BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS

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1 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS JASHA SOMMER-SIMPSON Abstract Given roupoids G and H as well as an isomorpism Ψ : Sd G = Sd H between subdivisions, we construct an isomorpism P : G = H I Ψ equals SdF or some unctor F, ten te constructed isomorpism P is equal to F It ollows tat te restriction o Sd to te cateory o roupoids is conservative Tese results do not old or arbitrary cateories Contents 1 Introduction 1 2 Notation and overview o proo 2 3 Construction o Sd C 4 4 Sd preserves coproducts and identiies opposite cateories 8 5 Sd C encodes objects and arrows o C 10 6 Sd G encodes composition in G up to opposites 17 7 Construction o ψ and proo o te main teorem 27 Reerences 40 Appendix A Proo o Propositions 6119 and Introduction Te cateorical subdivision unctor Sd : Cat Cat is deined as te composite Π Sd s N o te nerve unctor N : Cat sset, te simplicial barycentric subdivision unctor Sd s : sset sset, and te undamental cateory unctor Π : sset Cat (let adjoint to N) Te cateorical subdivision unctor Sd as deiciencies: or example, it is neiter a let nor a rit adjoint Neverteless, Sd bears similarities to its simplicial analo Sd s, and as many interestin properties For example, it is known tat te second subdivision Sd 2 C o any small cateory C is a poset [4, c 13] In eneral, tere exist small cateories B and C suc tat Sd B is isomorpic to Sd C but B is not isomorpic to C or to C op In tis paper we sow tat suc examples do not occur in te cateory o small roupoids: i Sd G is isomorpic to Sd H or roupoids G and H, ten tere exists an isomorpism between G and H In broad strokes, te arument is as ollows Tere is a canonical identiication o objects in Sd G wit non-deenerate simplices in te nerve NG Any isomorpism between Sd G and Sd H induces a bijection between te 0-simplices o N G and te 0-simplices o N H, and similarly or 1-simplices Te objects and morpisms Date: December 15,

2 2 JASHA SOMMER-SIMPSON o a cateory correspond (respectively) to te 0-simplices and 1-simplices in te nerve o tat cateory, so or any isomorpism Ψ : Sd G Sd H tere are induced bijections Ob (ψ) : Ob (G ) Ob (H ) and Mor (ψ) : Mor (G ) Mor (H ) We will sow tat te restriction o tese bijections to any connected component o G determines a possibly-contravariant unctor into H Proceedin one component at a time and usin te act tat any roupoid is isomorpic to its opposite, we can use tese maps ψ to construct a covariant isomorpism between G and H 2 Notation and overview o proo In tis section we introduce notation used trouout te paper We ten ive a sketc o te proo Notation We write [n] or te totally-ordered poset cateory avin objects te non-neative inteers 0,, n I i and j are inteers satisyin 0 i < j n, we write [i < j] to indicate te morpism rom i to j in te cateory [n] Given an object c in a small cateory C, we sall write c or te unctor [0] C tat represents c Given a sequence 1,, n o morpisms in C satisyin dom n = cod n 1 or 0 < i n, we will write n 1 or te unctor [n] C tat represents te iven sequence n 1 Overview In Section 3, we introduce te simplex cateory and te nerve unctor N We deine te notion o deenerate simplices, and introduce Sd C as a cateory wose objects are te non-deenerate simplices o N C In Section 4, we demonstrate tat or any small cateory C, tere is an isomorpism Sd C = Sd(C op ) wic sends c to c and n 1 to 1 n For any small roupoid G, tere is an automorpism α G o 1 n We also sow tat C is connected i and only i Sd C is connected, and tat tere is an isomorpism Sd( i C i ) = i (Sd C i ) or any set {C i } o small cateories Tese acts will be useul or reducin te problem to te case o connected roupoids, and or obtainin covariant isomorpisms rom contravariant ones In Section 5, we identiy cateorical properties o Sd C tat encode some structural aspects o C Tis will be useul later wen we consider isomorpisms Sd B = Sd C and sow tat te identiied cateorical properties are preserved Given an object y in Sd C, we write mt n (y) or te set SdG tat sends c to c and n 1 to 1 1 x:[n] C Sd C (x, y) o morpisms taretin y and avin domain equal to some (non-deenerate) n-simplex We write mt(y) or te set n N mt n(y) o all morpisms taretin y in Sd C We sow tat or any non-deenerate n-simplex y : [n] C, rearded as an object in Sd C, te morpisms o Sd C taretin y are in bijection wit te monomorpisms o taretin [n] Tus, an object o Sd C is an n-simplex i and only i it is te taret o 2 n+1 1 morpisms in Sd C Given an object y in Sd C, we deine te aces o y to be te objects x suc tat tere exists a morpism x y We sow tat te proper aces

3 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 3 o a non-deenerate 1-simplex are precisely te 0-simplices dom and cod Given a 2-simplex in Sd C, we sow tat is let-inverse to i and only i tere are exactly two morpisms in te set mt 1 ( ), te domains o wic are te 1-simplices and I is not inverse to, ten mt 1 ( ) contains tree morpisms, te domains o wic are,, and Supposin tat G is a small roupoid, we sow in section 6 ow properties o Sd G encode te structure o G up to opposites Supposin tat and are non-identity morpisms in G, te local structure o SdG near and determines weter a iven 1-simplex satisies one o te equations = or = Te two propositions stated at te end o section 6 (and proved in te Appendix) are tis paper s most diicult results Section 7 concerns maps between subdivisions o cateories We suppose tat G and H are roupoids, and tat Ψ : Sd G Sd H is an isomorpism Te results rom Sections 5 and 6 are used to sow tat te structure o G encoded by Sd G must matc te structure o H encoded by Sd H For any objects x and y in Sd G, te map Ψ induces a bijection between Sd G (x, y) and Sd H (Ψ x, Ψ y), ence aces o y are sent to aces o Ψ y Because te cardinality o mt(y) is equal tat o mt(ψ y), Ψ sends n- simplices to n-simplices It ollows tat te elements o mt n (y) are sent bijectively to elements o mt n (Ψ y) We deine te map ψ so tat Ψ c = ψ c and ψ id a = id ψa and Ψ = ψ are satisied or eac object c and eac non-identity morpism in G We sow tat tis map ψ : G H is an isomorpism between te (undirected) raps tat underlie G and H Te structure o Sd G near iven 1-simplices,, and is te same as tat o Sd H near ψ, ψ and ψ Tereore, or any,, and in G, we ave = or = i and only i ψ = (ψ ) (ψ ) or ψ = (ψ ) (ψ ) Wit elp rom a roup-teoretic result due to Bourbaki, we sow tat i G is a connected sinle-object roupoid (tat is, a roup) ten te map ψ : G H is a possibly-contravariant isomorpism We ive an analoous result concernin connected roupoids tat ave multiple objects Workin wit arbitrary (non-connected) roupoids G and H, we suppose tat G equals te coproduct G 1 G 2, and tat ψ is contravariant on G 1 and covariant on G 2 Tere exist some roupoids H 1 and H 2 suc tat H = H 1 H 2, and suc tat te composite isomorpism Sd G 1 Sd G 2 = Sd(G 1 G 2 ) Ψ Sd(H 1 H 2 ) = Sd H 1 Sd H 2 restricts to isomorpisms Ψ 1 : Sd G 1 Sd H 1 and Ψ 2 : Sd G 2 Sd H 2 We ave maps ψ 1 and ψ 2, eac o wic is a restriction o ψ; by usin te act tat G 1 is isomorpic to its opposite cateory, we can lip te variance o ψ 1 to obtain a covariant isomorpism ψ 1 : G 1 H 1, and tus a covariant isomorpism ψ 1 ψ 2 between G and H

4 4 JASHA SOMMER-SIMPSON Appendix A is te combinatorial eart o tis paper It is devoted to te proo o Section 6 s two most involved results, wic concern te way tat Sd G encodes te relationsip between endomorpisms in a iven roupoid G 3 Construction o Sd C Tis section deines te nerve unctor, explains wat (non)deenerate simplices in te nerve are, and ives a construction o te cateorical subdivision unctor Sd : Cat Cat 31 Te Nerve o a Cateory For eac non-neative inteer n, let [n] denote te poset cateory wose objects are te inteers 0,, n and wose morpisms are iven by te usual orderin on Z Te simplex cateory is te ull subcateory o Cat wose objects are te posets [n] Note tat is a concrete cateory: or any m and n, te morpisms [m] [n] in can be identiied wit te order-preservin unctions {0,, m} {0,, n} Note also tat epimorpisms in are just order-preservin surjections, and monomorpisms in are just order-preservin injections Te nerve NC o a cateory C is te restriction rom Cat op to op o te omunctor Cat(, C ) Te nerve construction can be made into a unctor N : Cat sset by sendin a unctor F : B C to te natural transormation NB NC iven by Cat(, F ) Te elements o Cat([n], C ) are called te n-simplices o NC Given a morpism µ : [m] [n] in, write µ or te unction Cat([n], C ) Cat([m], C ) deined by µ x = x µ or eac n-simplex x o NC A simplex y : [m] C o NC is said to be deenerate i tere exists some simplex x o NC and some non-identity epimorpism σ in satisyin σ x = y I tis simplex x is not deenerate, ten it is called te non-deenerate root o y Proposition 311 Given an m-simplex y : [m] C o NC, te non-deenerate root o y is unique Proo Tis is a corollary o te standard eneral unique decomposition o a morpism in as a composite o an epimorpism and a monomorpism Suppose tat x 1 : [n 1 ] C and x 2 : [n 2 ] C are distinct simplices satisyin σ 1x 1 = y = σ 2x 2 or some epimorpisms σ 1 and σ 2 We will prove tat one o x 1 and x 2 must be deenerate At least one o σ 1 and σ 2 must be a non-identity morpism, or oterwise x 1 = y = x 2 Assume witout loss o enerality tat σ 2 id, and let ν 1 : [m] [n 1 ] be rit-inverse to σ 1 Ten tere is equality x 1 = (σ 1 ν 1 ) x 1 = ν 1σ 1x = ν 1y = ν 1σ 2x 2 = (σ 2 ν 1 ) x 2 By unique decomposition we ave σ 2 ν 1 = ν σ or some monomorpism ν and some epimorpism σ, resultin in te equality x 1 = (σ 2 ν 1 ) x 2 = (ν σ ) x 2 = σ (ν x 2 ) Because σ 2 id, we must ave σ id, and tus te above demonstrates tat x 1 is deenerate

5 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 5 Te above proposition olds or simplices in arbitrary simplicial sets, not just or simplices in te nerve o a cateory We will use te notation λ(y) to denote te non-deenerate root o y No 0-simplex o NC is deenerate, and eac 0-simplex [0] C may be rearded as pickin out a sinle object o C For a positive inteer m, eac m-simplex x : [m] C is identiied wit te sequence (312) x(m) m 1 x(m 1) x(1) x(0) o morpisms obtained by settin eac i equal to te imae under x o te morpism [i 1 < i] in [m] Suc an m-simplex x is non-deenerate i and only i all o te arrows i are non-identity morpisms o C 32 Construction o Sd C Let C be a small cateory Te objects o Sd C are te non-deenerate simplices o NC Given objects x : [m] C and y : [n] C o Sd C, te morpisms x y in Sd C are te pairs (σ, ν) o morpisms in satisyin cod σ = [m] and cod ν = [n] suc tat (1) ν is a monomorpism, (2) σ is an epimorpism, and (3) tere is equality σ x = ν y Te identity morpism x x is iven by te pair (id ν [m], id [m] ) [k] [n] Condition (3) above requires tat dom σ = [k] = dom ν or some σ y k, as in te commutative diaram to te rit Note tat or eac morpism (σ, ν) : x y in Sd C, tere is a multivalued unction [m] x C {0,, m} {0,, n} iven by ν σ 1 Here σ 1 denotes te subset o {0,, m} {0,, k} tat is inverse to σ (as a binary relation), and ν σ 1 is te composite o binary relations ν and σ 1 Composition o morpisms in Sd C is induced by composition o suc multivalued unctions See Fiure 1 on pae 6 or visualization o a test morpism in Sd C Fiure 2 (on pae 7) depicts two composable morpisms in Sd C, and Fiure 3 depicts te composite Example 321 Let B be te cateory ( ) wit tree objects and two non-identity arrows, and let C be te cateory ( ) opposite to B We see tat B and C are not isomorpic, yet te barycentric subdivisions Sd B and Sd C are bot isomorpic to te cateory ( ) wit ive objects and our non-identity arrows We will later prove tat Sd C = Sd(C op ) or any cateory C Te ollowin example is due to Jonatan Rubin: 1 Example 322 Consider N and Z as poset cateories Tere is no isomorpism between N and Z, yet te subdivisions Sd N and Sd Z are isomorpic Generally, i T is a totally-ordered set (rearded as a poset cateory) ten te n-simplices o Sd T are all te linearly-ordered subsets o T avin n + 1 elements I x and y are objects in Sd T and i x and y are te correspondin subsets o T, ten tere exists a morpism x y in Sd T i and only i x is a subset o y Tus Sd T is te poset ( ) {F T 0 < F < }, o inite non-empty subsets o T, ordered by inclusion 1 Private communication

6 6 JASHA SOMMER-SIMPSON I T and T are totally-ordered sets ten any set bijection between T and T induces an isomorpism Sd T = Sd T o cateories, reardless o weter te iven bijection between T and T is order-preservin 33 Concernin SdF Let B and C be small cateories, and let F be a unctor B C Te unctor Sd F sends eac 0-simplex y : [0] B to te 0-simplex F y : [0] C Usin te notation rom Section 2, tis means tat b is sent to F (b) or eac object b o B For larer m, SdF sends eac m-simplex (312) to te subsequence o F ( x(m) ) F ( m) F ( x(m 1) ) F ( x(1) ) F ( 1) F ( x(0) ) consistin o non-identity arrows Formally, eac m-simplex x : [m] B is sent by SdF to te non-deenerate root o F x Multivalued unctions can be used to calculate were SdF sends te morpisms o Sd B For te purposes o tis paper, a precise ormulation o SdF s value on morpisms o Sd B is not necessary Example 331 Below is a depiction o a sample morpism x y in Sd C [4] 1 d c a b a y ν σ [2] id a d a a (σ,ν) [1] d a x Fiure 1 Here x : [1] C picks out te morpism, and y : [4] C picks out te sequence 1 o arrows in C Te dased lines illustrate ow and compose to, and ow 1 and compose to id a Te injection ν : [2] [4] as imae {0, 2, 4}, and σ : [2] [1] is te surjection satisyin σ(0) = 0 = σ(1) We ave assumed tat is not let-inverse to, and tat none o, 1,, are identity morpisms Te dased lines determine a multivalued unction rom {0, 1} to {0, 1, 2, 3, 4}

7 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 7 Example 332 Below, two composable morpisms x y and y z in Sd C are depicted end-to-end k 1 l 1 l k 1 z k 1 k y x Fiure 2 Te multivalued unctions associated to tese morpisms x y and y z can be used to calculate te composite arrow in Sd C Tis composite is depicted below k 1 l 1 l k 1 z x Fiure 3

8 8 JASHA SOMMER-SIMPSON Example 333 Te iure below depicts a sample calculation Te unctor SdF : Sd B Sd C is applied to a morpism x y o Sd B F y y i j k F ω 2 F id F j F k F F F j F k λ(f y) j F j i F ( ) id λ(f x) x F x ω 1 F j F ( ) Fiure 4 Te iven morpism x y is sent to a morpism λ(f x) λ(f y) in Sd C, were λ(f x) is te non-deenerate root o x (and similarly or λ(f y)) Te epimorpisms ω 1 and ω 2 are te unique maps in satisyin ω 1λ(F x) = F x and ω 2λ(F y) = F y We suppose ere tat F i is an identity morpism in C, and tat F is not let inverse to F 4 Sd preserves coproducts and identiies opposite cateories Tis section states two lemmas tat will be used later in te paper In particular, we sow in Lemma 41 tat te unctor Sd does not distinuis between a cateory C and its opposite cateory C op, and in Lemma 44 tat Sd preserves coproducts Te latter lemma allows or reduction o tis paper s main teorem to te case o connected roupoids Te ormer ilits a undamental issue: or any small cateory C we ave an isomorpism Sd C Sd(C op ) Tereore, a naive attempt to construct a map B C rom a iven isomorpism Sd B Sd C could result in contravariance As mentioned in te introduction, tis pitall can be avoided wen workin wit roupoids, as any roupoid is isomorpic to its opposite via inversion Lemma 41 Let C be a small cateory Tere is a canonical isomorpism (42) Sd C = Sd(C op ) between te subdivision o C and te subdivision o te opposite cateory

9 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 9 Proo Te claimed isomorpism sends eac m-simplex (312) in Sd C to te m- simplex m 1 x(m) x(m 1) x(1) x(0) in Sd(C op ) I µ is a arrow [m] [n] in, write µ or te arrow [m] [n] deined by µ (m i) = n µ(i) Te claimed map (42) is deined on morpisms o Sd C by (σ, ν) (σ, ν ) Tus, or any statement about te relationsip between Sd C and C, tere is a dual statement about Sd C and C op Example 43 For any small roupoid G we ave te inversion isomorpism G op G deined on morpisms by 1 Te subdivision o tis isomorpism is a map Sd(G op ) Sd G, wic can be composed wit te map Sd G Sd(G op ) rom Lemma 41 to obtain an isomorpism α G : Sd G Sd G Tis unctor α G sends eac m-simplex (312) o Sd G to te m-simplex x(m) 1 m x(m 1) x(1) 1 1 x(0) Note tat Sd is aitul, as any unctor F : B C is determined by were SdF sends te 0 and 1-simplices o Sd B Unless te roupoid G is a discrete cateory, te map α G is not equal to te subdivision o any automorpism G G Tus, or any automorpism ξ o a non-discrete roupoid G we ind an automorpism α G Sd ξ o te cateory Sd G Tereore, we ave te inequality in te case were G is not discrete Aut(Sd G ) 2 Aut(G ) Lemma 44 Te unctor Sd preserves coproducts A cateory C is connected i and only i its subdivision Sd C is connected Proo Subdivision Sd is iven by te composite Π Sd s N Here Π : sset Cat is let adjoint to N, and Sd s : sset sset denotes barycentric subdivision o simplicial sets Te unctors Π and Sd s are let adjoints, and te nerve unctor N preserves coproducts, so Sd does too Tereore, i Sd C is connected ten C must be connected, or oterwise we would ave C = i C i and ence Sd C = i Sd C i Suppose now tat C is a connected cateory To prove tat Sd C is connected we will introduce some new notation For an object c in C, let c denote te 0-simplex [0] C sendin 0 to c For a morpism in C, let denote te 1-simplex [1] C tat sends te morpism [0 < 1] to For any non-identity morpism in C tere are arrows dom (id [0],δ) and cod (id [0],δ ) in Sd C, were δ and δ are te monomorpisms [0] [1] avin respective imaes {0} and {1} Tus, or any sequence o objects in C, tere is a sequence a b c a b c o objects in SdC Tereore, all 0-simplices are in te same connected component To complete te proo, let δ : [0] [m] demote te morpism tat sends 0 to 0, and

10 10 JASHA SOMMER-SIMPSON note tat or any m-simplex x in Sd C tere is a morpism (id [0], δ) wose domain is te 0-simplex δ x and wose codomain is x Tus all object o Sd C belon to a sinle connected component Note tat te sinle-object cateory [0] is isomorpic to its subdivision Sd [0] It ollows rom te previous lemma tat any discrete cateory C is isomorpic to its own subdivision 5 Sd C encodes objects and arrows o C Tis section concerns te relationsip between a cateory and its subdivision We identiy cateorical properties o Sd C tat correspond to certain structures in te cateory C Te identiied properties o Sd C are preserved by isomorpism: supposin tat Sd B = Sd C or some cateory B, te structures o C will appear te same was in B For example, B is a roupoid i and only i C is a roupoid Notation 501 As in Section 3, or any objects x and y in Sd C, we reard te morpisms x y in Sd C as pairs (σ, ν) o morpisms in satisyin (1) ν is a monomorpism, (2) σ is an epimorpism, and (3) tere is equality σ x = ν y As in Section 4, or any object a o C we let a : [0] C denote te 0-simplex tat represents a Given a morpism in C, write : [1] C or te 1- simplex tat represents We extend tis notation as ollows: iven a sequence 1,, m o morpisms in C satisyin dom n+1 = cod n or 1 n < m, write m 1 or te m-simplex x : [m] C iven by te diaram x(m) m 1 x(m 1) x(1) x(0) in C, were i is equal to x([i 1 < i]) or eac i above Tis notation is inspired by te bar construction on roups For a natural number m, identiy eac nonempty subset S o {0,, m} wit te monomorpism in avin domain [ S 1 ] {0,1} [2] [1], codomain [m], and imae S For example, i x equals or x some morpisms : a b and : b c, ten we can identiy {0, 1} and {0, 2} wit injections [1] [2] to obtain equalities {0, 1} x = C and {0, 2} x = Recall tat te non-deenerate root o an m-simplex y : [m] C is te unique non-deenerate simplex λ(y) : [n] C tat satisies ω(y) λ(y) = y or some surjection ω(y) : [m] [n] Settin y = m 1, we obtain λ(y) by omittin identity arrows rom te sequence 1,, m Te map ω(y) is uniquely determined by te equations (502) ω(y)(0) = 0 and (503) ω(y)(i + 1) = { ω(y)(i) ω(y)(i) + 1 i i+1 is an identity morpism i i+1 is a non-identity morpism were (503) above olds or all i < m An m-simplex y : [m] C may be rearded as an object o Sd C precisely wen it is non-deenerate

11 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 11 Deinition 504 Given an object y o Sd C, let mt(y) denote te set o morpisms taretin y in Sd C { Mor (Sd C ) cod = y} Proposition 505 Given an m-simplex y, rearded as an object o Sd C, tere is a bijection between te set o nonempty subsets o {0,, m} and te set mt(y) o morpisms taretin y Tis bijection sends a subset S o {0,, m} to te morpism (ω(s y), S) : λ(s y) y in Sd C Proo Identiyin eac subset S wit te monomorpism [ S 1 ] [m] avin imae S, it suices to sow tat te map { monomorpisms o taretin [m] } mt(y) iven by S (ω(s y), S) is bijective Tis pair (ω(s y), S) is to be rearded as a morpism rom λ(s y) to y, witnessed by te equality ω(s y) λ(s y) = S y Note tat ω(s y) is an identity map i and only i S y is non-deenerate Te iven unction is injective: i S 1 S 2 ten (ω(s1y), S 1 ) (ω(s2y), S 2 ) To ceck surjectivity, suppose tat (σ, ν) is a morpism z y in Sd C Writin S = im(ν) so tat S y = ν y, we obtain te equality S y = σ z wic demonstrates tat z is te non-deenerate root o S y Tereore we ave z = λ(s y) and σ = ω(s y) Lemma 506 An object y o Sd C is an m-simplex i and only i te set mt(y) as cardinality 2 m+1 1 Proo Every object o Sd C is a m-simplex or some m Tere are 2 m+1 1 nonempty subsets o {0,, m}, and just as many morpisms taretin eac m- simplex Tus te cateorical structure o Sd C encodes te speciic dimension o simplices Note tat i x is an m-simplex and y is an n-simplex suc tat tere exists some non-identity morpism x y in Sd C, ten we must ave inequality m < n Tis is because i (σ, ν) is an epi-mono pair satisyin σ x = ν y, ten we ave dom σ = [k] = dom ν or some k satisyin m k n We obtain a strict inequality m < n i eiter σ or ν is a non-identity morpism Example 507 (Automorpisms o Sd [n]) Te cateory [n] is a inite poset Tereore, as discussed in Example 322, tere is a canonical isomorpism between te subdivision Sd [n] and te poset o non-empty subsets o {0,, n}, wit order iven by subset inclusion Under tis isomorpism, eac 0-simplex k o Sd [n] is sent to te sinleton subset {k} o {0,, n} It ollows tat eac automorpism o Sd[n] is determined by a permutation o te set 0,, n o 0-simplices Tereore, we ave an isomorpism Aut ( Sd[n] ) = Sn+1 between te roup o automorpisms o Sd[n] and te symmetric roup on n + 1 elements For comparison, te roup Aut([n]) is trivial or eac natural number n Deinitions 508 Let x and y be objects in Sd C suc tat te om-set Sd C (x, y) is non-empty Ten we say tat x is a ace o y I, in addition, x and y are distinct, ten x is a proper ace o y Te cateory o aces o y, written y, is te ull subcateory o Sd C wose objects are te aces o y

12 12 JASHA SOMMER-SIMPSON Te aces o an m-simplex y are precisely te simplices λ(s y) Tis is because, by Proposition 505, te maps into y are precisely o te orm (ω(s y), S) or S a non-empty subset o {0,, m} Te proper aces o y are tose simplices λ(s y) were S is a proper non-empty subset o {0,, m} Here is an example Let,, and be distinct non-identity morpisms in C, and assume tat = Write y =, and set a = dom, b = cod = dom, and c = cod I te objects a, b, c are all distinct ten te cateory o aces y is as in Fiure 5 I a equals c and is distinct rom b, ten y is as in Fiure 6 I a = b = c ten y is as in Fiure 7 b b y y y c a a a Fiure 5 Fiure 6 Fiure 7 Note tat identity arrows ave been omitted rom te above diarams Proposition 509 Let : [1] C be a 1-simplex in Sd C Ten te proper aces o are cod and dom Proo By Proposition 505, te proper aces o are te non-deenerate roots λ({0} ) and λ({1} ) Every 0-simplex is non-deenerate, so we ave λ ({0} ) = {0} = dom and λ ({1} ) = {1} = cod Corollary 5010 A non-identity morpism is an endomorpism i and only i te 1-simplex as just one proper ace Note tat Sd C does not distinuis wic ace o corresponds to domain and wic to codomain For example, write or te arrow 0 1 in [1], and consider te cateory Sd [1] pictured to 0 1 te rit Note tat te non-identity automorpism o Sd [1] switces 0 wit 1 Tus, Sd introduces symmetry, illustrated enerally by te canonical isomorpism Sd C = Sd(C op ) At best, we can ope or Sd C to encode C up to opposites 51 Encodin trianles We ave seen tat te proper aces o 1-simplices correspond to teir domain and codomain; we now o up one dimension to see ow 2-simplices in Sd C codiy relationsips amon morpisms in C Notation 511 Given an object y o Sd C, write mt n (y) or te set { Mor (Sd C ) cod = y and dom is an n-simplex} o morpisms taretin y tat ave source equal to some non-deenerate n-simplex

13 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 13 Some o te comin proos will require explicit calculation o te sets mt 0 (y) and mt 1 (y) or 2-simplices y Below we state some eneral acts tat will make suc calculation easier Elements o mt n (y) are morpisms (ω(s y), S) : λ(s y) y suc tat λ(s y) is an n-simplex Note tat te sets mt n (y) partition mt(y) Also, note tat λ(s y) as dimension less tan or equal to S y, wic in turn as dimension equal to S 1 Tereore, to ind mt n (y) it is enou to consider only tose S wit S 1 n Indeed, i S 1 < n ten dim(λ(s y)) dim(s y) S 1 < n, ence te dimension o λ(s y) is less tan n, and te morpism (ω(s y), S) cannot be in mt n (y) Finally, note tat i y is an m-simplex and i S is equal to te ull set {0,, m}, ten S y = y is non-deenerate, in wic case we ave equality dim(λ(s y)) = dim(y) = m Tereore, to ind mt n (y) in te case were n < m, it is ood enou to consider te proper subsets S o {0,, m} Te next order o business will be to sow ow Sd encodes inversion Tis will allow us to determine by lookin at Sd C weter te cateory C is a roupoid Moreover, i G is a roupoid and is a 1-simplex in Sd G, bein able to ind 1 will elp determine weter iven 1-simplices and satisy = Proposition 512 Let, be non-identity morpisms in C satisyin dom = cod Set y equal to te 2-simplex I is let-inverse to ten mt 0 (y) as our elements and mt 1 (y) as two elements On te oter and, i te composite is not an identity arrow ten mt 0 (y) and mt 1 (y) ave tree elements eac Proo For any object x o Sd C, te om-set Sd C (x, x) contains only one morpism We ave te identity {0, 1, 2} y = y, tereore te map rom Proposition 505 restricts to a bijection between te set o non-empty proper subsets o {0, 1, 2} and te union mt 0 (y) mt 1 (y) By countin subsets o {0, 1, 2} we ind mt 0 (y) + mt 1 (y) = 6 For eac sinleton subset {i} we ave a morpism wit domain {i} y, ence mt 0 (y) as at least tree elements Te subsets {0, 1} and {1, 2} correspond to morpisms y and y, tus mt 1 (y) as size at least two It remains to consider te subset {0, 2} We ave {0, 2} y =, and tus te bijection rom Proposition 505 sends {0, 2} to an arrow λ( ) y in Sd C I te composite is an identity arrow ten λ( ) is a 0-simplex On te oter and, i is not let-inverse to ten we ave a tird element o mt 1 (y) Given te identiication o objects in Sd C wit simplices o NC, we may tink o eac 2-simplex in Sd C as witnessin a trianle-saped commutative diaram in C Deinition 513 Let,, and be morpisms in C, and let y be a 2-simplex o NC Say y is o te orm i y represents te composite o some pair amon

14 14 JASHA SOMMER-SIMPSON,, yieldin te tird morpism Explicitly, y is o te orm ollowin is satisied: i one o te 1) = and y =, 2) = and y =, 3) = and y =, 4) = and y =, 5) = and y =, 6) = and y = Te expression and does not encode te order o,, and ; te orms are loically equivalent We will sometimes label vertices o tese trianles For example, i y : [2] C represents te commutative trianle (514) c b in C, ten we will say tat y is o te orm c a Note tat i tere is a nondeenerate 2-simplex o te orm ten te two morpisms y[0 < 1] and a b y[1 < 2] must be non-identity, were [i < j] denotes te morpism i j in te cateory [2] Tereore, i y is 2-simplex in Sd C o te orm, ten y satisies eiter 1) y = and is let inverse to, or 2) y = and is let inverse to Te ollowin Lemma sows ow tis trianle notation summarizes inormation concernin 2-simplices in Sd C id Lemma 515 Let y : [2] C be a non-deenerate 2-simplex Ten y is o te orm or some non-identity arrows,, and i and only i tere are tree morpisms in te set mt 1 (y), and tese morpisms ave respective domains,, and On te oter and, y is o te orm i and only i tere are two morpisms in te set mt 1 (y), and tese morpisms ave respective domains and id Proo Let y be a non-deenerate 2-simplex [2] C By deinition, y is o te orm or some morpisms,, and in C Assume witout loss o enerality tat y = Ten we ave =, aain by deinition Let a, b, c satisy a = dom and cod = b = dom and cod = c as in te commutative trianle (514) above By Proposition 505, te non-identity elements o mt(y) are in bijection wit te proper non-empty subsets o {0, 1, 2}

15 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 15 Suppose irst tat all tree o,, and are non-identity morpisms Ten te non-identity elements o mt(y) are te morpisms displayed below: a (id [0],{0}) y (id [1],{0,1}) y b (id [0],{1}) y (id [1],{1,2}) y c (id [0],{2}) y (id [1],{0,2}) y On te oter and, i is let-inverse to ten ten we must ave a = c and = id a In tis case, writin σ or te unique epimorpism [1] [0], te non-identity elements o mt(y) are te morpisms displayed below: a (id [0],{0}) y b (id [0],{1}) y a (id [0],{2}) y (id,{0,1}) y (id,{1,2}) y a (σ,{0,1}) y Tus we ave mt 0 (y) = 4 and mt 1 (y) = 2 i is let-inverse to, wereas mt 0 (y) = 3 = mt 1 (y) i is non-identity Example 516 By Lemma 506 we ave mt 0 (y) + mt 1 (y) = 6 or any nondeenerate 2-simplex y I y is o te orm c b a and satisies mt 0 (y) = 3, ten te morpisms in mt 0 (y) ave respective domains a, b, and c, as in Fiures 5 trou 7 Suppose instead tat y is o te orm b a a, id a satisyin mt 0 (y) = 4 Lettin σ denote te unique epimorpism [1] [0], te elements o mt 0 (y) are iven by te pairs (σ, {0, 2}) : a y, (id [0], {0}) : a y, (id [0], {1}) : b y, and (id [0], {2}) : a y

16 16 JASHA SOMMER-SIMPSON I a and b are distinct, ten y appears as in Fiure 8 I a = b but, ten y is as in Fiure 9 I = ten y is as in Fiure 10 a y y y b a a Fiure 8 Fiure 9 Fiure 10 Lemma 517 (Inverse criterion) A non-identity morpism in C is sel-inverse i and only i tere exists a 2-simplex o te orm in Sd C For distinct non-identity morpisms and in C, is te two-sided inverse to i and only i tere exist two distinct 2-simplices o te orm Proo I tere is a 2-simplex o te orm id id id ten it must be equal to In tis case is let inverse to (by Proposition 512) Conversely, i is sel-inverse ten te 2-simplex is o te orm Suppose now tat and are distinct and tat Sd C contains two 2-simplices o te orm Tese 2-simplices must equal and, so is id let inverse to and vice versa Conversely, i is te two-sided inverse to ten and are o te orm id Tus, te subdivision Sd C encodes weter te morpisms in C are invertible Example 518 (Automorpisms o Sd D 3 ) Tis example illustrates ow te cateorical structure o a subdivision Sd C mit ail to distinuis between te composites and o endomorpisms and in C Let r and s be enerators or te six-element diedral roup D 3 D 3 = r, s r 3, s 2, rsrs Let φ denote te automorpism D 3 D 3 tat sends r to r 2 and s to s Let α D3 denote te map rom Section 4 deined by m m Below is a comparison o te maps α D3 and Sd φ wit te composite α D3 (Sd φ) α D3 : SdD 3 SdD 3 Sd φ : SdD 3 SdD 3 α D3 Sd φ : D 3 D 3 r r 1 r r 1 r r s s s s s s rs rs rs sr rs sr Note in particular tat te composite α D3 Sd φ sends r to r and s to s, but does not send rs to rs id

17 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 17 Given 1-simplices and correspondin to composable endomorpisms and in a roupoid G, we will sow in te ollowin section tat te cateorical structure o Sd G allows to pick out te set {, } o 1-simplices correspondin to te composites o and O course, tese composites may or may not be distinct 6 Sd G encodes composition in G up to opposites Tis section builds on te previous one As Lemma 602 will demonstrate, te assumption tat te morpisms and are invertible will make it easier to count 2-simplices o te orm Tereore, we now restrict our attention rom small cateories to small roupoids We will sow ow te cateorical structure o Sd G determines wic triples (,, ) satisy one o te equations = and = Deinition 601 Let, be non-identity arrows in a roupoid G, and let y be a 2-simplex in Sd G Say tat y is a iller or te trianle i y is o te orm or o te orm or some morpism in G Suc an is called a tird id side o Because a illers o are objects in SdG, any suc iller y must be nondeenerate Te tird sides o a iven trianle are classiied as ollows Lemma 602 (Possible tird sides) Let and be non-identity morpisms in a roupoid G, and suppose tat is a tird side o te trianle Ten must equal one o te ollowin six composites: (603),, 1, 1, 1, 1 Proo Assumin tat and are invertible, te composites (603) are te values o correspondin to eac o te cases in Deinition 513 Explicitly, any iller o must be one o te six diarams [2] G displayed below: Note tat some or all o te ormal composites (603) may be undeined, dependin on ow te domain and codomain o and matc up 61 Composites in Groupoids Suppose tat and are endomorpisms o an object in a roupoid G, and tat te composites and are distinct Te cateorical structure o Sd G need not distinuis between te 1-simplices and, as in Example 518 on pae 16 It is possible, owever, to pick te 1-simplices and out rom amon te oter 1-simplices in Sd G

18 18 JASHA SOMMER-SIMPSON Generally, iven non-identity morpisms and in G satisyin dom = cod or dom = cod, te local structure o Sd G near and determines weter a iven 1-simplex satisies = or = Tis will be key in provin unctorality o te map ψ mentioned in te introduction To acieve tis result, we introduce some terminoloy concernin relationsips between arrows in G Eac deinition below can be ormulated in terms o te proper aces { dom, cod } o 1-simplices Deinitions 611 (1) End-to-end morpisms: or or Morpisms and are end-to-end i neiter nor is an endomorpism and te intersection {dom, cod } {dom, cod } as one element Note tat tis implies te tree dots are necessarily distinct (2) Ends-to-ends morpisms: or Morpisms and are ends-to-ends i neiter nor is an endomorpism and tere is equality {dom, cod } = {dom, cod } Tis means tat te intersection {dom, cod } {dom, cod } as two elements (3) End-to-endo morpisms: or Morpisms and are end-to-endo i is not an endomorpism, is an endomorpism, and te intersection {dom, cod } {dom, cod } as one element (4) Endo-to-endo morpisms: Morpisms and are endo-to-endo i tey are bot endomorpisms o a common object (5) Unrelated morpisms Morpisms and are unrelated i te sets {dom, cod } and {dom, cod } ave no elements in common For example, non-endomorpisms and in G are end-to-end i and only i and ave one ace in common A non-identity morpism is an endomorpism i and only i te 1-simplex as exactly one proper ace Deinition 612 Let and be end-to-end morpisms We say tat and are sequential i dom = cod or dom = cod We say tat and are coinitial i dom = dom, and tat and are coterminal i cod = cod Deinition 613 Let and be ends-to-ends morpisms We say tat and are parallel i dom = dom and cod = cod We say tat and are opposed i dom = cod and dom = cod Observe tat end-to-end morpisms are sequential i and only i tey can be composed in some order Similarly, ends-to-ends morpisms are opposed i and only i tey can be composed in eiter order Note tat unrelated morpisms are never composable, and tat end-to-endo and endo-to-endo morpisms are always composable Below are criteria or te composability o end-to-end and ends-toends morpisms

19 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 19 Proposition 614 Let and be end-to-end morpisms in G Tere exists a unique iller or te trianle i and only i and are sequential Tere is more tan one iller or i and only i and are coinitial or coterminal Proo It will suice to sow tat i and are coinitial or coterminal ten tere are exactly two illers or, and tat i and are sequential, ten tere is exactly one iller or Given distinct elements i and j o te set {0, 1, 2}, we will write [i < j] or te morpism rom i j in te cateory [2] I y is a 2-simplex in SdG, ie a unctor [2] G tat sends [0 < 1] and [1 < 2] to non-identity arrows, ten: (1) te domain o y[0 < 1] equals te domain o y[0 < 2], (2) te codomain o y[1 < 2] equals te codomain o y[0 < 2], and (3) y[1 < 2] and y[0 < 1] are composable Given a 2-simplex y in SdG tat ills, we must ave = y[i < j] and = y[k < l] or some distinct morpisms [i < j] and [k < l] in [2] Functors preserve domain and codomain; by lookin at te source and taret o and, we can rule out combinations o i, j, k, l Suppose irst tat and are coinitial Ten any iller y o must satisy eiter = y[0 < 1] and = y[0 < 2], or = y[0 < 2] and = y[0 < 1] To see tis, note tat y sends coinitial pairs in [2] to coinitial pairs in G, and similarly or sequential and coterminal pairs Write = y[i < j] and = y[k < l], and note tat and are neiter sequential nor coterminal By contraposition, [i < j] and [k < l] are neiter sequential nor coterminal Tus te preimaes o and must be [0 < 1] and [0 < 2] I = y[0 < 1] and = y[0 < 2] ten we must ave (y[1 < 2]) =, ence y is te 2-simplex 1 I = y[0 < 2] and = y[0 < 1] ten we ave (y[1 < 2]) =, ence y is te 2-simplex 1 Tus, tere are exactly two illers or Te arument is similar supposin tat and are coterminal O all endto-end morpism pairs in te cateory [2], only [1 < 2] and [0 < 2] are neiter coinitial nor sequential Tereore, i y ills ten we must ave = y[1 < 2] and = y[0 < 2], or vice versa Tese two possibilities correspond to te cases y = 1 and y = 1 Te coterminal case is dual to te coinitial case in a sense made precise by te isomorpism SdG Sd(G op ) rom Lemma 41, wic sends eac n-simplex 1 n in SdG to te n-simplex n 1 in Sd(G op ) Finally, suppose tat and are sequential I dom = cod and i y ills ten we ave = y[1 < 2] and = y[0 < 1], ence y equals Similarly, i dom = cod and i y ills ten we ave = y[1 < 2] and = y[0 < 1], ence y equals In eiter case, tere is only one possible iller y Note tat te above result can ail i and are not invertible Corollary 615 Let and be end-to-end morpisms in G Ten and are sequential i and only i tere is a unique tird side o te trianle Tis tird side is necessarily equal to te composite o and

20 20 JASHA SOMMER-SIMPSON Proo Suppose irst tat and are sequential Te previous proposition sows tat tere is a unique 2-simplex iller, and tus a unique tird side, or te iven trianle For te reverse implication, suppose tat and are not sequential; we will sow tat tere are two distinct tird sides or te iven trianle Note tat nonsequential end-to-end morpisms must be eiter coinitial or coterminal Supposin irst tat and are coinitial, we ave two 2-simplex illers or, namely 1 and 1 Te correspondin tird sides are 1 and 1, wic must be distinct because cod cod Similarly, i and are coterminal ten we ave tird sides 1 and 1 o Tese tird sides must be distinct because dom dom Te ollowin result is analoous to te previous proposition, concernin endsto-ends morpisms Proposition 616 Let and be distinct ends-to-ends morpisms in G Tere are our illers or i and only i and are parallel Tere are two illers or te trianle i and only i and are opposed Proo It will suice to sow tat i and are parallel ten tere are exactly our illers or, and tat i and are opposed, ten tere are exactly two iller or As in te previous proo, we will write [i < j] or te morpism i j in [2] Suppose irst tat and are parallel, and tat y ills Ten we must ave = y[i < j] and = y[k < l] or some distinct morpisms [i < j] and [k < l] in [2] Moreover, te morpisms [i < j] and [k < l] cannot be composable because and are not composable Tis means tat i l and j k Tereore, i y ills ten tere are our possibilities: (1) = y[0 < 1] and = y[0 < 2], (2) = y[0 < 1] and = y[0 < 2], (3) = y[0 < 2] and = y[1 < 2], or (4) = y[0 < 2] and = y[1 < 2] Tus, tere are at most our illers Te cases (1)-(4) above correspond (respectively) to te 2-simplices y 1 = 1, y 2 = 1, y 3 = 1, y 4 = 1 To prove tat tere are exactly our illers or, we must sow tat te above illers y i are all distinct By lookin at te values y(0), y(1), y(2), we see tat te only pairs amon te illers y i tat could be equal are y 1, y 2 and y 3, y 4 We ave y 1 y 2 because y 1 [0 < 1] = = y 2 [0 < 1],

21 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 21 and similarly we ave y 3 y 4 because y 3 [1 < 2] y 4 [1 < 2] exactly our illers or Tus, tere are Suppose now tat and are opposed Te 2-simplices and are o te orm and, respectively We must sow tat tese are tese are te only two 2-simplices tat ill By te proo o Lemma 602, wic lists all potential illers o, any iller wic is not equal to or must be equal to one o te our illers 1, 1, 1, 1 But and are opposed, so we ave dom dom and cod cod Tereore none o te composites 1, 1, 1, and 1 are valid, ence none o te our illers above are deined It ollows tat tere are exactly two illers or Corollary 617 I and are opposed ends-to-ends morpisms in G, ten tere are exactly two tird sides o, namely and Proposition 618 Let and be end-to-endo morpisms in G Ten is te composite o and i and only i (1) and are parallel ends-to-ends morpisms, and (2) is a tird side o te trianle 1 Proo Tere are two cases: eiter cod = dom = cod, or dom = dom = cod Assume irst tat cod = dom = cod, as in te diaram to te rit One implication is clear: te composite is parallel to because is an endomorpism, and i we set = ten te 2-simplex 1 is witness to bein a tird side o 1 For te reverse implication, suppose tat is a tird side o 1 and tat is parallel to Ten and 1 are opposed ends-to-ends morpisms By te deinition o tird sides, tere must exist some non-deenerate 2-simplex o te orm 1 in SdG Tus, is a tird side o te trianle 1, and it ollows rom Corollary 617 above tat is equal to 1 or to 1 But is an endomorpism o te object cod, wereas 1 is an endomorpism o dom Because dom cod, we must ave = 1 and tereore = I we suppose instead tat dom = dom = cod, ten te proo ollows by a similar arument Te remainder o tis section establises a result analoous to Propositions , pertainin to te case were and are endo-to-endo For te time bein we will drop te composition symbol, writin (or example) or te composite and 2 or te composite I is sel-inverse, ten we will write 2 = id Te lemma below concerns composites 2

22 22 JASHA SOMMER-SIMPSON Lemma 619 (Square criterion) Let and be non-identity endomorpisms in G Ten 2 = i and only i SdG contains a non-deenerate 2-simplex o te orm Proo I 2 = ten te 2-simplex is o te orm suppose tat y : [2] G is o te orm possible 2-simplices o te orm Conversely, By Deinition 513, wic lists all, we must ave 2 = or = or = I = or i = ten we ave = id, wic contradicts our assumption tat is non-identity y 2 a Fiure 11 Let : a a be some endomorpism tat is not selinverse, and set y = Ten y is iven by te diaram above Remark 6110 Given a non-deenerate 2-simplex y : [2] G, te cateory o aces y is iven by one o te Fiures 5 trou 11 Tis is to say, tese seven Fiures classiy te possible cateories o aces o 2-simplices Tis is true or roupoids, but not or arbitrary cateories For example, in an arbitrary cateory we mit ave 2 = or some non-identity endomorpism Notation 6111 Given morpisms,, and in G, we will write tere exists a non-deenerate 2-simplex o te orm n i in Sd G Te notation means tat tere are exactly n distinct non-deenerate 2-simplices o tat orm Similarly, means tat no suc 2-simplices exist, and n means tat tere are at least n suc 2-simplices Lemma 6112 Suppose,,, and are endomorpisms in G, te morpisms,, and are distinct, and a 2-simplex y is simultaneously o te orm and Ten = Proo Te morpism is one o te y[i < j] or some 0 i < j 1 Tereore equals or or I equals or, ten or two distinct pairs i < j, k < l we ave y[i < j] = y[k < l] since y is o te orm On te oter and, since y is o te orm or,, all distinct, tis is impossible Tereore = Lemma 6113 Let and be non-identity endomorpisms in G, and suppose tat 1 and 2 1 I = 1, ten = 3 i and only i 2 4 I 1, ten = 3 i and only i 2 2

23 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 23 Proo Because is non-identity and 1 and 2 1, te morpisms id,, 2, and 3 are all distinct Te cases = 1 and 1 above correspond to weter or not 4 equals id Tere are ive distinct illers o 2, displayed below: Tese illers are o te orm 2 (respectively) 3, 2 3, 2, 2, and I 3 = = 1, ten tere are our distinct 2-simplices o te orm 2 namely 2, 2, 1 2, and 2 1 Conversely, i = 1 and tere are exactly our distinct 2-simplices o te orm 2 o tese 2-simplices must be o te orm 2 3, ten at least one Tis 2-simplex is simultaneously o te orm 2 and 2, so it ollows by te previous Lemma tat = 3 3 On te oter and, i = 3 1 ten tere are two distinct 2-simplices o te orm 2, namely 2 and 2 Conversely, i 1 and tere are exactly two distinct 2-simplices o te orm 2, ten tese 2-simplices must be equal to 2 and 2 (or oterwise te previous Lemma would ive = = 3 or = 1, contradictin our assumptions) Te 2-simplices 2 and 2 are simultaneously o te orm 2 te previous lemma tat equals 3 and 2 Te corollary below ollows directly rom te lemma above 3, and it ollows rom Corollary 6114 (Cube criterion) Let and be non-identity endomorpisms in G, and suppose tat 1 and 2 1 Ten = 3 i and only i eiter = 1 and 2 4, or 1 and 2 2 Given a 1-simplex in Sd G suc tat is an endomorpism satisyin 2 id and 3 id, te previous results can be used (or example) to ind te 1-simplices 2 and 3 Recall rom Lemma 602 tat i a iven 2-simplex is o te orm must be one o te composites (6115),, 1, 1, 1, 1,, ten

24 24 JASHA SOMMER-SIMPSON We sall now deine notation to set te stae or Lemma 6117, wic will state tat under certain conditions on and, te number o 2-simplices o te orm is equal to te number o composites above tat are equal to Notation 6116 Let and be non-identity endomorpisms o some object in G Suppose tat Write C(, ) or te set o quadruples (k, s, l, t) were (1) (k, l) is equal to (, ) or (, ), and (2) (s, t) is equal to (1, 1) or (1, 1) or ( 1, 1) Te six elements o C(, ) are all distinct, wereas te six composites (6115) mit not all be distinct Te evaluation map ev : C(, ) Mor (G ), deined by ev(k, s, l, t) = k s l t, ives a correspondence between te elements o C(, ) and te composites (6115) Given tis correspondence, we can tink o C(, ) as a set o ormal composites We will later make use o te sets C(, ) to keep track o Te ollowin Lemma sows tat, i we assume and are endomorpisms satisyin 2 and 2, tere is a bijection between C(, ) and te set o 2-simplex illers or, sendin eac ormal composite (k, s, l, t) to to a iller wose tird side equals k s l t We will later use te set C(, ) to simpliy a countin arument tat involves keepin track o te relationsip between 2-simplex illers and tird sides Lemma 6117 Let and be non-identity endomorpisms o some object in G Suppose tat and 2 and 2 For any morpism in G, te number o quadruples (k, s, l, t) C(, ) satisyin k s l t = is equal to te number o 2-simplices o te orm Proo Note tat or any non-identity and satisyin 2 and 2 as above, i is a tird side or ten must be distinct rom and Every 2-simplex iller o must be one o te six 2-simplices,, 1, 1, 1, 1 displayed in te proo o Lemma 602 Given te present assumptions on and, tese 2-simplices are all distinct For example, is distinct rom 1 because 1 is a non-identity morpism, and 1 is distinct rom 1 because 2 Tus we ave a bijection (, 1,, 1) (, 1,, 1) 1 (, 1,, 1) 1 (, 1,, 1) (, 1,, 1) 1 (, 1,, 1) 1 between C(, ) and te set o 2-simplex illers or We will let ζ : C(, ) {2-simplex illers or }

25 BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS 25 denote tis bijection Note tat ζ sends eac quadruple (k, s, l, t) to a 2-simplex o te orm Te map ζ restricts to an injection k s l t ζ : {γ C(, ) ev(γ) = } {2-simplices o te orm so tere are at least as many 2-simplices o te orm }, as tere are quadruples (k, s, l, t) satisyin k s l t = Because ζ is a bijection, we can prove tat ζ is a surjection by notin ev(γ) = wenever ζ(γ) is o te orm Indeed, ζ(γ) is o te orm, so i ζ(γ) is also o te orm ten equality ev(γ) = ev(γ) ollows rom Lemma 6112 We now ave a way to keep track o 2-simplices o te orm by usin ormal composites o,, 1, and 1 We will later deine an equivalence relation on ormal composites by (k, s, l, t) (k, s, l, t ) ev(k, s, l, t) = ev(k, s, l, t ) Tis equivalence relation ives a rap structure on te set C(, ), wit edes between equivalent elements Suc raps will be used to make easier te computations tat underlie te proos o Propositions 6119 and 6120 below Tese proos, ound in Appendix A, are te combinatorial eart o tis paper Under te assumption tat and satisy, 1, 2, and 2, Proposition 6119 below ives conditions tat are necessary and suicient or commutativity = Assumin tat (6118), 1, 2, 2, 2 1, 1 2, Proposition 6120 establises criteria necessary and suicient or a iven endomorpism to satisy one (or bot) o te equations = and = As usual, we aim to encode relationsip amon,, and in terms o te structure o te cateory Sd G in a neiborood o its objects,, and Proposition 6119 (Commutativity criterion) Let and be non-identity endomorpisms in G satisyin and 1 and 2 and 2 Ten equals i and only i or every every tird side o te trianle tere are an even number o 2-simplices o te orm To prove te above, we deine a rap G(, ) wose vertices are te ormal composites C(, ); te rap is deined to ave ede between distinct ormal composites γ 1 and γ 2 wenever ev(γ 1 ) equals ev(γ 2 ) By Lemma 6117, te number o 2-simplices o te orm is equal to te size o te connected component o G(, ) wose elements (k, s, l, t) satisy = k s l t By a combinatorial arument,

26 26 JASHA SOMMER-SIMPSON we sow tat equals i and only i every connected component o G(, ) as even cardinality A ull proo is in Appendix A Proposition 6120 Let and be non-identity endomorpisms o some object in G satisyin and 1 and 2 and 2 and 2 1 and 1 2 Let be anoter non-identity endomorpism o te same object in G Te cases below ive criteria under wic is equal to or to Cases 1-2 apply wen 2 = 2, and cases 3-4 apply wen 2 2 All possibilities are exausted Case 1: Suppose tat 2 = id = 2 I =, ten = = i and only i 6 I, ten equals or i and only i 3 Case 2: Suppose tat 2 = 2 and 2 id and 2 id I =, ten = = i and only i 2 I, ten equals or i and only i 1 or 3 Case 3: Suppose tat 2 2, and eiter 2 = id or 2 = id I =, ten = = i and only i 4 I, ten equals or i and only i 2 or 3 Case 4: Suppose tat 2 2 and 2 id and 2 id I =, ten = = i and only i and 1 and 1 I and 2 = 2, ten equals or i and only i 2 1 and 2 1 I and 2 2, ten equals or i and only i one o te ollowin is satisied: (1) 1 and 1 1 (2) and 1 1, or and two o te ollowin tree existential statements old: 2 See Appendix A or proo, 2 1, 1 2