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cal c an object u ob c displaystyle u in text ob mathcal c and an object x ob x u displaystyle x in text ob mathcal x u the automorphism group of a gerbe is defined as the automorphism group l aut _ x u x displaystyle l underline text aut _ mathcal x u x notice this is well defined whenever the automorphism group is always the same given a covering u u i x i i displaystyle mathcal u u_ i to x _ i in i there is an associated class c l _ h 3 x l _ displaystyle c underline l in h 3 x underline l representing the isomorphism class of the gerbe x displaystyle mathcal x banded by l displaystyle l for example in topology many examples of gerbes can be constructed by considering gerbes banded by the group u 1 displaystyle u 1 as the classifying space b u 1 k z 2 displaystyle b u 1 k mathbb z 2 is the second eilenberg maclane space for the integers a bundle gerbe banded by u 1 displaystyle u 1 on a topological space x displaystyle x is constructed from a homotopy class of maps in x b 2 u 1 x k z 3 displaystyle x b 2 u 1 x k mathbb z 3 which is exactly the third singular homology group h 3 x z displaystyle h 3 x mathbb z it has been found 6 that all gerbes representing torsion cohomology classes in h 3 x z displaystyle h 3 x mathbb z are represented by a bundle of finite dimensional algebras end v displaystyle text end v for a fixed complex vector space v displaystyle v in addition the non torsion classes are represented as infinite dimensional principal bundles p u h displaystyle pu mathcal h of the projective group of unitary operators on a fixed infinite dimensional separable hilbert space h displaystyle mathcal h note this is well defined because all separable hilbert spaces are isomorphic to the space of square summable sequences ℓ 2 displaystyle ell 2 the homotopy theoretic interpretation of gerbes comes from looking at the homotopy fiber square x s f b 2 u 1 displaystyle begin matrix mathcal x to downarrow downarrow s xrightarrow f b 2 u 1 end matrix analogous to how a line bundle comes from the homotopy fiber square l s f b u 1 displaystyle begin matrix l to downarrow downarrow s xrightarrow f bu 1 end matrix where b u 1 k z 2 displaystyle bu 1 simeq k mathbb z 2 giving h 2 s z displaystyle h 2 s mathbb z as the group of isomorphism classes of line bundles on s displaystyle s examples edit c algebras edit there are natural examples of gerbes that arise from studying the algebra of compactly supported complex valued functions on a paracompact space x displaystyle x 7 pg 3 given a cover u u i displaystyle mathcal u u_ i of x displaystyle x there is the čech groupoid defined as g i j u i j u i displaystyle mathcal g left coprod _ i j u_ ij rightrightarrows coprod u_ i right with source and target maps given by the inclusions s u i j u j t u i j u i displaystyle begin aligned s u_ ij hookrightarrow u_ j t u_ ij hookrightarrow u_ i end aligned and the space of composable arrows is just i j k u i j k displaystyle coprod _ i j k u_ ijk then a degree 2 cohomology class σ h 2 x u 1 displaystyle sigma in h 2 x u 1 is just a map σ u i j k u 1 displaystyle sigma coprod u_ ijk to u 1 we can then form a non commutative c algebra c c g σ displaystyle c_ c mathcal g sigma which is associated to the set of compact supported complex valued functions of the space g 1 i j u i j displaystyle mathcal g _ 1 coprod _ i j u_ ij it has a non commutative product given by a b x i k j a x i j b x j k σ x i j k displaystyle a b x i k sum _ j a x i j b x j k sigma x i j k where the cohomology class σ displaystyle sigma twists the multiplication of the standard c displaystyle c algebra product algebraic geometry edit let m displaystyle m be a variety over an algebraically closed field k displaystyle k g displaystyle g an algebraic group for example g m displaystyle mathbb g _ m recall that a g torsor over m displaystyle m is an algebraic space p displaystyle p with an action of g displaystyle g and a map π p m displaystyle pi p to m such that locally on m displaystyle m in étale topology or fppf topology π displaystyle pi is a direct product π u g u u displaystyle pi _ u g times u to u a g gerbe over m may be defined in a similar way it is an artin stack m displaystyle mathcal m with a map π m m displaystyle pi colon mathcal m to m such that locally on m in étale or fppf topology π displaystyle pi is a direct product π u b g u u displaystyle pi _ u colon mathrm b g times u to u 8 here b g displaystyle bg denotes the classifying stack of g displaystyle g i e a quotient g displaystyle g of a point by a trivial g displaystyle g action there is no need to impose the compatibility with the group structure in that case since it is covered by the definition of a stack the underlying topological spaces of m displaystyle mathcal m and m displaystyle m are the same but in m displaystyle mathcal m each point is equipped with a stabilizer group isomorphic to g displaystyle g from two term complexes of coherent sheaves edit every two term complex of coherent sheaves e e 1 d e 0 displaystyle mathcal e bullet mathcal e 1 xrightarrow d mathcal e 0 on a scheme x sch displaystyle x in text sch has a canonical sheaf of groupoids associated to it where on an open subset u x displaystyle u subseteq x there is a two term complex of x u displaystyle x u modules e 1 u d e 0 u displaystyle mathcal e 1 u xrightarrow d mathcal e 0 u giving a groupoid it has objects given by elements x e 0 u displaystyle x in mathcal e 0 u and a morphism x x displaystyle x to x is given by an element y e 1 u displaystyle y in mathcal e 1 u such that d y x x displaystyle dy x x in order for this stack to be a gerbe the cohomology sheaf h 0 e displaystyle mathcal h 0 mathcal e must always have a section this hypothesis implies the category constructed above always has objects note this can be applied to the situation of comodules over hopf algebroids to construct algebraic models of gerbes over affine or projective stacks projectivity if a graded hopf algebroid is used in addition two term spectra from the stabilization of the derived category of comodules of hopf algebroids a γ displaystyle a gamma with γ displaystyle gamma flat over a displaystyle a give additional models of gerbes that are non strict moduli stack of stable bundles on a curve edit consider a smooth projective curve c displaystyle c over k displaystyle k of genus g 1 displaystyle g 1 let m r d s displaystyle mathcal m _ r d s be the moduli stack of stable vector bundles on c displaystyle c of rank r displaystyle r and degree d displaystyle d it has a coarse moduli space m r d s displaystyle m_ r d s which is a quasiprojective variety these two moduli problems parametrize the same objects but the stacky version remembers automorphisms of vector bundles for any stable vector bundle e displaystyle e the automorphism group a u t e displaystyle aut e consists only of scalar multiplications so each point in a moduli stack has a stabilizer isomorphic to g m displaystyle mathbb g _ m it turns out that the map m r d s m r d s displaystyle mathcal m _ r d s to m_ r d s is indeed a g m displaystyle mathbb g _ m gerbe in the sense above 9 it is a trivial gerbe if and only if r displaystyle r and d displaystyle d are coprime root stacks edit another class of gerbes can be found using the construction of root stacks informally the r displaystyle r th root stack of a line bundle l s displaystyle l to s over a scheme is a space representing the r displaystyle r th root of l displaystyle l and is denoted l s r displaystyle sqrt r l s 10 pg 52 the r displaystyle r th root stack of l displaystyle l has the property r l s r l displaystyle bigotimes r sqrt r l s cong l as gerbes it is constructed as the stack l s r sch s o p grpd displaystyle sqrt r l s operatorname sch s op to operatorname grpd sending an s displaystyle s scheme t s displaystyle t to s to the category whose objects are line bundles of the form m t α m α m m r l s t displaystyle left m to t alpha _ m alpha _ m m otimes r xrightarrow sim l times _ s t right and morphisms are commutative diagrams compatible with the isomorphisms α m displaystyle alpha _ m this gerbe is banded by the algebraic group of roots of unity μ r displaystyle mu _ r where on a cover t s displaystyle t to s it acts on a point m t α m displaystyle m to t alpha _ m by cyclically permuting the factors of m displaystyle m in m r displaystyle m otimes r geometrically these stacks are formed as the fiber product of stacks s b g m b g m b g m s b g m displaystyle begin matrix s times _ b mathbb g _ m b mathbb g _ m to b mathbb g _ m downarrow downarrow s to b mathbb g _ m end matrix where the vertical map of b g m b g m displaystyle b mathbb g _ m to b mathbb g _ m comes from the kummer sequence 1 μ r g m r g m 1 displaystyle 1 xrightarrow mu _ r xrightarrow mathbb g _ m xrightarrow cdot r mathbb g _ m xrightarrow 1 this is because b g m displaystyle b mathbb g _ m is the moduli space of line bundles so the line bundle l s displaystyle l to s corresponds to an object of the category b g m s displaystyle b mathbb g _ m s considered as a point of the moduli space root stacks with sections edit there is another related construction of root stacks with sections given the data above let s s l displaystyle s s to l be a section then the r displaystyle r th root stack of the pair l s s displaystyle l to s s is defined as the lax 2 functor 10 11 l s s r sch s o p grpd displaystyle sqrt r l s s operatorname sch s op to operatorname grpd sending an s displaystyle s scheme t s displaystyle t to s to the category whose objects are line bundles of the form m t α m t α m m r l s t t γ t m α m t r s displaystyle left m to t alpha _ m t begin aligned alpha _ m m otimes r xrightarrow sim l times _ s t t in gamma t m alpha _ m t otimes r s end aligned right and morphisms are given similarly these stacks can be constructed very explicitly and are well understood for affine schemes in fact these form the affine models for root stacks with sections 11 4 locally we may assume s spec a displaystyle s text spec a and the line bundle l displaystyle l is trivial hence any section s displaystyle s is equivalent to taking an element s a displaystyle s in a then the stack is given by the stack quotient l s s r spec b μ r displaystyle sqrt r l s s text spec b mu _ r 11 9 with b a x x r s displaystyle b frac a x x r s if s 0 displaystyle s 0 then this gives an infinitesimal extension of spec a μ r displaystyle text spec a mu _ r examples throughout algebraic geometry edit these and more general kinds of gerbes arise in several contexts as both geometric spaces and as formal bookkeeping tools azumaya algebras deformations of infinitesimal thickenings twisted forms of projective varieties fiber functors for motives differential geometry edit h 3 x z displaystyle h 3 x mathbb z and o x displaystyle mathcal o _ x gerbes jean luc brylinski s approach history edit this section needs more citations please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed january 2021 learn how and when to remove this message gerbes first appeared in the context of algebraic geometry they were subsequently developed in a more traditional geometric framework by brylinski brylinski 1993 one can think of gerbes as being a natural step in a hierarchy of mathematical objects providing geometric realizations of integral cohomology classes a more specialised notion of gerbe was introduced by murray and called bundle gerbes essentially they are a smooth version of abelian gerbes belonging more to the hierarchy starting with principal bundles than sheaves bundle gerbes have been used in gauge theory and also string theory 12 current work by others is developing a theory of non abelian bundle gerbes see also edit twisted sheaf azumaya algebra twisted k theory algebraic stack bundle gerbe string group references edit basic bundle theory and k cohomology invariants husemöller dale berlin springer 2008 isbn 978 3 540 74956 1 oclc 233973513 cite book cs1 maint others link 1 2 section 8 11 06ny gerbes the stacks project stacks math columbia edu retrieved 2020 10 27 giraud j jean 1971 cohomologie non abélienne berlin springer verlag isbn 3 540 05307 7 oclc 186709 section 7 8 00vs families of morphisms with fixed target the stacks project stacks math columbia edu retrieved 2020 10 27 section 21 11 0cjz second cohomology and gerbes the stacks project stacks math columbia edu retrieved 2020 10 27 karoubi max 2010 12 12 twisted bundles and twisted k theory arxiv 1012 2512 math kt block jonathan daenzer calder 2009 01 09 mukai duality for gerbes with connection arxiv 0803 1529 math qa edidin dan hassett brendan kresch andrew vistoli angelo 2001 brauer groups and quotient stacks american journal of mathematics 123 4 761 777 arxiv math 9905049 doi 10 1353 ajm 2001 0024 s2cid 16541492 hoffman norbert 2010 moduli stacks of vector bundles on curves and the king schofield rationality proof cohomological and geometric approaches to rationality problems progress in mathematics vol 282 pp 133 148 arxiv math 0511660 doi 10 1007 978 0 8176 4934 0_5 isbn 978 0 8176 4933 3 s2cid 5467668 1 2 abramovich dan graber tom vistoli angelo 2008 04 13 gromov witten theory of deligne mumford stacks arxiv math 0603151 1 2 3 cadman charles 2007 using stacks to impose tangency conditions on curves pdf amer j math 129 2 405 427 arxiv math 0312349 doi 10 1353 ajm 2007 0007 s2cid 10323243 bunk severin szabo richard j 2017 05 18 fluxes bundle gerbes and 2 hilbert spaces letters in mathematical physics 107 10 1877 1918 arxiv 1612 01878 bibcode 2017lmaph 107 1877b doi 10 1007 s11005 017 0971 x retrieved 2025 11 06 giraud jean 1971 cohomologie non abélienne springer isbn 3 540 05307 7 brylinski jean luc 1993 loop space characteristic classes and geometric quantization birkhäuser verlag isbn 0 8176 3644 7 external links edit introductory articles edit constructions with bundle gerbes stuart johnson an introduction to gerbes on orbifolds ernesto lupercio bernado uribe what is a gerbe by nigel hitchin in notices of the ams bundle gerbes michael murray moerdijk ieke introduction to the language of stacks and gerbes retrieved 2007 05 20 gerbes in topology edit homotopy theory of presheaves of simplicial groupoids zhi ming luo twisted k theory edit twisted k theory and k theory of bundle gerbes twisted bundles and twisted k theory karoubi applications in string theory edit stable singularities in string theory contains examples of gerbes in appendix using the brauer group branes on group manifolds gluon condensates and twisted k theory lectures on special lagrangian submanifolds very down to earth introduction with applications to mirror symmetry the basic gerbe over a compact simple lie group gives techniques for describing groups such as the string group as a gerbe retrieved from https en wikipedia org w index php title gerbe oldid 1374910213 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