Moore hyperrectangles on a space form a strict cubical omega-category

Mo ore h yp errectangles on a space fo rm a strict cubical omega-category Ronald Bro wn No v ember 5, 2021 Abstract A questio n of Jack Morav a is ans wered by generalising the notion of Moo re paths to that of Mo ore hyperr e ctangles, so obtaining a strict cubical ω -catego r y . This a lso has the structure of connections in the s ense of Brown and Higgins, but cancellation of connections do es not hold. In tro d uction W e recall in S ection 1 the notion of the space of Mo ore paths on a top ological space X . A v ariant of the defin ition is giv en in [Bro06]. Mo ore paths ha ve the adv an tage of giving a category of paths, w ith an asso ciativ e comp osition, and id entities, rather th an the common description in terms of maps I → X . Ho w ever, w hereas in higher dimen s ions the appropr iate and analogous op erations on m ap s of cub es I n → X hav e b een we ll used , see for example [BH81b ], there seems to hav e b een in higher dimensions n o d efinition analo gous to that of Mo ore paths. In this pap er we giv e such a definition in Section 2 and in Section 3 we giv e the la ws th at this structure satisfies. The f orm ulation of these is taken from [AABS02], but for a large part they go back to [BH77]. The cubical la ws were given in [Kan55]. The cubical approac h in th at pap er w as abandoned in fa vour of simplicial sets once the pr ob lems of the geometric realisation of the cartesian pro d uct w ere found , and Milnor had w r itten on the geometric r ealisatio n of the simplicial sets. The introdu ction of ‘connections’ in all dimensions wa s in [BH77, BH81a, BH81b] for the purp ose of discussing ‘commuta tive sh ells’. This w as extended to the category case in [Mos87, AA89, AABS02]. The general theory of cub ical sites is d ev elop ed in [GM03]. Maltsiniotis has sho wn in [Mal09] that cub ical sets with connections, in con trast to the stand ard case, ha v e go o d realisations of cartesian pr o ducts. The thesis [P at08] uses cu b ical sets with what h e calls pseudo c onne ctions for the th eory of derived functors, analogously to the simplicial case. 1 The p ap er [Gra02 ] uses an analogous pro cedure to this for the definition of a higher cate- gorical structur e, w ith cub es indexed on Z n and constan t on eac h v ariable outside of a certain ‘supp ort’, b u t do es not tak e the s u pp ort as part of the structure. An applicati on of the cub ical classifying sp ace of a crossed complex is in [FRS95]. It is suggested in r ecen t work that th e notion of Kan simplicial set can b e regarded as an ∞ -group oid, see for example [Lu r09]. In some wa ys this is curious as this is regarded as the start of such an idea is the f undamental group or group oid, which is made of classes of paths un d er homotop y r elativ e to th e end p oin ts. One w ou ld exp ect on the same prin ciple to tak e some form of homotop y classes of maps of m -paths. T he difficult y in this is sho wn th at by the f act that an absolute strict homotop y m -group oid has b een defin ed only for m = 1 , 2, in [BHKP02]. The pap er [BH81b] sh o ws that successful higher h omotop y group oids can b e defined for filtered spaces. This allo ws a r oute into algebraic top ology without setting up singular homology theory . In any case, the construction M ∗ ( X ) can b e seen as another candidate for a weak form of ∞ -group oid. 1 Mo ore paths Let R + = [0 , ∞ ) b e the nonnegativ e real line. F or a s p ace X let M ( X ) b e the sub space of X R + × R + of p airs ( f , r ) suc h that f is constan t on [ r , ∞ ). There are t wo maps ∂ − , ∂ + : M ( X ) → X , ∂ − ( f , r ) = f (0) , ∂ + ( f , r ) = f ( r ) . No w comp osition ◦ of Mo ore paths on M ( X ) is given by th e comp osition M ( X ) ∂ + × ∂ − M ( X ) φ − → X R + × R + × R + 1 × + − → X R + × R + where the fir s t term is th e pullb ac k, and φ sends pairs ( f , r ) , ( g , s ) ∈ M ( X ) such that f ( r ) = g (0) to triples ( h, r , s ) ∈ X R + × R + × R + suc h that h is constan t on [ r + s, ∞ ), h | [0 , r ] = f | [0 , r ] and h ( t ) = g ( t − r ) for t > r , and + is th e addition fun ction. So comp osition is cont inuous. W e also hav e an identit y function ε : X → M ( X ) giv en b y ε ( x ) = ( ˆ x, 0) where ˆ x is the constan t map on R + with v alue x . This comp osition giv es, as is well kn o wn, a cat egory str u cture ( M ( X ) , ∂ ± , ◦ , ε ). This struc- ture also h as a ‘rev er s e’ − : M ( X ) → M ( X ) given by − ( f , r ) = ( g , r ) w h ere g ( t ) = ( f ( r − t ) if 0 6 t 6 r, f (0) if t > r . Th u s ∂ − ( − a ) = ∂ + ( a ) , ∂ + ( − a ) = ∂ − a . W e no w discuss the relation w ith the fundamenta l group oid on a set C of base p oin ts in X . 2 By a h omotopy H of elements a 0 = ( f 0 , r 0 ) , a 1 = ( f 1 , r 1 ) of M ( X ) we mean a con tinuous map H : [0 , 1] → M ( X ) suc h that H (0) = a 0 , H (1) = a 1 , or, equiv alen tly , a map H : [0 , 1] × R + → X suc h th at H (0 , t ) = a 0 ( t ) , H (1 , t ) = a 1 ( t ) for t ∈ R + and there is a contin uous function s 7→ r ( s ) where 0 6 s 6 1 , r ( s ) ∈ R + , r (0) = r 0 , r (1) = r 1 and H ( s, t ) = H ( s, r ( s )) f or t > r ( s ) , 0 6 s 6 1. This homotop y is r el end p oints if H ( s, 0) = f 0 (0) , H ( s, r ( s )) = f 0 ( r 0 ) for all 0 6 s 6 1. The fundamental group oid π 1 ( X, C ) on the set of base p oints C ⊆ X is the the set of homotop y classes rel end p oin ts of elemen ts of M ( X ) with sour ce and target in C . F or more information on th e use of π 1 ( X, C ), but w ith a slight ly differen t construction, see [Bro06]. 2 Mo ore h yp errectangles Let M n ( X ) b e th e s u bspace of X ( R + ) n × ( R + ) n of pairs ( f , ( r )) wh ere ( r ) = ( r 1 , . . . , r n ) suc h that f ( t 1 , . . . , t i , . . . , t n ) = f ( t 1 , . . . , r i , . . . , t n ) for t i > r i , i = 1 , . . . , n. W e call ( r ) the shap e and f the action of the n -path ( f , ( r )). W e ha v e ∂ − i , ∂ + i : M n ( X ) → M n − 1 ( X ) giv en by ev aluating at 0 or r i in the i th p osition and omitting the r i . More precisely , ∂ α i ( f , ( r )) = ( f ′ , ( r ′ )) where ( r ′ ) = ( r 1 , . . . , ˆ r i , . . . , r n ) and f ′ ( r ′ ) = f ( r 1 , . . . , α ′ , . . . , r n ) where α ′ = 0 or r i according as α = − or +. T o defin e the degeneracies ε i : M n − 1 ( X ) → M n ( X ) w e set ε i ( f ′ , ( r ′ )) = ( f , ( r )) where ( r ) is obtained from ( r ′ ) b y putting 0 in the i th place, and f ( t 1 , . . . , t n ) = f ′ ( t 1 , . . . , ˆ t i , . . . , t n ). T o defi ne the connections Γ − i : M n − 1 ( X ) → M n ( X ) w e set Γ − i ( f ′ , ( r ′ )) = ( f , ( r )) wh ere ( r ) is obtained from ( r ′ ) b y rep eating r i (in the i th and ( i + 1)th place, an d mo ving the others along), and setting f ( t 1 , . . . , t n ) = f ′ ( t 1 , . . . , t i − 1 , max( t i , t i +1 ) , t i +2 , . . . , t n ) . Similarly w e get Γ + i using min ins tead of max. ( T his follo ws the con ven tions of [AABS02].) F or i = 1 , . . . , n the category structure ( M n ( X ) , , ∂ − i , ∂ + i , ◦ i , ε i ) is simp ly that given in section 1 but in the i th p lace. In this wa y w e giv e the family M ∗ ( X ) = { M n ( X ) } for n > 0 the stru cture of cubical ω - catego r y: the la ws for this and the connections are giv en in S ection 3. Th e pap er [BH81 c ] also sho ws ho w to obtain w hat w e no w call a globular ω -category fr om this cubical structur e, as a substru cture in wh ic h certain faces of a cub e hav e v arious lev els of degeneracy . How ev er this globular structure is not equiv alen t to the cubical structur e, as the pro of in [AABS02] requires the cancella tion la w for connections, whic h do es not h old here: see Remark 3.2. W e r efer also to [Bro08 ] for th e construction of a fundamenta l glo b ular ω -group oid ρ ( X ∗ ) of a filtered space X ∗ . 3 3 La ws In this section w e giv e the full structur e and la ws on the cubical set with connections and comp ositions M ∗ ( X ). W e tak e these from [AABS02]. Let K b e a cubical set, that is, a f amily of sets { K n ; n > 0 } with for n > 1 face m aps ∂ α i : K n → K n − 1 ( i = 1 , 2 , . . . , n ; α = + , − ) and degeneracy maps ε i : K n − 1 → K n ( i = 1 , 2 , . . . , n ) satisfying the u sual cubical relations: ∂ α i ∂ β j = ∂ β j − 1 ∂ α i ( i < j ) , (3.1)(i ) ε i ε j = ε j +1 ε i ( i 6 j ) , (3.1)(i i) ∂ α i ε j =      ε j − 1 ∂ α i ( i < j ) ε j ∂ α i − 1 ( i > j ) id ( i = j ) (3.1)(i ii) W e say that K is a c ubic al set with c onne ctions if for n > 0 it has additional structure maps (called c onne ctions ) Γ + i , Γ − i : K n → K n +1 ( i = 1 , 2 , . . . , n ) satisfying the r elations: Γ α i Γ β j = Γ β j +1 Γ α i ( i < j ) (3.2)(i ) Γ α i Γ α i = Γ α i +1 Γ α i (3.2)(i i) Γ α i ε j = ( ε j +1 Γ α i ( i < j ) ε j Γ α i − 1 ( i > j ) (3.2)(i ii) Γ α j ε j = ε 2 j = ε j +1 ε j , (3.2)(i v ) ∂ α i Γ β j = ( Γ β j − 1 ∂ α i ( i < j ) Γ β j ∂ α i − 1 ( i > j + 1) , (3.2)(v) ∂ α j Γ α j = ∂ α j +1 Γ α j = id, (3.2)(vi) ∂ α j Γ − α j = ∂ α j +1 Γ − α j = ε j ∂ α j . (3.2)(vii ) The conn ections are to b e thought of as extra ‘degeneracies’. (A d egenerate cub e of type ε j x has a p air of opp osite faces equ al and all other faces degenerate. A cub e of t y p e Γ α i x has a pair of adjacen t faces equal and all other faces of t yp e Γ α j y or ε j y .) Cubical complexes with this, and other, str u ctures ha v e also b een considered by E v r ard [Evr ]. The pr ime example of a cubical set with conn ections is the singu lar cubical complex K X of a space X . Here for n > 0 K n is the set of singular n -cub es in X (i.e. conti nuous maps I n → X ) and the conn ection Γ α i : K n → K n +1 is induced by the map γ α i : I n +1 → I n defined b y γ α i ( t 1 , t 2 , . . . , t n +1 ) = ( t 1 , t 2 , . . . , t i − 1 , A ( t i , t i +1 ) , t i +2 , . . . , t n +1 ) where A ( s, t ) = max( s, t ) , min( s, t ) as α = − , + r esp ectiv ely . Here are pictur es of γ α 1 : I 2 → I 1 where the internal lin es sh o w lines of constancy of the m ap on I 2 . 4 γ − 1 = γ + 1 = 2 1   / / The complex K X h as some further r elev an t structur e, n amely the comp osition of n -cub es in the n different directions. Accordingly , w e define a c ubic al c omplex with c onne ctions and c omp ositions to b e a cubical set K with connections in wh ic h eac h K n has n partial comp ositions ◦ j ( j = 1 , 2 , . . . , n ) satisfying the follo wing axioms. If a, b ∈ K n , then a ◦ j b is defined if and only if ∂ − j b = ∂ + j a , and then ( ∂ − j ( a ◦ j b ) = ∂ − j a ∂ + j ( a ◦ j b ) = ∂ + j b ∂ α i ( a ◦ j b ) = ( ∂ α j a ◦ j − 1 ∂ α i b ( i < j ) ∂ α i a ◦ j ∂ α i b ( i > j ) , (3.3) The inter change laws . If i 6 = j then ( a ◦ i b ) ◦ j ( c ◦ i d ) = ( a ◦ j c ) ◦ i ( b ◦ j d ) (3.4) whenev er b oth sides are d efi ned. (Th e diagram  a b c d  j i   / / will b e used to ind icate that b oth sides of th e ab ov e equatio n are defin ed and also to denote the unique comp osite of the four elemen ts.) If i 6 = j then ε i ( a ◦ j b ) = ( ε i a ◦ j +1 ε i b ( i 6 j ) ε i a ◦ j ε i b ( i > j ) (3.5) Γ α i ( a ◦ j b ) = ( Γ α i a ◦ j +1 Γ α i b ( i < j ) Γ α i a ◦ j Γ α i b ( i > j ) (3.6)(i ) Γ + j ( a ◦ j b ) =  Γ + j a ε j a ε j +1 a Γ + j b  j +1 j   / / (3.6)(i i) Γ − j ( a ◦ j b ) =  Γ − j a ε j +1 b ε j b Γ − j b  j +1 j   / / (3.6)(i ii) These last tw o equations are the tr ansp ort laws 1 . 1 Recall from [BS76] th at the term c onne ction was chosen b ecause of an an alogy with path- connections in differential geometry . In particular, the transp ort law is a v ariation or sp ecial case of the transp ort law for a path-connection. 5 It is easily v erified that the cub ical Mo ore complex M ∗ X of a space X satisfies these axioms with our ab o ve d efinitions. In this conte xt the trans p ort law for Γ − 1 ( a ◦ b ) can b e illustr ated by the picture a b b a Remark 3.1 That the ab o ve la w s for these structures apply to M ∗ ( X ) is easy to c hec k. It is imp ortan t that th e shap e tuples ( r 1 , . . . , r n ) are p art of the stru cture. Th u s if ∂ + 1 ( f , ( r )) = ∂ − 1 ( g , ( s )) then this implies that r i = s i , 2 6 i 6 n as w ell as f ( r 1 , t 2 , . . . , t n ) = g (0 , t 2 , . . . , t n ) for 0 6 t i 6 r i , 2 6 i 6 n. This may seem a strong condition, but ‘comp osition is the inv erse of sub division’, and this enables one to obtain multiple comp ositions as th e inv erse of ‘sub dividing’ an elemen t ( f , ( r )) ∈ M n ( X ). ✷ Remark 3.2 In [AABS02] a cubical ω -category with connections G = { G n } is defi ned as a cubical set with conn ections and comp ositions suc h that eac h ◦ j is a category structure on G n with iden tit y elements ε j y ( y ∈ G n − 1 ), and in addition Γ + i x ◦ i Γ − i x = ε i +1 x, Γ + i x ◦ i +1 Γ − i x = ε i x. (2.7) Ho w ever th is cancellation la w do es not hold for M ∗ ( X ). Thus the equiv alence b et ween globular and cubical categories d ev elop ed in [AABS02 ] do es not app ly to M ∗ ( X ), nor do es the exact relations b et ween ‘commutat ive shells’ and ‘thin elemen ts’ deve lop ed in [Hig0 5 ]. ✷ Remark 3.3 There are also r everses − i : M n ( X ) → M n ( X ) , 1 6 i 6 n defined as in Section 1. A problem with our construction is that a path [0 , 1] → M n ( X ) is not n ecessarily an elemen t of M n +1 ( X ). In p articular the easily defin ed h omotop y rel end p oin ts of p aths a ◦ − a ≃ 0 ∂ − a is not an elemen t of M 2 ( X ). ✷ 4 T e n sor pro ducts The tensor pro d uct K ⊗ L of cubical sets is also defined in [Kan55] and extended to cubical sets with connections in [BH87, AABS02]. W e see the con venience of cub es in the curr en t accoun t since since if a = ( f , ( r )) ∈ M m ( X ) and b = ( g , ( s )) ∈ M n ( Y ) then their tensor pro d uct a ⊗ b ∈ M m + n ( X × Y ). It is giv en by a ⊗ b = ( h, (( r ) ◦ ( s ))) w h ere ( r ) ◦ ( s ) = ( r 1 , . . . , r m , s 1 , . . . , s n ) and h = f × g : ( R + ) m × ( R + ) n → X × Y with the us ual ident ifi cation of ( R + ) m × ( R + ) n and ( R + ) m + n . 6 References [AA89] Al-Agl, F. Asp e cts of multiple c ate gories . Ph.D. th esis, Universit y of W ales, Ba n gor (1989 ). [AABS02] Al-Agl, F. A., Bro wn, R. and Steiner, R. ‘Multiple categories: the equiv alence of a globular and a cubical approac h’. A dv. Math. 170 (1) (2002) 71–118. [Bro06] Bro wn, R. T op olo gy and Gr oup oids . Prin ted by Bo oksurge LLC, Ch arleston, S. Carolina, third ed ition (2006). [Bro08] Bro wn, R. ‘A new higher homotop y group oid: the fund amen tal globular ω -group oid of a fi ltered sp ace’. Homolo gy, Homotopy Appl. 10 (1) (2008) 327– 343. [BHKP02] Bro wn, R., Hardie, K. A., Kamps, K. H. an d Po r ter, T. ‘A homotop y double group oid of a Hausd orff space’. The ory A ppl. Cate g. 10 (2002) 71–9 3 (electronic). [BH77] Bro wn, R. and Higgins, P . J. ‘Sur les complexes crois ´ es, ω -group o ¨ ıdes, et T - complexes’. C. R. A c ad. Sci. Paris S´ er. A- B 285 (16) (1977) A997– A999. [BH81a ] Br own, R. and Higgins, P . J. ‘On the alge b ra of cub es’. J. Pu r e Appl. Algebr a 21 (3) (1981 ) 233–260. [BH81b] Bro wn, R. and Higgins, P . J. ‘Colimit theorems for relativ e homotop y groups’. J. Pur e Appl. Algebr a 22 (1) (19 81) 11–41. [BH81c] Bro wn, R. and Higgins, P . J. ‘The equiv alence of ∞ -group oids and crossed com- plexes’. Cahiers T op olo gie G´ eom. Diff´ er entiel le 22 (4) (198 1) 371–38 6. [BH87] Bro wn, R. and Higgins, P . J. ‘T ensor pro ducts and homotopies for ω -group oids and crossed complexes’. J. Pur e Appl. Algebr a 47 (1) (1987) 1–33. [BS76] Bro wn, R. and Sp encer, C. B. ‘Double group oids and crossed mo dules’. Cah ie rs T op olo gie G ´ eom. Diff´ er entiel le 17 (4) (197 6) 343–362. [Evr] Evrard, M. ‘Homotopie des co m plexes simpliciaux et cub iqu es’. Pr eprint (1976). [FRS95] F enn, R., Rour k e, C. and Sanderson, B. ‘T runks and classifying spaces’. Appl. Cate g. Structur es 3 (4) (19 95) 321–3 56. [Gra02] Grandis, M. ‘An in trinsic homotop y theory for sim p licial complexes, with app lica- tions to image analysis’. Appl. Cate g . Structur es 10 (2) (2002) 99–155 . [GM03] Grandis, M. and Mauri, L. ‘Cubical sets and their site’. The ory Applic. Cate gories 11 (20 03) 185–2 01. 7 [Hig05] Higgins, P . J. ‘Thin element s and comm u tativ e shells in cubical ω -categories’. The ory Appl. Cate g . 14 (2005) No. 4, 60 –74 (electronic). [Kan55] Kan, D. M. ‘Abstract homotop y . I’. Pr o c . N at. A c ad. Sci. U.S.A. 41 (1955) 1092– 1096. [Lur09] Lurie, J . Higher top os the ory , Annal s of Mathematics Studies , V olume 170. Princeton Univ ersity Press, Prin ceton, NJ (2009). [Mal09] Maltsiniotis, G. ‘La cat ´ ego r ie cubique a vec connections est une cat ´ ego r ie test stricte’. (pr eprint) (2009 ) 1–16. [Mos87] Mosa, G. H. Higher dimensional algebr oids and cr osse d c omplexes . Ph.D. thesis, Univ ersity of W ales, Bangor (1987) . [P at08] P atc h k oria, I. Cubic al r esolutions and derive d f unctors . Diplome thesis, Iv ane Ja v akhishvil i Tbilisi State Univ ersit y (2008), arXiv: 0907.190 5. 8