ASSOCIATIVITY DATA IN AN (, 1)-CATEGORY

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1 ASSOCIATIVITY DATA IN AN (, 1)-CATEGORY EMILY RIEHL A popular sloan is tat (, 1)-cateories (also called quasi-cateories or - cateories) sit somewere between cateories and spaces, combinin some o te eatures o bot. Te analoy wit spaces is airly clear, at least to someone wo is appy to reard spaces as Kan complexes, wic are simplicial sets in wic every orn can be illed. Te analoy wit cateories is somewat more subtle. We coose to model (, 1)-cateories as quasi-cateories, wic are a particular type o simplicial set. Wen we reard te 0-simplices o an quasi-cateory as its objects and te 1-simplices as its morpisms, we can deine a wea composition law tat is well-deined, unital, and associative only up to omotopy. Tis means exactly tat wen we replace te 1-simplices by omotopy classes o 1-simplices we obtain an ordinary cateory, called te omotopy cateory o our quasi-cateory. However, a lot o data is lost wen we replace an quasi-cateory by its omotopy cateory. In particular, tere exists a 3-simplex tat witnesses te act tat a particular composite () o 1-simplices,, and (were some composite o and as already been cosen) is also a composite o (a cosen composite o and ) and. Tis is well-nown by tose wo are amiliar wit te construction o te omotopy cateory mentioned above, but a question remains: wat sort o associativity data is provided by te n-simplices, or n > 3? Eac n-simplex o an quasi-cateory exibits some composite o te n morpisms wic mae up its spine. We will arue tat tese simplices can be rearded as unbiased associaedra in te sense tat tey witness te commutativity o teir boundaries, wic are n 1-simplices o te same orm. In act, one may coose all but one o tese n 1-simplices to be any witnesses tat you want, subject to some obvious constraints. We will describe in detail te combinatorial analoies between tese constraints and Stase s associaedra in wat ollows, at least in low (n 6) dimensions. Beore turnin to tese concrete details, we sould acnowlede a more conceptual explanation or te experts in tis ield, due to Jacob Lurie, or te penomena we ll describe below. Te teory o quasi-cateories is equivalent to te teory o simplicial (or topoloical) cateories. One eature o tis equivalence is tat te mappin spaces in any quasi-cateory (owever one cooses to extract tem) are omotopy equivalent to mappin spaces in a simplicial cateory, were tere is a strictly associative multiplication. Transportin tis structure alon te omotopy equivalences will ive an A structure on te mappin spaces in te oriinal quasicateory. By deinition, A -spaces are alebras or te non-σ operad {K n } wit objects iven by Stase s associaedra, accountin or teir appearance in wat ollows. We bein by explainin te construction o te omotopy cateory o an quasicateory, empasizin te role played by te 2- and 3-simplices. Te reader wo Date: October 8,

2 2 E. RIEHL already nows tis story can comortably sip to te next section, ater main note o our convention or labelin 3-simplices. In te second section, we describe te low-dimensional ier associativity data tat is present in any quasi-cateory, beinnin wit n = 4 and ten movin on to n = 5 and n = Preliminaries, or ow is an (, 1)-cateory cateory-lie? We ll use standard notation or simplicial sets, wic arees wit Lurie s in [Lur08] or [Lur09], altou we preer te name quasi-cateory or is -cateory. In particular, we write n or te simplicial set op Set represented by te object [n] = {0, 1,..., n} o. I S : op Set is a eneric simplicial set, S n is te set o its n-simplices. We ave ace maps d i : S n S n 1 and deeneracy maps s i : S n S n+1 or 0 i n. A orn Λ n i is a simplicial subset o n (or also o its boundary n ). It is enerated by n n 1-simplices wic satisy te relations typical o te collection o all aces but te it o any n-simplex. We are more oten interested in relative orns, i.e., maps Λ n i S. Concretely, tese are speciied by elements σ 0,..., ˆσ i,... σ n S n 1 (te imaes o te eneratin n 1-simplices mentioned previously) satisyin te relations (1.1) d j σ = d 1 σ j or all 0 j < n, wit j, i. An quasi-cateory is a simplicial set S suc tat every inner orn can be illed, wic means tat or every collection o n 1-simplices as above wit 0 < i < n, tere exists σ i S n 1 and σ S n suc tat te aces o σ are precisely te σ j. Fix an quasi-cateory S. It is cateory-lie in te ollowin sense. We reard te vertices aa points aa 0-simplices as objects and edes aa arrows aa maps aa 1-simplices S 1 as morpisms rom d 1 to d 0 ; we write d 1 d 0. For any x S 0, s 0 x S 1 is a morpism rom x to x, wic we reard as te identity at x and depict wit an equals sin. We will depict 2-simplices lie tis Te vertices will always be in te same position, so we typically omit te numbers. Te it ede is te ede opposite vertex i. We say morpisms and wit te same endpoints are omotopic i tere exists a 2-simplex wit boundary or or wit and switced. In an quasi-cateory, tis deines an equivalence relation and urtermore any one o tese conditions implies all o te oters. Importantly, we ave a wea composition law deined as ollows. Morpisms and are composable wen te codomain d 0 equals te domain d 1, i.e., exactly

3 ASSOCIATIVITY DATA IN AN (, 1)-CATEGORY 3 wen determines a orn Λ 2 1 S (in words, a 2,1-orn in S). Any 2-simplex σ S 2 tat ills tis orn exibits te ede d 1 σ as a composite o and. As S is an quasi-cateory, some suc σ always exists. In an quasi-cateory, any two suc composites are omotopic, and urtermore i we replace or by omotopic maps, te composites are still omotopic. All o te unproven claims above ollow easily by illin 3,1- or 3,2-orns in S to obtain a 3-simplex. By constructin te orn appropriately, te missin ace tat is iven by its iller will be te desired 2-simplex (exibitin, e.., tat some arrow is a composite o two oter arrows, or tat some pair o arrows are omotopic). Let s aree to draw 3-simplices lie tis Te vertices will always be in te same position, so we typically omit te numbers. Usin te conventional numberin or simplices, te 3rd ace is te let ace, te 2nd ace is te bac ace, te 1st ace is te bottom ace, and te 0t ace is te rit ace. We can now prove tat composition is associative up to omotopy. Suppose we ave tree composable morpisms,, and and let us coose composites and, as well as 2-simplices exibitin tese cosen composites. We can eiter coose a composite () and sow tat it is a composite o and or coose a composite () and sow tat it is a composite o and. Te ormer stratey amounts to illin te 3,2-orn depicted by (1.2) wit te bac arrow equal to te cosen morpism (). Fillin tis orn ills in te bac ace, wic exibits () as a composite o and. Te latter stratey amounts to illin te 3,1-orn also depicted by (1.2) but wit te bac arrow equal to (). It ollows tat () and () are omotopic, as tey are bot composites o te same morpisms. We ave proven tat te simplicial set S is cateory-lie: iven S we can obtain an actual cateory wit objects te 0-simplices and morpisms omotopy classes o 1-simplices. Te acts stated above deine composition on omotopy classes and

4 4 E. RIEHL prove tat it is associative and unital. Tis cateory is called te omotopy cateory o S and is depicted in te literature by τ 1 S (in te unpublised wor o André Joyal) or S (by Jacob Lurie). However, a lot o data is lost wen we oret about S and wor instead wit its omotopy cateory, wic is wy quasi-cateories are objects o interest in teir own rit. 2. Hier associativity data Wit tese preliminaries aside, we are now inally able to address te ollowin question: wat sort o ier associativity data is present in any quasi-cateory? First, a deinition rom te teory o simplicial sets: te spine o an n-simplex σ S n consists o te edes 0 1, 1 2,..., n 1 n between te successive vertices o σ. In te lanuae o te previous section, te spine o σ is te maximal list o composable edes o σ. Wen n > 2, a n,i-orn in S as te same edes as any o its illers, so we may spea o te spine o a orn as well. For example, in (1.2), te spine is,,. Given a sequence,, o tree composable morpisms, any 3-simplex wit spine,, ives a piece o associativity data associated to tat sequence. Te aces o any suc simplex exibit composites and as well an an arrow d 1 d 0 tat is simultaneously exibited as a composite o wit and wit. Te wor o te previous section sows tat suc simplices exist in any quasicateory. Tis sort o associativity data is more natural tan tat exibited, e.., by a 3-simplex () () tat ills te 3,1-orn wit aces constructed rom te aces o (1.2). Tis is because te ormer is unbiased, wereas te latter empasizes te cosen composite and nelects to coose a composite o and. Wit tis in mind, te next level o associativity data sould consist o 4- simplices wit spine,,,, were tis is aain a sequence o composable morpisms. We will sow irstly tat suc simplices can be constructed in any quasi-cateory and secondly tat te aces o any suc simplex can be understood as some sort o unbiased associativity pentaon. We will ten explain ow te observations made or 4-simplices extend to 5- and 6-simplices, wic we explicitly relate to te (rater more complicated) associaedra K 5 and K 6, ater wic point we become tired o computin associaedra and stop dimensional associativity. First, we will construct a 4-simplex tat sould be tout o as a piece o associativity data or te sequence o composable morpisms,,,. We will construct tis simplex by illin a 4,2-orn. Te construction eels a bit less natural i we instead coose to ill a 4,1- or 4,3-orn. We will also describe suc a construction below to illustrate te dierences. Any 4-simplex σ as ive aces σ 0,..., σ 4, wic are temselves 3-simplices. We will construct our 4-simplex by illin te 4,2-orn iven by te 3-simplices depicted

5 ASSOCIATIVITY DATA IN AN (, 1)-CATEGORY 5 below, wic are temselves constructed by illin 3-dimensional orns: σ 0 () σ 1 ()() () σ 3 ()() () σ 4 () Te labels or te arrows are meant to indicate ow tese composites are cosen. We bein by coosin 2-simplices exibitin composites,, and o te 3 composable pairs contained in our list o morpisms. Next, we coose simplices exibitin te composites () and () tat will orm te 3rd ace o σ 1 and 0t ace o σ 3, respectively. Wit te data previously cosen, we can orm a 3,1- orn, wic we ill to obtain σ 0 and a 3,2-orn, wic we ill to obtain σ 4. Finally, we coose a 2-simplex exibitin a composite ()() o and. Tis allows us to construct a 3,1-orn, wic we ill to obtain σ 1 and a 3,2-orn, wic we ill to obtain σ 3. We must sow tat tese 3-simplices it toeter to orm a 4,2-orn in S. Te conditions tat we must cec are d 3 σ 4 = d 3 σ 3 d 1 σ 4 = d 3 σ 1 () d 0 σ 4 = d 3 σ 0 d 0 σ 3 = d 2 σ 0 () d 0 σ 1 = d 0 σ 0 d 1 σ 3 = d 2 σ 1 ()(). Eac o tese six conditions ass tat we ave areed on te 2-simplex tat we ave cosen to exibit te indicated composite. E.., te irst condition ass tat we ave cosen a 2-simplex exibitin a particular composite o and. Hence, te above procedure ives rise to a 4,2-orn, wic we can ill in S to obtain te desired 4-simplex σ. How can we understand tis 4-simplex? First, note tat its 5t ace is σ 2 ()() () () Te aces o σ 2 are exactly te tins we aven t cosen already: 2-simplices exibitin () as a composite o and, () as a composite o and, and ()() as a composite o bot and () and () and. Importantly, te ive aces o σ can be associated wit te edes o te associativity pentaon. Usin common notation or te latter (suc as in te context o a monoidal cateory), te associations are as ollows: (2.1) σ 0 α,,, σ 1 α,,, σ 2 α,,, σ 3 α,,, σ 4 α,,. Tese associations are easily reconized rom te spines o eac σ i. Tese ive simplices orm a simplicial spere in S, i.e., a map 4 S, 1 and σ ills tis 1 An n-dimensional simplicial spere in S is speciied by n + 1 n 1-simplices σ0,..., σ n satisyin te relations o (1.1), except droppin te requirement tat j, i.

6 6 E. RIEHL spere. In common parlance, a simplicial spere commutes exactly wen it can be illed. So σ witnesses te act tat te spere ormed by te 3-dimensional associativity data σ 0, σ 1, σ 2, σ 3, σ 4 commutes. In contrast to te situation o te associativity pentaon, tis commutativity is unbiased; we don t ave a natural way o composin 3 simplices so we don t attempt to combine tem in any particular order. For completeness sae, we will also say a ew words describin an alternate construction o associativity 4-simplices, were we instead construct and ill 4,1- or 4,3-orns. Tese situations are dual, in te sense tat one transorms into te oter wen we replace an quasi-cateory S by its opposite S op, also an quasi-cateory. 2 We will construct a 4,1-orn out o 3-simplices τ 0, τ 2, τ 3, and τ 4. In order or a iller τ to ave te spine,,,, te aces must ave te spines indicated τ 0,, τ 2,, τ 3,, τ 4,,. Te required relations are as ollows d 3 τ 4 = d 3 τ 3 d 0 τ 4 = d 3 τ 0 d 0 τ 2 = d 1 τ 0 () = () d 2 τ 4 = d 3 τ 2 () d 0 τ 3 = d 2 τ 0 () d 2 τ 3 = d 2 τ 2 (()). Te top our relations proscribe tat we must irst coose 2-simplices exibitin te composites,, and (implicit in te ourt relation). We ten coose 2-simplices exibitin te composites () and (). Tese coices allow us to construct and ill 3,1-orns to obtain τ 0 and τ 4. Te 1st ace o τ 0 exibits our cosen () as a composite o and. Te it relation says tat we must tae tis 2-simplex to be te 0t ace o τ 2. Finally, te last relation says we must coose a 2-simplex to ill te 2nd aces o τ 2 and τ 3 ; as our previously cosen ede () is already exibited as a composite o wit, tis is no obstacle. We ill te 3,1-orns just constructed to obtain te 3-simplices τ 2 and τ 3, as depicted below. τ 0 () τ 2 (()) () () τ 3 (()) () τ 4 () By te construction just iven, tese 3-simplices it toeter to orm a 4,1-orn in S, wic can be illed to obtain a 4-simplex τ. Te missin ace is τ 1 (()) () 2 Te opposite o a simplicial set S : op Set is obtained by precomposin wit te endounctor o tat reverses te order o eac object [n]. Tis unctor taes : [n] [m] to te unction op : [n] [m] deined by op (i) = m (n i).

7 ASSOCIATIVITY DATA IN AN (, 1)-CATEGORY 7 As beore, te aces o te 4-simplex τ can be associated to te edes o te associativity pentaon, and τ sould be tout o as a witness to te unbiased commutativity o tese 3-simplices dimensional associativity. Now we move up a level to consider 5-simplices tat can be tout o as a piece o associativity data or te sequence o composable morpisms,,,, l. Any 5-simplex as six aces, wic we identiy by teir spines. Given a 5-simplex σ wit spine,,,, l, its aces ave te spines indicated: σ 0,,, l σ 1,,, l σ 2,,, l σ 3,,, l σ 4,,, l σ 5,,,. As beore it is possible to mae a sequence o coices in suc a way as to construct a 5,1-, 5,2-, 5,3-, or 5,4-orn wose iller σ is suc an associativity 5-simplex. Rater tan describe te details o suc a construction, we switc perspectives somewat and consider toeter all o te relations satisied by te six aces tat orm te simplicial spere illed by σ. Tey are listed in te ollowin table: d 3 σ 5 = d 4 σ 3 d 2 σ 5 = d 4 σ 2 d 1 σ 5 = d 4 σ 1 d 0 σ 4 = d 3 σ 0 d 0 σ 3 = d 2 σ 0 d 0 σ 2 = d 1 σ 0 d 2 σ 4 = d 3 σ 2 d 1 σ 4 = d 3 σ 1 d 1 σ 3 = d 2 σ 1 d 4 σ 5 = d 4 σ 4 d 0 σ 5 = d 4 σ 0 d 0 σ 1 = d 0 σ 0 d 1 σ 2 = d 1 σ 1 d 2 σ 3 = d 2 σ 2 d 3 σ 4 = d 3 σ 3 Tese relations as tat certain aces o te 4-simplices σ i are equal, i.e., tat we coose certain 3-simplices consistently wen constructin te σ i. We ll identiy tese cosen 3-simplices wit teir spines or simplicity. Te requirement tat two 3-simplices are equal implies tat teir aces are te same, so we don t care to speciy wat tese 2-simplices are anymore. Te top tree rows o relations correspond, respectively, to te 3-simplices wit spines,,,,,,,, l,, l,, l,, l,, l,, l Tese correspond to te nine edes o te associaedron K 5 tat are sared by two pentaonal aces in te same manner o te correspondence iven in te previous section: namely, te 3-simplex wit spine x, y, z is associated to α x,y,z. Tis sould be compared wit (2.1). Te relations in te ourt row correspond, respectively, to te 3-simplices,, ;,, ; and,, l. We will say more about tese relations in a moment. Te inal row o relations correspond, respectively, to te 3-simplices wit spines,, l;,, l;,, l. Te morpism sould be te bac ede o te 3-simplex,, servin as a common composite or wit and wit, and similarly o course or te oter triples. Tese last two rows o relations correspond to te edes o te associaedron K 5 tat surround te tree naturality squares in te ollowin manner. For example, te relation d 4 σ 5 = d 4 σ 4 corresponds to te top and bottom arrows o te naturality square below; note tat tese arrows arise

8 8 E. RIEHL rom te same natural transormation α,,. l((())) l((())) (l)(()) (l)(()) Te relation d 1 σ 2 = d 1 σ 1 corresponds to te let and rit arrows o tis square, wic arise rom te natural transormation α,,l. Te reason tese aces o te associaedron are squares and not some oter sape as to do wit te act tat te inormation tey contain is someow te product o two K 3, wic are intervals eometrically. In te example above, one K 3 (or, extendin our analoy downward, one 3-simplex) contains te data associated wit te triple,, wile te oter contains te data associated wit te triple,, l. Te product o tese two intervals is a square. Wen we describe te associaedron K 6 in te next section, we will see tat it is productive to tin o some o its aces as (non-trivial) products in a similar manner. In conclusion, te 5-simplex σ can be viewed as data witnessin te commutativity o an unbiased 5-dimensional associaedron wose aces correspond to te lower-dimensional data described above dimensional associativity and beyond. We extend tis analoy one dimension urter in part to clariy ow te naturality squares o K 5 eneralize to ier dimensions. Te associaedron K 6 is te 4-dimensional polytope wose vertices correspond to eac way o multiplyin (non-associatively) an ordered list o six elements. Alternatively, eac vertex corresponds to a planar rooted binary tree wit six leaves. K 6 as 14 aces, wic are temselves polyedra. Seven o tese tae te orm o te associaedron K 5 wile te oter seven are pentaonal prisms (eometrically equal to K 4 K 3 ). As was te case or lower dimensions, one can construct a 6-simplex wit spine,,,, l, m by illin a 6,i-orn or some 0 < i < 6, were te aces o te orn are also constructed by illin orns. As usual, tese aces must satisy te relations (1.1). We aain omit te details o tis construction and instead describe te combinatorics o suc 6-simplices. A 6-simplex σ wit spine,,,, l, m will ave seven aces correspondin to te seven associaedral aces o K 6. Te aces σ 0,..., σ 6 satisy 21 relations, wic describe ow tese 5-simplices are lued toeter to orm te boundary o σ. Eac associaedral ace o K 6 is lued alon its pentaonal aces to eac oter ace eiter directly or via a connectin prism, and tese luins correspond bijectively to te 21 relations mentioned above. Additionally, te tree squares o an associaedral ace are lued to square aces o tree separate pentaonal prisms, and some o te pentaonal prisms are lued to eac oter directly alon te remainin square aces. We understand tis lower level luin to correspond to te coices tat ad to be made prior to constructin te 4-simplices tat will serve as aces o te σ i. Tis sould be compared wit te observation in te previous section tat we ad to coose a morpism, wic sould be te bac ede o te 3-simplex,, beore constructin te 3-simplex,, l.

9 ASSOCIATIVITY DATA IN AN (, 1)-CATEGORY 9 Te conclusion is te same as we ave asserted previously: a 6-simplex σ can be viewed as data witnessin te commutativity o te unbiased 6-dimensional associaedron wose aces correspond to te lower-dimensional data described above. Reerences [Lur08] J. Lurie. Wat is an -cateory? Notices o te AMS, 55(8): , [Lur09] J. Lurie. Hier Topos Teory. Princeton University Press, Department o Matematics, University o Cicao, 5734 S. University Ave., Cicao, IL address: eriel@mat.ucicao.edu