no de will b e painted. 6.5. Definition. 26 Recall that the upp er facets C K ( t ; r 1 , . . . , r t ) corresp ond to upp er trees suc h as: u ( t ; r 1 , . . . , r t ) = 0 r 1 z}|{ . . . r 2 z}|{ . . . r t z}|{ . . . n − 1 • • • . . . • ? ? ? ? ? ? ? ? ? ? : : : : / / / / 4 4 4 4 * * * * * * * ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? * * * * * * * * * * * * * * These are a ssigned a h yp erplane H w 0 ,...,w n − 1 0 ( u ( t ; r 1 , . . . , r t )) determined b y the equation x r 1 + x ( r 1 + r 2 ) + · · · + x ( r 1 + r 2 + ··· + r t − 1 ) = X 1 ≤ i
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