SOME EXAMPLES OF SUBDIVISION OF SMALL CATEGORIES MARCELLO DELGADO Abstrct. The purpose of this pper is to build up the bsic conceptul frmework nd underlying motivtions tht will llow us to understnd ctegoricl subdivision, nd then work through some specific exmples. Contents 1. Subdivision of Simplicil Sets 1 2. The Nerve of him... 2 3. Fundmentl Ctegory 2 4. A Whole New Section 3 5. Concrete Exmples 4 Acknowledgements 7 References 7 1. Subdivision of Simplicil Sets The ide of subdividing ctegory rises from the notion of brycentric subdivision of simplicil complex. A brycentric subdivision decomposes n n-simplex into simplicil complex of n! n-simplices whose geometric reliztion is homotopy equivlent to the originl simplicil complex. It is with this ide in mind tht we will define the subdivision of simplicil set. First, we hve the notion of geometric simplicil complex, spce built from simplices. We cn move towrds n bstrct definition of simplicil complex. Definition 1.1 (Abstrct Simplicil Complex). A simplicil complex is set X nd collection of its finite subsets S (clled the simplices of X) with the property tht for ny simplex K of X, every subset of K is lso simplex of X. This gives us the fce nd degenercy mps from clss [2], whereby we cn include lower dimensionl simplices in higher dimensionl ones s fces nd we cn project from higher dimensionl simplex onto one of its fces. The key thing to tke wy from this bstrction from the geometric simplicil complexes from topology is the importnce of the fce nd degenercy mps, in prticulr their composition properties. Especilly when working with ordered simplicil complexes (which we cn do by imposing prtil ordering on X so tht the simplices re the totlly ordered subsets), these fce nd degenercy mps become mps of finite totlly ordered sets. This motivtes the following pir of definitions. Dte: July 29. 2010. 1
2 MARCELLO DELGADO Definition 1.2 ( ). Let [n] represent the finite poset {0 < 1 <... < n}. Then let denote the ctegory tht hs s its objects the finite posets [n] for ll nonnegtive integers n nd s its rrows the monotonic functions between these posets. Definition 1.3 (Simplicil Sets). A simplicil set is contrvrint functor K : op Set. The ctegory of simplicil sets is the ctegory of ll such contrvrint functors, [ op, Set], nd it is denoted sset. While bstrct, we note tht ech object of picks out set which serves s the set of simplices of tht dimension nd we inherit the fce nd degenercy mps with the composition properties we love from the monotonic mps in nd by contrvrincy. In the wy we hve subdivision of simplicil complexes, we cn define subdivision on simplicil sets, functor Sd : ssets ssets, which is explined in detil in My s notes [2]. With this functor, we cn define subdivision on smll ctegories by composing nd precomposing it with functors to nd from Ct, respectively. These re the fundmentl ctegory nd nerve functors. 2. The Nerve of him... Here, we will construct the nerve functor N : Ct sset. The nerve functor gives us wy to turn smll ctegories into simplicil sets in firly intuitive wy. First, we note tht every poset, nd in prticulr the posets [n], cn be seen s ctegories with t most one rrow between ny pir of objects nd monotonic mps between posets cn then be seen s functors. Thus, we hve n inclusion : Ct. For given smll ctegory C, define NC n to be the set of ll covrint functors [n] C. For n rrow µ : [m] [n], define µ := µ : NC n NC m. This collection of sets NC n defines simplicil set, where the objects [n] of re mpped to the corresponding set NC n nd rrows µ in re mpped to µ. Now suppose we hve functor F : C D. We get nturl trnsformtion F : NC ND given by F n = F : NC n ND n. Thus, we hve functor N : Ct sset given on objects by C NC nd on rrows by F F. Definition 2.1 (Nerve Functor). The covrint functor N : Ct sset is clled the nerve functor. 3. Fundmentl Ctegory Now we describe the fundmentl ctegory functor, τ 1 : ssets Ct, llowing us to ssocite ctegory to simplicil set in cnonicl wy, nd it will turn out tht this functor is left djoint to the nerve functor. Strting with simplicil set K, tke s the objects of τ 1 K the elements of K 0, the 0-simplices of K. For every 1-simplex y, hve n rrow y : d 1 y d 0 y. Tck on ll forml composites tht mke sense (i.e. form yz if d 0 z = d 1 y). Finlly, we impose the following two reltions: s 0 x = 1 x for x K 0 nd d 1 z = d 0 z d 2 z for z K 2 This gives us well-defined ctegory [2]. Definition 3.1 (Fundmentl Ctegory Functor). The functor τ 1 : sset Ct is clled the fundmentl ctegory functor.
SOME EXAMPLES OF SUBDIVISION OF SMALL CATEGORIES 3 Now tht we hve ll these pieces in plce, we re in position to define ctegoricl subdivision. Definition 3.2 (Subdivision). We define the subdivision functor Sd : Ct Ct s Sd := τ 1 Sd N, where the Sd in the composition is the subdivision functor on simplicil sets. While this definition mkes conceptul sense, it is not computtionlly friendly. To see concrete exmples of ctegoricl subdivision, we need more constructive look t subdivision. 4. A Whole New Section So in order to understnd the construction of the subdivision functor, we first need to introduce the concept of left fiber. Definition 4.1 ([1]). [Left Fiber] Let F : C D be functor nd let T ob D. The left fiber F/T of F over T is the ctegory of pirs (X, µ), where X ob C nd µ : F X T with rrows f : (X, µ) (X, µ ) corresponding to rrows f : X X in C s.t. µ F f = µ. Digrmmticlly, F f F X F X µ µ T Let s consider the objects of, the posets [n], s ctegories. Then we hve n inclusion functor : Ct. If we consider n rbitrry smll ctegory C in Ct, we cn look t the left fiber of over C: /C. The objects of this ctegory re the ll the rrows [p] C for ll [p], which re exctly ll the the simplices of NC. In the following discussion, we will use [n] to represent the imge of [n] under, but for ny rrows f : [m] [n], f : [m] [n] will represent the imge of f in Ct. If we re lredy using subscripts for something else, then we ll use superscript : f. Now tht we hve this ctegory, consider q simplex X NC q nd surjective (monotonic) mp s : [q + 1] [q] in. If d, d : [q + 1] [q] re the two right inverses of s, then we sy tht d, d : ([q], X) ([q + 1], Xs) re elementry equivlent, denoted d d. Clerly, is symmetric nd reflexive. Let be the miniml equivlence reltion on the rrows of /C generted by. The we cn define [ /C] := ( /C)/. Now tht we hve this, we cn define the subdivision of smll ctegory C: Definition 4.2 (Subdvision). For smll ctegory C, the subdivision of C, Sd(C), is the full subctegory of [ /C] tken on the nondegenerte simplices of C, where nondegenrte simplex is one in which none of the q composed rrows is n identity rrow for some object in C. This functor is equl to the functor given erlier in terms of simplicil sets, i.e. Sd = τ 1 sd N [1].
4 MARCELLO DELGADO 5. Concrete Exmples Here in this section, we will compute few concrete exmples of subdivisions. We will strt with some trivil exmples nd work up to some interesting ones, including one infinite exmple. We ll see multiple times instnces of how two consecutive subdivisions yields poset. It s worth noting t this point tht when we compute these exmples, since our result will only depend on nondegenerte simplices, we will omit them in the erly stges of the computtion (despite the fct tht they should be there in the ctegories we re looking t t tht stge) becuse they ll be irrelevnt in the finl result. Further, when we re looking for rrows in /C, we do not need to consider the σ i mps. To see this, suppose we hve n rrow in /C corresponding to some σ i : [q + 1] [q]. Then this tells us tht the (q + 1)-simplex we re considering fctors through s q-simplex, nd so one of its composble rrows is n identity. Thus, the (q + 1)-simplex is degenerte, nd doesn t pper in Sd(C). Finlly, we will only consider the rrows δ i. The reson for this is tht ny rrows in cn be written s composition of the σ i nd δ i, nd so the rrows in /C corresponding to these compositions re just the composition of the rrows for the ssocited δ i nd σ i (remembering of course tht the σ i will become irrelevnt). Hving this, let s consider our first concrete exmple. Consider the ctegory 1 consisting of single object nd its identity rrow. First, let s mke list of ll the nondegenerte simplices of 1. We only hve one nondegenerte 0-simplex, : [0] 1, nd no nondegenerte simplices of higher dimension. Since there re no other simplices, then there re no nonidentity rrows in /C, nd since there re no nontrivil composble pirs, then we hve no elementry equivlent pirs. Thus, Sd(1) = 1. Now consider the ctegory 1, consisting of single object which hs one nonidentity rrow :. In this exmple, we ll ssume tht f f = 1. Now, we hve one 0-simplex, : [0] 1, nd one nondegenerte 1-simplex, : [1] 1. Agin, there re no other nondegenerte simplices. Thus, we re considering ctegory with two objects: nd. Our potentil rrows re, : [0] [1], which in /C will be of the form δ0, δ1 : X for some object (simplex) X. We re looking for simplices X : [0] 1 such tht X =. picks out the codomin (by deleting 0) of, which is 0, so X = 0. Anlogously, we hve n rrow δ1 : 0 since picks out 0 (by deleting 1). nd cnnot be right inverses for the sme mp, so they re not elementry equivlent. Therefore, we get tht Sd(1 ) = 0 We lso note the following useful heuristic: If we view 1 s 0 0, then it s subdivision is 0 δ0 δ 1 0. Our next exmple is the ctegory 2: 0 1. There re three nondegenerte simplices: 0 : [0] 2, 1 : [0] 2, nd : [1] 2. Since there is no nonidentity rrow [0] [0], we cn ssume tht ll nonidentity rrows in Sd(2) will be either of the form 0 or 1. Following the led of the previous exmple, we see tht
SOME EXAMPLES OF SUBDIVISION OF SMALL CATEGORIES 5 these two rrows re : 1 nd : 0. Agin, these re not elementry equivlents, so we hve tht Sd(2) = 0 δ1 δ 1 1. Our next exmple is the ctegory C tht is just corner of rrows: 0 The heuristic tells us tht our subdivision should look like this: 0 2 b 1 In C, we hve three 0-simplices nd two 1-simplieces: 0 : [0] C : [1] C 1 : [0] C b : [1] C 2 : [0] C 2 b 1. There re no nonidentity mps [0] [0]. Any mp [1] [1] is either constnt mp (giving us degenercy) or the identity (by monotonicity), so we only need to consider rrows [0] [1]. There re no rrows 0 b or 2 since the imges of 0 nd b re disjoint nd the imges of 2 nd re disjoint. Thus, we hve four possible rrows in Sd(C), ll of which turn out to ctully be rrows. Thus, we hve tht Sd(C) = 0 δ 0 1 2 b Or, if we linerize it, we get 0 δ1 δ 1 δ 2 b 0 1, which is exctly wht we expected. Note tht Sd(C) = Sd 2 (2), which is poset. Now let s consider the ctegory P consisting if two objects 0 nd 1 nd two prllel nonisomorphism rrows, b : 0 1. Digrmmticlly, P = 0 1. This ctegory hs two 0-simplices, 0 : [0] P nd 1 : [0] P, nd two 1-simplices, : [1] P nd b : [1] P. There re no other nondegenerte simplices. This gives us four relevnt rrows in [ /P]: : 1, : 0, : 1 b, nd : 0 b. Since none of these rrows re elementry equivlent pirs, we hve tht Sd(P) = 0 1 b It s worth noting tht Sd(P) = Sd 2 (1 ), nd tht it s poset. If we look t the geometric reliztion of 1, N(1 ), we observe tht it s homeomorphic to S 1 nd tht Sd 2 (1 ) is the 4-point model tht is wekly homotopy equivlent to S 1. b
6 MARCELLO DELGADO Now let s exmine n exmple with nontrivil composition of rrows. Consider the ctegory T : 0 1 Let s mke list of ll the nondegenerte simplices: b 0-simplices 1-simplices 2-simplices 0 : [0] T : [1] T b : [2] T 1 : [0] T b : [1] T 2 : [0] T b : [1] T Here, we distinguish between the 1-simplex corresponding to the composite b nd between the 2-simplex consisting of the two composble rrows 0 1 b 2 which we denote b. So our subdivided ctegory will hve these seven objects. Now let s looks t rrows in our new ctegory. We hve six rrows going from the 0-simplices to the 1-simplices which re the mps tht compose with the degenercy mps δ i to give us mps tht pick out the domin nd codomin of ech rrow s in ech of the previous exmples. This gives us the six rrows : 1 : 0 : 2 b 2 b : 1 b : 2 b : 0 b We lso get three more rrows tht pick out the fces of the commuttive tringle: : b b : b b δ 2 : b Thus, we get the following digrm for /T (restricted to nondegenerte simplices): 0 δ 2 1 b δ1 b 2 b We leve out the composite rrows nd we don t clim tht this digrm commutes since we hve not shown, for exmple, tht δ 2 = for the two composite rrows 0 b. Wht we would like to show is tht these three pirs of composites re elementry equivlent, tht is: δ 2 δ 2 There is only single mp s : [2] [0]. By composition with ny of the six mps listed bove, we get 1 [0] since there re no other mps [0] [0], which proves elementry equivlence. Thus, the digrm bove commutes in Sd(T ). We cn rerrnge this digrm nd get the following digrm which commutes becuse the
SOME EXAMPLES OF SUBDIVISION OF SMALL CATEGORIES 7 unlbeled composites re elementry equivlent nd re identified in [ /T ]: 1 δ 2 δ 0 b b 0 b 2 This should remind us very strongly of the result of brycentric subdivision of 2-simplex in lgebric topology. For our lst finite exmple, consider the groupoid G : 0 1, where b = 1 0 nd b = 1 1. We observe tht G hs two nondegenerte simplices of ech dimension: the 0-simplices 0 nd 1, the 1-simplices nd b, nd for ech [q] for q > 2 the two compositions of length q b... nd bb.... As for the rrows, there re two wys to include string of lternting s nd bs of length q into one of length q + 1: dd the pproprite letter to the beginning ( ) or to the end (δ q+1 ). It will turn out [1] tht ll composites will work out to be ppropritely elementry equivlent, so tht we get Sd(G) = δ 2 0 b b 1 b b bb δ 2 And our finl exmple is our only infinite exmple. Consider the ctegory A We know how ech corner of rrows subdivides by previous exmple. Since we hve no nontrivil compositions of rrows nd since ech corner subdivides into two corners, we get tht Sd(A) = A. δ 3 δ 3 Acknowledgements Thnks gin to my mentors Clire nd Emily, without with which this pper would be lot shorter. References [1] Mtis del Hoyo On the subdivision of smll ctegories Topology nd its Applictions. Volume 155, Issue 11, 1 June 2008, Pges 1189-1200 [2] J. P. My. Subdivisions of Smll Ctegories. http://mth.uchicgo.edu/ my/finite/subdivide.pdf b