Double loop spaces braided monoidal categories and algebraic type of space

Clemens Berger

Description

Double loop spaces, braided monoidal categories and algebraic 3-type of space Clemens Berger 15 march 1997 Abstract We show that the nerve of a braided monoidal category carries a nat- ural action of a simplicial E2-operad and is thus up to group completion a double loop space. Shifting up dimension twice associates to each braided monoidal category a 1-reduced lax 3-category whose nerve realizes an ex- plicit double delooping whenever all cells are invertible. We deduce that lax 3-groupoids are algebraic models for homotopy 3-types. Introduction The concept of braiding as a refinement of symmetry is the starting point of a rich interplay between geometry (knot theory) and algebra (representation theory). The underlying structure of a braided monoidal category reveals an interest of its own in that it encompasses two at first sight different geometrical objects : double loop spaces and homotopy 3-types. The link to double loop spaces was pointed out by J. Stasheff [38] and made precise by Z. Fiedorowicz [15], who proves that double loop spaces may be characterized (up to group completion) as algebras over a contractible free braided operad. The link to homotopy 3-types goes back to A. Grothendieck's pursuit of stacks [20] and was taken up by O. Leroy [29], who shows that a weak form of 3-groupoid, we call here lax 3-groupoid, models homotopy 3-types; the laxness stems precisely from “relaxing” a strict commutativity constraint to an interchange (i.

• braided monoidal


• e2-operad


• sk ?


• pure braid


• group completion


• cellular e2-preoperad


• thus


• sk induced

Sujets

• Berger
• Baxter
• Braid group

Informations

• Publié par: pefav
• Nombre de lectures: 15
• Langue: English

Extrait

Doubleloopspaces,braidedmonoidalcategories andalgebraic3-typeofspace ClemensBerger 15march1997 Abstract Weshowthatthenerveofabraidedmonoidalcategorycarriesanat- uralactionofasimplicialE 2 -operadandisthusuptogroupcompletiona doubleloopspace.Shiftingupdimensiontwiceassociatestoeachbraided monoidalcategorya1-reducedlax3-categorywhosenerverealizesanex- plicitdoubledeloopingwheneverallcellsareinvertible.Wededucethat lax3-groupoidsarealgebraicmodelsforhomotopy3-types. Introduction Theconceptofbraidingasarefinementofsymmetryisthestartingpointof arichinterplaybetweengeometry(knottheory)andalgebra(representation theory).Theunderlyingstructureofabraidedmonoidalcategoryrevealsan interestofitsowninthatitencompassestwoatfirstsightdifferentgeometrical objects:doubleloopspacesandhomotopy3-types.Thelinktodoubleloop spaceswaspointedoutbyJ.Stasheff[38]andmadeprecisebyZ.Fiedorowicz [15],whoprovesthatdoubleloopspacesmaybecharacterized(uptogroup completion)asalgebrasoveracontractiblefreebraidedoperad.Thelinkto homotopy3-typesgoesbacktoA.Grothendieck’spursuitofstacks[20]andwas takenupbyO.Leroy[29],whoshowsthataweakformof3-groupoid,wecall herelax3-groupoid,modelshomotopy3-types;thelaxnessstemspreciselyfrom “relaxing”astrictcommutativityconstrainttoaninterchange(i.e.braiding) cell.Lax3-groupoidsarecalledsemi-strict3-groupoidsbyBaez-Neuchl[2],and GraygroupoidsbyGordon-Power-Street[17]. Ourmainconcernhereisto“tietogether”thesetwoaspectsofthesame structure.Inparticular,wehopethismighthelptoconstructageneralscheme relatingiteratedloopspacestohomotopyn-types.Thepointisthatthecom- binatorialstructureofiteratedloopspacesisbynowquitewellunderstood(cf. [8],[3]),whereasthesameisnottrueforhomotopyn-typeswhenn ≥ 4. Thetextisdividedintothreeparts: • PartOneprovestheexistenceofadoubledeloopingforbraidedmonoidal categoriesinthe“realm”ofsimplicialE 2 -operads.Insomeprecisesense,the 1

braidgroupsarecontainedinthecombinatorialstructureofthesymmetric groupsbymeansoftheweakBruhatorder. • PartTwofullyembedsthecategoryofbraidedmonoidalcategoriesinto thecategoryoflax3-categoriesthusprovidinganexplicitdoubledeloopingfor braidedmonoidalgroupoids.Themaintoolisacosimpliciallax3-categorical object. • PartThreeprovestheequivalenceofthehomotopycategoriesofsimplicial 3-typesandlax3-groupoidsonthebasisofthefollowingobservation:thefun- damentalgroupoidofthedoubleloopsonasimplyconnected3-typeisbraided monoidal. IwouldliketotaketheopportunitytothankthemembersofSydney’sCat- egorySeminarfortheirhospitalityduringmyvisitofthesouthernhemisphere. Thefollowingtextowesquitealottoallofthem. 1Braidedmonoidalcategories Throughout,weshalladoptthefollowingconventionsandnotations: ‘Monoidal’alwaysmeans‘strictmonoidal’.–Abraidingisnotassumedtobe invertible.–Theclassofn-cellsofa(multiple)category C iswritten C n .–The symmetricgrouponasetIisdenotedby S I .For S { 1,...,n } wewrite S n andthe permutationwhichmaps(1,...,n)to(a 1 ,...,a n )isrepresentedby[a 1 ,...,a n ]. Definition1.1.Abraidedmonoidalcategoryisamonoidalcategory( C ,,U) endowedwithabinaturalfamilyofmorphismsc A,B :AB → BA,called braidings,suchthatforallA,B,C ∈C 0 wehave i)c A,U =c U,A =1 A (unitarity), ii)c A  B,C =(c A,C 1 B ) ◦ (1 A c B,C ) c A,BC =(1 B c A,C ) ◦ (c A,B 1 C )(transitivity). Naturalityandtransitivityofthebraidingsimplythecommutativityofthe socalledYang-Baxterhexagon: c A,C 1 B ACB ✲ CAB ❅✒� 1 A c B,C � ❅ 1 C c A,B ❘❅� ABCCBA ✒� ❅ c A,B 1 C ❅ � c B,C 1 A � ❘❅1 B c A,C BAC ✲ BCA Thepurposeofthischapterisanewproofofthefollowingtheorem: 2