Operads of higher transformations for globular sets and for higher magmas

Transcription

1 Volume 3, Number 1, July 215, Operads of higher trasformatios for globular sets ad for higher magmas Camell Kachour I dedicate this work to Adre Joyal. Abstract. I this article we discuss examples of fractal ω-operads. Thus we show that there is a ω-operadic approach to explai existece of the globular set of globular sets1, the reflexive globular set of reflexive globular sets, the ω-magma of ω-magmas, ad also the reflexive ω-magma of reflexive ω-magmas. Thus, eve though the existece of the globular set of globular sets is ituitively evidet, may other higher structures which fractality are less evidet, could be described with the same techology, usig fractal ωoperads. We have i mid the o-trivial questio of the existece of the weak ω-category of the weak ω-categories i the globular settig, which is described i [9] with the same techology up to a cotractibility hypothesis. Itroductio This article is the secod i a series of three articles (see [8, 9]). Here we give some relevat examples of ω-operads havig the fractal property i the sese of [8], exhibited by relevat higher structures where cotractibility i the sese of Batai s article [2] is ot ivolved. The atural directio we propose allows us to Keywords: Higher categories; higher operads; weak higher trasformatios. Mathematics Subject Classificatio [21]: 3B15, 3C85, 18A5, 18C2, 18D5, 18D5, 18G55, 55U35, 55U4. Received: 4 May 215, Accepted: 9 July 215 ISSN Prit: Olie: Shahid Beheshti Uiversity 1 Globular sets are also called ω-graphs by the Frech School. 89

2 9 Camell Kachour cosider four examples of higher structures: globular sets, reflexive globular sets, ω-magmas, ad reflexive ω-magmas. We use two coglobular objects i the category T-Gr p,c of poited T-graphs over costat globular sets to build these examples: the object C ad a subobject G of C, both described i the last sectio of [8]. Thaks to the object G, we freely geerate the coglobular object of ω-operads for globular sets ad the coglobular object of ω-operads for reflexive globular sets. Thaks to the object C, we freely geerate the coglobular object of ω-operads for ω-magmas ad the coglobular object of ω-operads for reflexive ω-magmas. It is the easy to show that the ω-operad BG of globular sets, the ω-operad B G u of reflexive globular sets, the ω-operad BM of ω-magmas, the ω-operad B M u of reflexive ω-magmas, all have the fractal property. Usig the same techology related to the stadard actio i T-CAT 1 (see [8]), we deduce the existece of a globular set of globular sets, a reflexive globular set of reflexive globular sets, a ω-magma of ω-magmas, ad a reflexive ω-magma of reflexive ω-magmas. We suspect that the ω-operad BC of Batai (see [2, 5, 13]), which algebras are his defiitio of weak higher categories, has the fractal property. Thus the weak ω-category of weak ω-categories should have a similar descriptio as those of this article, up to the cotractibility of a specific ω-operad. This importat fact of higher category theory is described i [9]. 1 Preamble We are goig to describe four examples of fractal ω-operads i the sese of [8]. Not oly is the ω-operad of globular sets actually fractal, but so are the ω-operad of reflexive globular sets, the ω-operad of ω-magmas, ad the ω-operad of reflexive ω-magmas. Remark 1.1. It is importat to ote that our techology also applies i low dimesios. For example, as a exercise, the reader ca check easily that graphs form a graph, or reflexive graphs form a reflexive graph, by iterpretig i the laguage of this article. Ideed we ca also show easily that the 1-operad of graphs, ad the 1-operad of reflexive graphs are also fractals. Also the 2-operad of 2-graphs is fractal as well, where we cosider a fiite 3-trucated coglobular objects build with the 2-operad of 2-graphs, the 2-operad of morphisms of 2-graphs, ad the 2-operad of trasformatios of 2-graphs. However this 2-operad of 2-graphs ca be also used to show that graphs do form a 2-graphs. But i that case this costructio does t show the fractality structure 2 of such 2-operad of 2-graphs. May such low dimesios facts (such that the existece of the category of categories, the 2 I the sese of [8].

3 Operads of higher trasformatios for globular sets ad higher magmas 91 strict 2-category of strict 2-categories, the bicategory of bicategories 3, etc.) ca be described easily to show more the pertiece of the techology developed i this article. Remark 1.2. I [17] Dimitri Tamarki has show that DG-categories form a DGcategory. We do ot kow if such proof is similar to the techology developed i our article. If it is the case, such result might be still true for a higher versio of DG-categories 4. But we do ot wat to give more details of such subject, because we are ot eough expert about DG-categories. 2 The coglobular object of graphical ω-operads The category ω-gr of globular sets has trivial higher trasformatios. First we are goig to describe these higher trasformatios as presheaves o appropriate small categories G, ad the see that they form a globular set that we call the globular set of globular sets. It is the combiatorial descriptio of these small categories G which allows a straightforward proof of Propositio 2.2. This propositio basically says that these higher trasformatios are algebras for adapted 2-coloured ω-operads. Cosider the classical globe category G s 1 1 s s 1 t 1 t 2 1 t 1 subject to the relatios 5 o cosources s +1 ad cotargets t +1. For each each 1 we are goig to build other small categories G resultig i a coglobular object i CAT δ 1 δ G 1 δ 2 G 1 G 1 2 G 1 G κ 1 κ 2 1 where G is the globe category. The category G is called the -globe category. 3 For these everyday cases, we use adapted trucatios of operads build i [9]. 4 If such higher versio makes sese. 5 which are describe i [8]

4 92 Camell Kachour The 1-globe category G 1 is preseted by α s 1 t 1 1 α 1 s 1 t 1 1 s 2 1 t α 2 s 2 1 t α 1 1 s 1 t 1 s 1 t 1 α subject to α +1 s +1 = s +1 α, α +1 t +1 = t +1 α. The 2-globe category G 2 is preseted by β α s 1 t 1 1 β 1 α 1 s 1 t 1 1 s 1 2 t 2 1 t β 1 s α 1 1 β 1 α 1 1 s 1 t 1 s 1 t 1 β α where i particular we have a arrow ξ 1 : 1. Arrows s +1, t +1, α, β, ad ξ 1 satisfy the followig relatios α +1 s +1 = s +1 α, α +1 t +1 = t +1 α, β +1 s +1 = s +1 β, β +1 t +1 = t +1 β, ξ 1 s 1 = α ad ξ 1 t 1 = β. More geerally the -globe category G is give by the category s 1 t 1 1 s 1 2 t s 1 t 1 β 2 α 2 β α β 1 α 1 β α s 1 t 1 β 1 α 1 1 s 2 1 t s 1 t 1

5 Operads of higher trasformatios for globular sets ad higher magmas 93 where i particular we have a arrow ξ 1 : 1, ad also for each 1 p 2, we have arrows p α p β p. The arrows s +1, t +1, α, β, α p, β p, ad ξ 1 satisfy the followig relatios α +1 s +1 = s +1 α, α +1 t +1 = t +1 α, β +1 s +1 = s +1 β, β +1 t +1 = t +1 β. α p s p p 1 = β p s p p 1 = α p 1 ad α p t p p 1 = β p t p p 1 = α p 1, ad we put α := α ad β := β, ξ 1 s 1 2 = α 2 ad ξ 1 t 1 2 = β 2. δ 1 The cosources ad cotargets fuctors G G 1 κ 1 are such that δ 1 seds G to G, ad κ 1 seds G to G δ 2 1. The cosources ad cotargets fuctors G 1 G 2 sed G to G, ad G to G. Also δ 2 1 seds the symbols α to the symbols α, ad κ 2 1 seds the symbols α to the symbols β. Now we cosider the case 3. The cosource ad cotarget fuctors G 1 δ 1 G κ 2 1 are costructed as follows. First we remove the cell ξ 1 ad the cell β 2 from G, ad we obtai the category G 1. Clearly we have a isomorphism of categories G 1 G 1 (which seds α 2 to ξ 2 ), ad also the embeddig G 1 δ 1 G. The compositio of this embeddig with the last isomorphism δ 1 gives G 1 G. The cotarget fuctor is built similarly: First we remove the cell ξ 1 ad the cell α 2 from G, ad we obtai the category G + 1. Clearly we have a isomorphism of categories G + 1 G 1 (which seds β 2 to ξ 2 ), ad also the embeddig G + κ 1 1 G. The composite of this embeddig with the last

6 94 Camell Kachour isomorphism gives G 1 G. It is easy to see that these fuctors δ 1 ad verify the cosource/cotarget coditios as for the globe category G above. We deote the category of sets by Set ad the category of large sets by SET. Whe we apply the cotravariat fuctor [ ; Set]() to the coglobular object i CAT G op δ 1 κ 1 G op 1 δ 1 2 κ 2 1 G op 2 G op 1 δ 1 G op it is easy to see that we obtai the globular set of globular sets 6 [G op ; Set]() σ 1 β 1 [G op 1 ; Set]() [G op 1 ; Set]() σ 1 β 1 [G op ; Set]() Remark 2.1. If istead we apply to it the cotravariat fuctor [ ; Set] we obtai the globular category of globular sets, which is useful i [9], for example to describe the globular category of the strict ω-categories. A object of the category of presheaves [G op ; Set] is called a (, ω)-graph 7. For istace, if 3, the the source fuctor σ 1 is described as follows. Take a (, ω)-graph X : G op Set with uderlyig ( 1)-trasformatio X(ξ 1 ) : X( ) X( 1 ), ad the σ 1(X)(ξ 2 ) : X( ) X( 2 ) is the uderlyig ( 2)-trasformatio of σ 1(X) defied by: Similarly for the target fuctors β 1. σ 1(X)(ξ 2 ) = X(s 1) X(ξ 1 ). 6 For each N, [G op ; Set]() meas the set of objects of the presheaf category [G op ; Set]. 7 Do ot cofuse (, ω)-graphs with the (, )-graphs that we defied i [7]. They are completely differet objects. I [7], (, )-graphs are a kid of globular set which plays a cetral role i defiig a algebraic approach of (, )-categories.

7 Operads of higher trasformatios for globular sets ad higher magmas 95 Now we are goig to give a operadic approach to the globular set of globular sets by usig the techology of Sectio 2 of [8]. Let us deote by G the coglobular object i T-Gr p,c G δ 1 G 1 δ 1 2 G 2 G 1 δ 1 G κ 1 κ 2 1 built just by removig all cells µ p ad ν p from the object C i T-Gr p,c (described i Sectio 4 of [8]): C δ 1 C 1 δ 1 2 C 2 C 1 δ 1 C κ 1 κ 2 1 If we apply to it the free fuctor M : T-Gr p,c T-CAT c (see Sectio 2 of [8]) we obtai a coglobular object 8 i T-CAT c B G δ 1 κ 1 B 1 G δ 1 2 κ 2 1 B 2 G B 1 G δ 1 B G which produces the globular object B σ G -Alg 1 B 1 G -Alg B 1 G -Alg σ1 B G -Alg β 1 i CAT. We the have the followig easy propositio. Propositio 2.2. The category B G-Alg is the category [Gop ; Set] of globular sets, B 1 G-Alg is the category [Gop 1 ; Set] of (1, ω)-graphs, ad for each iteger 2, B G-Alg is the category [G op ; Set] of (, ω)-graphs. Let us deote this coglobular object i T-CAT c by B G. Its stadard actio is give by the followig diagram i T-CAT 1 : β 1 Coed(B G ) Coed(Alg(.)) Coed(A op G ) Coed(Ob(.)) Ed(A,G ) It is the stadard actio of higher trasformatios specific to the basic globular set structure. The moochromatic ω-operad Coed(B G ) of coedomorphisms plays a 8 Here B G is the iitial ω-operad.

8 96 Camell Kachour cetral role for globular sets, ad we call it the white operad. It is straightforward that B G has the fractal property because it is iitial i the category T-CAT 1 of ω-operads ad thus has a uique morphism B G! G Coed(B G ) of ω-operads. If we compose it with the stadard actio of the globular sets Coed(B G ) Coed(Alg(.)) Coed(A op G ) Coed(Ob(.)) Ed(A,G ) we obtai a morphism of ω-operads B G G Ed(A,G ) which expresses a actio of the ω-operad BG of globular sets o the globular object B G -Alg() i SET of (, ω)-graphs ( N). This gives a globular set9 structure o (, ω)-graphs ( N). 3 The fuctor of cotractible uits We deote the category of the reflexive globular sets by ω-grr (see [16]). We have the adjuctio ω-grr U ad the geerated moad of reflexive globular sets is deoted by (R, η, µ). Objects of ω-grr are usually writte as (G, (1 p ) p< )), where the operatios (1 p ) p< is a chose reflexive structure o the globular set G. Also we cosider the moad T o the globular sets whose algebras are strict ω-categories. I this paragraph we will build the fuctor of the cotractible uits for poited T-graphs (see [13]). It plays the same role for the poited T-graphs as the previous fuctor R plays for the globular sets. First we must defie poited T-graphs with cotractible uits which are, for the poited T-graphs, what reflexive globular sets are for globular sets. Throughout this paper we will work with poited T-graphs ad with T-categories over costat globular sets (see [5]), ad a subscript c o categories will mea that we work with costat globular sets. For istace the category T-Gr p,c of poited T-graphs is adored with a c to idicate that objects of this category are the poited T-graphs over costat globular sets. Now cosider a object (C, d, c; u) of the category T-Gr p,c. Here u deotes a chose poit of (C, d, c); that is, the data of a morphism 9 Some mathematicias might prefer to say large globular set or globular class. R Gr

9 Operads of higher trasformatios for globular sets ad higher magmas 97 (G, η G, id G ) (u,1 G) (C, d, c) i T-Gr c where G desigates the uderlyig costat globular set of arities of (C, d, c) (or i other words, desigates the set of colours of (C, d, c)). Remark 3.1. A p-cell of G is deoted by g(p) ad this otatio has the followig meaig. The symbol g idicates the colour, ad the symbol p remids us that we must see g(p) as a p-cell of G because, while G is just a set, we are thikig of it as a costat globular set. I order to defie poited T-graphs with cotractible uits we are goig first to defie a itermediate structure o T-graphs. Cosider a T-graph (C, d, c) ad, for each N, we deote the set of -cells of the T-graph (C, d, c) by C(). Cosider also the reflexive globular set (T(G), (1 p ) p< )) such that the operatios 1 p are freely geerated by the moad T. We say that the T-graph (C, d, c) is equipped with a reflexive structure if its uderlyig globular set C is equipped with a reflexive structure i the usual sese ad d is a morphism of reflexive globular sets. Note that G is also equipped with a trivial reflexivity structure (G, (1 p ) p< )) such that the operatios 1 p are defied by 1 p (g(p)) = g(), forcig c to be a morphism of reflexive globular sets as well. We deote reflexive T-graphs, where the operatios 1 p are those of C, by (C, d, c; (1 p ) p< ). A morphism betwee two reflexive T- graphs is just a morphism of T-graphs which preserves reflexivity, ad the category of reflexive T-graphs over costat globular sets is deoted by T-Grr c. A poited T-graph (C, d, c; p) over a costat globular set G has cotractible v uits if it is equipped with a moomorphism R(G) C such that u factorise as R(G) G η(g) u ad such that the iduced T-graph T(G) d R(G) c G is reflexive. That is, the restrictio of d to R(G) is a morphism of reflexive globular sets. We deote poited T-graphs with cotractible uits by (C, d, c; p, v, (1 p ) p< ). A morphism (C, d, c; p, v, (1 p ) p< ) (f,h) C (C, d, c ; p, v, (1 p ) p< ) of poited T-graphs with cotractible uits is give by a morphism of poited T-graphs (see [13]) (C, d, c; p) (f,h) (C, d, c ; p )

10 98 Camell Kachour such that fv = v R(h), ad (R(G), d, c) (R(G ), d, c ) is a morphism of reflexive T-graphs. Thus morphisms betwee two T-graphs equipped with cotractible uits preserve this structure of cotractibility o the uits. The category of poited T-graphs with cotractible uits is deoted by UT-Gr p,c. It is easy to see that UT-Gr p,c is locally presetable, because it is based o the locally presetable category T-Gr p,c, ad equipped with a structure of cotractibility o the uits, whose operatios 1 p m o the uits ad their axioms, show easily that UT-Gr p,c is also projectively sketchable 1. Also we ca easily prove that the forgetful fuctor (f,h) UT-Gr p,c U T-Gr p,c is a right adjoit by usig basic techiques comig from logic as i [5]. Thus we ca apply Propositio of [3] which shows the moad T U iduced by this adjuctio has rak. Also U is moadic by the Beck theorem o moadicity. We write R for the left adjoit of U : U UT-Gr p,c T-Gr p,c. Furthermore, we have the geeral fact (which ca be foud i [1, 11]). R Propositio 3.2 (G.M. Kelly). Let K be a locally fiitely presetable category, ad Md f (K) the category of fiitary moads o K ad strict morphisms of moads. The Md f (K) is itself locally fiitely presetable. If T ad S are object of Md f (K), the the coproduct T S is algebraic, which meas that K T K S is K equal to K T S ad the diagoal of the pullback square K T K S p 1 K K S p 2 U K T V K is the forgetful fuctor K T S K. Furthermore the projectios K T K S K K T ad K T K S K S are moadic. K 1 Good refereces for sketch theory are [1, 3, 12, 14].

11 Operads of higher trasformatios for globular sets ad higher magmas 99 Remark 3.3. Accordig to Steve Lack, this result ca be easily geeralised to moads havig raks i the cotext of locally presetable category. With the fuctor V defied i Sectio 2 of [8], we have the followig diagram : UT-Gr p,c T-Gr p,c T-CAT c p 1 T-CAT c p 2 V UT-Gr p,c U T-Gr p,c Applyig the above propositio, we see that UT-CAT c := UT-Gr p,c T-Gr p,c T-CAT c is a locally presetable category 11, ad also that the forgetful fuctor UT-CAT c O T-Gr p,c is moadic. Deote by F the left adjoit to O. I Sectios 4 ad 6, we apply this fuctor F to the coglobular objects G ad C of T-Gr p,c, to obtai respectively, the coglobular object of higher operads for reflexive globular sets, ad the coglobular object of higher operads for reflexive ω-magmas. 4 The coglobular objects of the reflexive graphical ω-operads By takig the globe category G (see Sectio 2) as basis, we build the reflexive globe category G,r as follow. For each N we add ito G the formal morphism such that 1 +1 s +1 = 1 +1 t +1 = 1. For each p < we deote 1 p := 1 p p+1 1p+1 p For each each 1 we are goig to build similar categories G,r i order to obtai a coglobular object 11 UT-CAT c is the category of coloured ω-operads with chose cotractible uits where i particular morphisms of this category preserve cotractibility of the uits.

12 1 Camell Kachour i 1 i 1 2 i 1 δ 1 δ G 1 2,r G 1,r G 2,r κ 1 κ 2 1 δ G 1 1,r G,r i CAT, equipped with coreflexivity fuctors i +1. Each category G,r is called the reflexive -globe category. It is built as were the categories G ( 1) where ow we just replace G ad G by G,r ad G,r. For each 1, the cosource ad the cotarget fuctors are defied as were those for the G 1,r δ 1 G,r G 1 δ 1 G of Sectio 2, where i additio δ 1 seds, for all p, the reflexivity morphism 1 p p+1 to the reflexivity morphism 1p p+1, ad κ 1 seds the reflexivity morphism 1 p p+1 to the reflexivity morphism 1 p p+1. Also, if 2, δ +1 ad κ +1 sed, for all p,, ad the reflexivity the reflexivity morphism 1 p p+1 to the reflexivity morphism 1p p+1 morphism 1 p p+1 to the reflexivity morphism 1 p p+1. These fuctors δ 1 ad do ideed satisfy the cosource ad cotarget coditios. For each 1, the coreflexivity fuctor G,r i 1 G 1,r is built as follows: the coreflexivity fuctor ι 1 seds, for all q, the object q to q, the object q to q, the cosource morphisms s q+1 q ad s q+1 q to s q+1, the cotarget morphisms t q+1 q ad t q q+1 to t q+1 q, ad the fuctor morphisms α q to 1 q. Also the coreflexivity fuctor i 1 2 seds, for all q, the object q to q, the object q to q, the cosource morphism s q+1 q to the cosource morphism s q+1 q, the cosource morphism s q+1 q to the cosource morphism s q q+1, the cotarget morphism t q+1 q to the cotarget morphism t q+1 q, the cotarget morphism t q q+1 to the cotarget morphism t q+1 q, the fuctor morphisms α q to the fuctor morphisms αq, the fuctor morphisms βq to the fuctor morphisms β q, ad the atural trasformatio morphism ξ 1 to α 1 1. Also, for each 3, the coreflexivity fuctor i 1 seds, for all q, the object q to q, the object q to q, the cosource morphism s q+1 q to the cosource

13 Operads of higher trasformatios for globular sets ad higher magmas11 morphism s q+1 q cotarget morphism t q+1, the cosource morphism s q+1 q to the cosource morphism s q+1 q, the q, the cotarget morphism to the fuctor q to the cotarget morphism t q+1 t q+1 q to the cotarget morphism t q q+1, the fuctor morphisms α q morphisms α q, the fuctor morphisms βq to the fuctor morphisms βq. Also, if p 3, it seds the p-trasformatio α p to the p-trasformatio α p, the p-trasformatio β p to the p-trasformatio β 12 p, the ( 2)-trasformatio α 2 ad β 2 to the ( 2)-trasformatio ξ 2, ad fially the ( 1)-trasformatio ξ 1 to ξ With this costructio it is ot difficult to show that fuctors i 1 satisfy the coreflexivity idetities i 1 δ 1 = 1 G r 1 = i 1. ( 1) Whe we apply the cotravariat fuctor [ ; Set]() to the coglobular object i CAT i 1 i 1 2 i 1 G op,r δ 1 G op 1,r δ 1 2 κ 1 κ 2 1 G op 2,r G op 1,r δ 1 G op,r we obtai the reflexive globular set of reflexive globular sets: [G op,r; Set]() ι 1 σ 1 β 1 [G op 1,r ; Set]() ι 1 [G op 1,r ; Set]() σ 1 β 1 [G op,r ; Set]() A object of the category of presheaves [G op ; Set] is called a reflexive (, ω)-graph. For istace, if 3, the reflexivity fuctor ι 1 ca be described as follows. If X : G op 1 Set is a ( 1, ω)-graph ad X(ξ 2 ) : X( ) X( 2 ) is its uderlyig ( 2)-trasformatio, the ι 1 (X)(ξ 1 ) : X( ) X( 1 ) 12 By covetio we put α = α ad β = β. I fact this covetio is atural because from our poit of view of -trasformatios, 1-trasformatios are the usual atural trasformatios, ad a -trasformatio should be see as the uderlyig fuctio F actig o the -cells of a fuctor F.

14 12 Camell Kachour is the ( 1)-trasformatio defied by ι 1 (X)(ξ 1 ) = X(1 2 1 ) X(ξ 2). Now we are goig to give a operadic approach of the reflexive globular set of reflexive globular sets by usig the techology of Sectio 2 of [8]. Cosider the coglobular object G i T-Gr p,c as i 2. If we apply the free fuctor F : T-Gr p,c UT-CAT c (see Sectio 3) to it we obtai a coglobular object i T-CAT c : BG δ 1 u BG 1 δ 1 2 u BG 2 u B 1 κ 1 κ 2 1 G u δ 1 B G u It is importat to otice that the ω-operad BG u is iitial i UT-CAT 1. Also this coglobular object B G u produces the followig globular object i CAT B G u Alg σ 1 β 1 B 1 G u Alg B 1 G u Alg σ 1 β 1 B G u Alg ad we have: Propositio 4.1. The category B G u -Alg is the category [G op,r ; Set] of reflexive globular sets, B 1 G u -Alg is the category [G op 1,r ; Set] of reflexive (1, ω)-graphs, ad for each iteger 2, B G u -Alg is the category [G op,r; Set] of reflexive (, ω)-graphs. We deote this coglobular object i T-CAT c by B G u. Accordig to the results of Sectio 2 of [8], we obtai the diagram Coed(B G u ) Coed(Alg(.)) Coed(A op G u ) Coed(Ob(.)) Coed(A op,g u ) i T-CAT 1 that we call the stadard actio of the reflexive globular sets. It is a specific stadard actio. The moochromatic ω-operad Coed(B G u ) of coedomorphisms plays a cetral role for reflexive globular sets, ad we call it the blue operad. Also we have the followig result: Propositio 4.2. B G u has the fractal property. Proof. The uits of the ω-operad Coed(B G u ) are give by the idetity morphisms B G u 1 B B G u. We are goig to exhibit a morphism of ω-operads B +1 [1 B ;1 B ] +1 G u B G u

15 Operads of higher trasformatios for globular sets ad higher magmas13 which is the cotractibility of the uit 1 B with itself. First cosider the morphism of G +1 c +1 B G u of T-Gr c, which seds u m to u m, v m to v m, α m to α m, β m to β m, α p to α p, β p to β p, α to ξ, β to ξ, ad ξ +1 to γ([u ; u ] +1; 1 ξ ). This map c +1 equips BG u with a operatio system of the type G +1. Now BG u has cotractible uits, so by the uiversality of the map η +1, we obtai a uique morphism [1 B ; 1 B ] +1 of ω-operads. B +1 [1 B ;1 B ] +1 G u η +1 G +1 c +1 B G u This ( + 1)-cell [1 B ; 1 B ] +1 has as arity the degeerate tree 1 +1(1()). Now we just eed to prove that the followig diagram commutes serially, which will show that the source ad target of [1 B ; 1 B ] +1 are the uit 1 B : δ +1 B +1 G u B G u κ +1 [1 B ;1 B ] +1 1 B B G u But we have the followig diagram which, o the left side commutes serially, ad o the right side commutes: BG δ +1 u κ +1 η G δ +1 κ +1 [1 B ;1 B ] +1 G u B +1 G +1 η +1 c +1 B G u The morphism c +1 is a morphism of T-Gr p,c, as are the morphisms δ +1 ad o the bottom of this diagram. Their combiatorial descriptios show easily κ +1 that we have the equalities c +1 δ +1 = c +1 κ +1 = η. So we have the equalities [1 B ; 1 B ] +1 δ +1 = [1 B ; 1 B ] +1 κ +1 = 1 B. This shows that the ω-operad Coed(B G u ) has cotractible uits, ad thus we have a uique morphism B G u! Coed(B G u ) of ω-operads which expresses the fractality of B G u.

16 14 Camell Kachour If we compose the morphism! globular sets with the stadard actio of the reflexive Coed(B G u ) Coed(Alg(.)) Coed(A op G u ) Coed(Ob(.)) Coed(A, )op we obtai a morphism of ω-operads B G u G u Ed(A, ) which expresses a actio of the ω-operad BG u of reflexive globular sets o the globular object B G u -Alg() i SET of the reflexive (, ω)-graphs ( N). This gives a reflexive globular set structure o reflexive (, ω)-graphs ( N). 5 The coglobular objects of magmatic ω-operads Cosider ow the case P = M (magmatic). That is, we deal with the category T-CAT c of ω-operads. We apply the free fuctor T-Gr p,c M T-CAT c to the coglobular object C i T-Gr p,c of the higher trasformatios (see Sectio 3 of [8]) ad we obtai a coglobular object B M of ω-operads i T-CAT c B M δ 1 κ 1 B 1 M δ 1 2 κ 2 1 B 2 M B 1 M δ 1 B M If we write BT-CAT 1 for the category of ω-operads equipped with a chose compositio system i Batai s sese, whose morphisms are those which preserve these compositio systems, the it is importat to ote that the ω-operad BM is iitial i BT-CAT 1. Also, this coglobular object B M produces the followig globular object i CAT B M σ -Alg 1 B 1 M -Alg B M 1 -Alg σ1 BM -Alg β 1 I particular, BM is the ω-operad for ω-magmas, ad, for all >, algebras for BM are what we call (, ω)-magmas. The stadard actio associated to B M is give by the followig diagram i T-CAT 1 : β 1

17 Operads of higher trasformatios for globular sets ad higher magmas15 Coed(B M ) Coed(Alg(.)) Coed(A op M ) Coed(Ob(.)) Ed(A,M ) We call this the stadard actio of ω-magmas, which is a specific stadard actio of higher trasformatios. The moochromatic ω-operad Coed(B M ) of coedomorphisms plays a cetral role for ω-magmas. We call it the yellow operad. Also we have the followig result: Propositio 5.1. B M has the fractal property. Proof. I 7 we built a compositio system for Coed(B M ). Therefore, just usig the uiversality of BM i BT-CAT 1 we get the result. If we compose the morphism! M B M! M Coed(B M ) with the stadard actio associated to B M, we obtai a morphism of ω-operads B M Id Ed(A,M ) which expresses a actio of the ω-operad BM of ω-magmas o the globular object B M -Alg() i SET of (, ω)-magmas ( N), ad thus gives a ω-magma structure o (, ω)-magmas ( N). 6 The coglobular object of reflexive magmatic ω-operads Cosider the case P = M u (magmatic with cotractible uits). That is, we deal with the category UT-CAT c of ω-operads with chose cotractible uits (see Sectio 3). We apply the free fuctor (see Sectio 3) T-Gr p,c F UT-CAT c to the coglobular object C of higher trasformatios i T-Gr p,c ad we obtai a coglobular object B M u of ω-operads i T-CAT c BM δ 1 u BM 1 δ 1 2 u BM 2 u B 1 κ 1 κ 2 1 M u δ 1 B M u If we write UBT-CAT 1 for the category of ω-operads equipped with a compositio system i Batai s sese ad which have chose cotractible uits, where the

18 16 Camell Kachour morphisms are those which preserve these compositio systems ad cotractibility of the uits, the it is importat to ote that the ω-operad BM u is iitial i UBT-CAT 1. Also this coglobular object B M u produces the followig globular object i CAT B M u -Alg σ 1 β 1 B 1 M u -Alg B 1 M u -Alg σ 1 β 1 B M u -Alg I particular BM u is the ω-operad for reflexive ω-magmas (see [7]). The stadard actio associated to B M u is give by the followig diagram i T-CAT 1 : Coed(B M u ) Coed(Alg(.)) Coed(A op M u ) Coed(Ob(.)) Ed(A,Mu ) It is a specific stadard actio of higher trasformatios. The moochromatic ω-operad Coed(B M u ) of coedomorphisms plays a cetral role for reflexive ω- magmas. We call it the gree operad. Also we have the followig result: Propositio 6.1. B M u has the fractal property. Proof. I 7 we built a compositio system for Coed(B M u ), ad cotractibility of its uits is proved as i Sectio 4. Thus we just use the uiversality of BM u i UBT-CAT 1 to coclude. If we compose the morphism! Mu B M u! Mu Coed(B M u ) with the stadard actio associated to B M u, we obtai a morphism of ω-operads B M u C u Ed(A,Mu ) which expresses a actio of the ω-operad BM u of reflexive ω-magmas o the globular object B M u -Alg() i SET of reflexive (, ω)-magmas ( N), ad thus gives a reflexive ω-magma structure o reflexive (, ω)-magmas ( N). Remark 6.2. I [9] we use the same combiatorics for the C but with differet moads of arities: I fact we will use moads T (for all 2) o Glob 2 of the strict -trasformatios istead, where Glob 2 is the product of the category of globular sets with itself i CAT. If we deote (1, 1) the termial object i Glob 2, the all free strict higher trasformatios T (1, 1) will play the role of arities domai for

19 Operads of higher trasformatios for globular sets ad higher magmas17 these C 13. It give us a similar coglobular object C for the higher trasformatios which fits completely the cotrol of cohereces cells for its geerated higher operads for the strict ad the weak higher trasformatios. We could have proposed this differet coglobular object for buildig this article, but the author believe that higher structures ivolved here are more simple ad also more easily described by usig the simpler coglobular object C with arities domai T(1) + T(1), that we use all alog this article. We suspect that the ω-operad BC of Batai which algebras are his defiitio of weak ω-categories is fractal (see [6] ad [9]). We also suspect that the ω-operad BS 14 u which algebras are the strict ω-categories is fractal as well (see [9]): Suprisigly, we see i [9] that these two questios of fractality, for the strict case ad for the weak case, share i fact the same level of difficulty. 7 Compositio systems I this sectio we describe a compositio system i Batai s sese for the yellow operad Coed(B M ) described i Sectio 5, ad for the gree operad Coed(B M u ) described i Sectio 6. B P, or B for short, deotes the coglobular object B M (see Sectio 5) or the coglobular object B M u (see Sectio 6) i T-CAT c. Also, we deote by B B the 3-coloured ω-operad i T-CAT B p c which is obtaied by the pushout of B p κ p B δ p B i T-CAT c, where δ p = δ 1...δ p p+1 ad κp =...κ p p+1. For itegers p < we are goig to defie a morphism C µ p B B p B 13 Istead of the other useful arities domai T(1) + T(1) of this article. These arities domai T (1, 1) follow the spirit of costructios of weak higher trasformatios as i [4], where cohereces must be cotrolled by strict higher trasformatios. 14 I [6] ad [9], we show that the ω-operad of the strict ω-categories ca be preseted i a completely similar way as the Batai s operad B C : It is the iitial object i the category of the ω-operads which are strictly cotractible ad equipped with a compositio system. This also explai the similar otatio for this operad with those of Batai.

20 18 Camell Kachour i T-Gr p,c which, depedig o the uiversality property required, gives us a uique morphism B µ p B B p B i T-CAT c, that we still call µ p because there is o risk of cofusio. The uiversal map C η B gives us such morphism µ p. The key poit i defiig these morphisms µ p is first to describe the differet compositios p for the strict higher trasformatios. If < p <, we kow that for two strict -trasformatios σ ad τ, we have (σ p τ)(a) := σ(a) 1 p 1 τ(a) whose operadic iterpretatio is give by the cell γ(µ 1 p 1 ; σ 1 morphism p 1 τ). The the C µ p B B p B i T-Gr p,c seds the pricipal cell τ of C to the ( 1)-cell γ(µ 1 p 1 ; σ 1 p 1 τ) of B B, seds for each i N the i-cell F B p i of C to the i-cell F i of B B, ad B p seds the i-cell G i of C to the i-cell H i of B B. This morphism of T-Gr B p p,c is boudary preservig i a evidet sese. For p = it is a bit more complex. We are i the situatio of the pushout diagram B κ B δ B B i B. 2 B First we describe the compositio for the strict case i order to fid the cells that we eed i our ω-operad. Cosider the followig diagram i ω-cat, the strict ω-category of strict ω-categories: i 1 C F G τ D H τ E K

21 Operads of higher trasformatios for globular sets ad higher magmas19 Here C, D ad E are -cells (that is, strict ω-categories), F, G, H ad K are 1-cells (that is, strict ω-fuctors) ad τ ad σ are -cells (that is, strict -trasformatios). This picture describes τ ad σ with 2-cells, but the reader must see them as -cells. Also, τ ad σ are such that: s (σ) = C, t (σ) = s (τ) = D, ad t (τ) = E. If a C(), the F τ(a) G is a ( 1)-cell of D ad it iduces the followig commutative square of ( 1)-cells i E: H (F (a)) 1 H (τ(a)) H (G (a)) σ(f ) K (F (a)) K 1 (τ(a)) σ(g ) K (G (a)) This gives (σ τ)(a) = σ(g (a)) 1 H 1 (τ(a)) = K 1 (τ(a)) 1 σ(f (a)) which gives the two pricipal ( 1)-cells of B B B that we eed: The we have two choices of γ 1 (µ 1 ; γ(σ; G ) 1 γ(h 1 ; τ)) ad γ 1 (µ 1 ; γ(k 1 ; τ) 1 γ(σ; F )). C µ o B B p B which sed the pricipal cell τ of C to γ 1 (µ 1 ; γ(σ; G ) 1 γ(h 1 ; τ)) or to γ 1 (µ 1 ; γ(k 1 ; τ) 1 γ(σ; F )), ad for both cases which sed, for each i N, the i-cell F i of C to the i-cell γ(f i ; H i ) of B B, ad the i-cell G i of B C to the i-cell γ(g i ; K i ) of B B B. These morphisms of T-Gr p,c are boudary preservig i a evidet sese. Thaks to the uiversal property of η, we obtai the followig uique 15 morphisms of ω-operads µ p ad µ (the dotted arrows). 15 We say uique for µ because we choose oe presetatio of µ amog the two choices as above which are possible for µ. However it is importat to ote that uder the hypothesis of cotractibility of [9], these two choices will coect with a coherece cell or will equalise, depedig o the cotractibility ivolved.

22 11 Camell Kachour B µ p B B p B η µ p C B µ B B B η µ C With the idetity morphisms of operads B 1 B B C c Coed(B ) µ p u µ p 1 B Thus, aticipatig Sectios 5 ad 6, we have the followig coclusio: Propositio 7.1. The ω-operads of coedomorphisms Coed(B M ) ad Coed(B M u ), both have a compositio system. Ackowledgemet I am grateful to Adré Joyal who helped me improve my operadic approach to weak higher trasformatios. I am grateful to Steve Lack ad Mark Weber who explaied me some techical poits that I was ot able to uderstad by myself. Also, thaks to the referee for his/her commets which improved the paper. Refereces [1] J. Adámek ad J. Rosický, Locally Presetable ad Accessible Categories, Cambridge Uiversity Press, [2] M. Batai, Mooidal globular categories as a atural eviromet for the theory of weak--categories, Adv. Math. 136 (1998), [3] F. Borceux, Hadbook of Categorical Algebra, Vol. 2, Cambridge Uiversity Press, [4] K. Kachour, Défiitio algébrique des cellules o-strictes, Cah. Topol. Géom. Différ. Catég. 1 (28), [5] C. Kachour, Operadic defiitio of the o-strict cells, Cah. Topol. Géom. Différ. Catég. 4 (211), [6] C. Kachour, Aspects of Globular Higher Category Theory, Ph.D. Thesis, Macquarie Uiversity, 213.

23 Operads of higher trasformatios for globular sets ad higher magmas111 [7] C. Kachour, Algebraic defiitio of weak (, )-categories, To appear i Theory Appl. Categ. (215). [8] C. Kachour, ω-operads of coedomorphisms ad fractal ω-operads for higher structures, Categ. Geeral Alg. Structures Appl. 3(1) (215), [9] C. Kachour, Steps toward the weak ω-category of the weak ω-categories i the globular settig, To appear i Categ. Geeral Alg. Structures Appl. 4(1) (215). [1] G.M. Kelly, A uified treatmet of trasfiite costructios for free algebras, free mooids, colimits, associated sheaves, ad so o, Bull. Aust. Math. Soc. 22 (198), [11] S. Lack, O the moadicity of fiitary moads, J. Pure Appl. Algebra 14 (1999), [12] L. Coppey ad Ch. Lair, Leços de théorie des esquisses, Uiversité Paris VII, (1985). [13] T. Leister, Higher Operads, Higher Categories, Lodo Math. Soc. Lect. Note Series, Cambridge Uiversity Press 298, 24. [14] M. Makkai ad R. Paré, Accessible Categories: The Foudatios of Categorical Model Theory, America Mathematical Society, [15] G. Maltsiiotis, Ifii groupoïdes o strictes, d après Grothedieck, maltsi/ps/ifgrart.pdf (27). [16] J. Peo, Approche polygraphique des -catégories o-strictes, Cah. Topol. Géom. Différ. Catég. 1 (1999), [17] D. Tamarki, What do DG categories form?, Compos. Math. 143(5) (27), Camell Kachour, Departmet of Mathematics, Macquarie Uiversity, Sydey, Australia. camell.kachour@gmail.com