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cohomology in degree 2 they can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group gerbes provide a convenient if highly abstract language for dealing with many types of deformation questions especially in modern algebraic geometry in addition special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them gerbe is a french and archaic english word that literally means wheat sheaf definitions edit gerbes on a topological space edit a gerbe on a topological space s displaystyle s 1 318 is a stack x displaystyle mathcal x of groupoids over s displaystyle s that is locally non empty each point p s displaystyle p in s has an open neighbourhood u p displaystyle u_ p over which the section category x u p displaystyle mathcal x u_ p of the gerbe is not empty and transitive for any two objects a displaystyle a and b displaystyle b of x u displaystyle mathcal x u for any open set u displaystyle u there is an open covering u u i i i displaystyle mathcal u u_ i _ i in i of u displaystyle u such that the restrictions of a displaystyle a and b displaystyle b to each u i displaystyle u_ i are connected by at least one morphism a canonical example is the gerbe b h displaystyle bh of principal bundles with a fixed structure group h displaystyle h the section category over an open set u displaystyle u is the category of principal h displaystyle h bundles on u displaystyle u with isomorphism as morphisms thus the category is a groupoid as principal bundles glue together satisfy the descent condition these groupoids form a stack the trivial bundle x h x displaystyle x times h to x shows that the local non emptiness condition is satisfied and finally as principal bundles are locally trivial they become isomorphic when restricted to sufficiently small open sets thus the transitivity condition is satisfied as well gerbes on a site edit the most general definition of gerbes are defined over a site given a site c displaystyle mathcal c a c displaystyle mathcal c gerbe g displaystyle g 2 3 129 is a category fibered in groupoids g c displaystyle g to mathcal c such that there exists a refinement 4 c displaystyle mathcal c of c displaystyle mathcal c such that for every object s ob c displaystyle s in text ob mathcal c the associated fibered category g s displaystyle g_ s is non empty for every s ob c displaystyle s in text ob mathcal c any two objects in the fibered category g s displaystyle g_ s are locally isomorphic note that for a site c displaystyle mathcal c with a final object e displaystyle e a category fibered in groupoids g c displaystyle g to mathcal c admits a local section if and only if ob g e displaystyle text ob g_ e neq varnothing in which case g c displaystyle g to mathcal c automatically satisfies the first axiom motivation for gerbes on a site edit one of the main motivations for considering gerbes on a site is to consider the following naive question if the čech cohomology group h 1 u g displaystyle h 1 mathcal u g for a suitable covering u u i i i displaystyle mathcal u u_ i _ i in i of a space x displaystyle x gives the isomorphism classes of principal g displaystyle g bundles over x displaystyle x what does the iterated cohomology functor h 1 h 1 g displaystyle h 1 h 1 g represent meaning we are gluing together the groups h 1 u i g displaystyle h 1 u_ i g via some 1 cocycle gerbes are a technical response for this question they give geometric representations of elements in the higher cohomology group h 2 u g displaystyle h 2 mathcal u g it is expected this intuition should hold for higher gerbes cohomological classification edit one of the main theorems concerning gerbes is their cohomological classification whenever they have automorphism groups given by a fixed sheaf of abelian groups l _ displaystyle underline l 5 2 called a band for a gerbe x displaystyle mathcal x on a site c displaystyle mathcal 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 sect...