Fibred categories (orfibered categories) are abstract entities inmathematics used to provide a general framework fordescent theory. They formalise the various situations ingeometry andalgebra in whichinverse images (orpull-backs) of objects such asvector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for everycontinuous map from a topological spaceX to another topological spaceY is associated thepullbackfunctor taking bundles onY to bundles onX. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular inalgebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to definestacks, which are fibered categories (over a site) with "descent". Fibrations also play an important role in categorical semantics oftype theory, and in particular that ofdependent type theories.

Fibred categories were introduced byAlexander Grothendieck (1959,1971), and developed in more detail byJean Giraud (1964,1971).

There are many examples intopology andgeometry where some types of objects are considered to existon orabove orover some underlyingbase space. The classical examples include vector bundles,principal bundles, andsheaves over topological spaces. Another example is given by "families" ofalgebraic varieties parametrised by another variety. Typical to these situations is that to a suitable type of amap${\displaystyle f:X\to Y}$ between base spaces, there is a correspondinginverse image (also calledpull-back) operation${\displaystyle f^{\ast }}$ taking the considered objects defined on${\displaystyle Y}$ to the same type of objects on${\displaystyle X}$. This is indeed the case in the examples above: for example, the inverse image of a vector bundle${\displaystyle E}$ on${\displaystyle Y}$ is a vector bundle${\displaystyle f^{\ast }(E)}$ on${\displaystyle X}$.

Moreover, it is often the case that the considered "objects on a base space" form a category, or in other words have maps (morphisms) between them. In such cases the inverse image operation is often compatible with composition of these maps between objects, or in more technical terms is afunctor. Again, this is the case in examples listed above.

However, it is often the case that if${\displaystyle g:Y\to Z}$ is another map, the inverse image functors are notstrictly compatible with composed maps: if${\displaystyle z}$ is an objectover${\displaystyle Z}$ (a vector bundle, say), it may well be that

${\displaystyle f^{\ast }(g^{\ast }(z))\ne (g\circ f{)}^{\ast }(z).}$

Instead, these inverse images are onlynaturallyisomorphic. This introduction of some "slack" in the system of inverse images causes some delicate issues to appear, and it is this set-up that fibred categories formalise.

The main application of fibred categories is indescent theory, concerned with a vast generalisation of "glueing" techniques used in topology. In order to support descent theory of sufficient generality to be applied in non-trivial situations in algebraic geometry the definition of fibred categories is quite general and abstract. However, the underlying intuition is quite straightforward when keeping in mind the basic examples discussed above.

There are two essentially equivalent technical definitions of fibred categories, both of which will be described below. All discussion in this section ignores theset-theoretical issues related to "large" categories. The discussion can be made completely rigorous by, for example, restricting attention to small categories or by usinguniverses.

If${\displaystyle \varphi :F\to E}$ is afunctor between twocategories and${\displaystyle S}$ is an object of${\displaystyle E}$, then thesubcategory of${\displaystyle F}$ consisting of those objects${\displaystyle x}$ for which${\displaystyle \varphi (x)=S}$ and those morphisms${\displaystyle m}$ satisfying${\displaystyle \varphi (m)={\text{id}}_S}$, is called thefibre category (orfibre)over${\displaystyle S}$, and is denoted${\displaystyle F_S}$. The morphisms of${\displaystyle F_S}$ are called${\displaystyle S}$-morphisms, and for${\displaystyle x,y}$ objects of${\displaystyle F_S}$, the set of${\displaystyle S}$-morphisms is denoted by${\displaystyle {\text{Hom}}_S(x,y)}$. The image by${\displaystyle \varphi }$ of an object or a morphism in${\displaystyle F}$ is called itsprojection (by${\displaystyle \varphi }$). If${\displaystyle f}$ is a morphism of${\displaystyle E}$, then those morphisms of${\displaystyle F}$ that project to${\displaystyle f}$ are called${\displaystyle f}$-morphisms, and the set of${\displaystyle f}$-morphisms between objects${\displaystyle x}$ and${\displaystyle y}$ in${\displaystyle F}$ is denoted by${\displaystyle {\text{Hom}}_f(x,y)}$.

A morphism${\displaystyle m:x\to y}$ in${\displaystyle F}$ is called${\displaystyle \varphi }$-cartesian (or simplycartesian) if it satisfies the following condition:

if${\displaystyle f:T\to S}$ is the projection of${\displaystyle m}$, and if${\displaystyle n:z\to y}$ is an${\displaystyle f}$-morphism, then there isprecisely one${\displaystyle T}$-morphism${\displaystyle a:z\to x}$ such that${\displaystyle m\circ a=n}$.

Acartesian morphism${\displaystyle m:x\to y}$ is called aninverse image of its projection${\displaystyle f=\varphi (m)}$; the object${\displaystyle x}$ is called aninverse image of${\displaystyle y}$by${\displaystyle f}$.

The cartesian morphisms of a fibre category${\displaystyle F_S}$ are precisely the isomorphisms of${\displaystyle F_S}$. There can in general be more than one cartesian morphism projecting to a given morphism${\displaystyle f:T\to S}$, possibly having different sources; thus there can be more than one inverse image of a given object${\displaystyle y}$ in${\displaystyle F_S}$ by${\displaystyle f}$. However, it is a direct consequence of the definition that two such inverse images are isomorphic in${\displaystyle F_T}$.

A functor${\displaystyle \varphi :F\to E}$ is also called an${\displaystyle E}$-category, or said to make${\displaystyle F}$ into an${\displaystyle E}$-category or a categoryover${\displaystyle E}$. An${\displaystyle E}$-functor from an${\displaystyle E}$-category${\displaystyle \varphi :F\to E}$ to an${\displaystyle E}$-category${\displaystyle \psi :G\to E}$ is a functor${\displaystyle \alpha :F\to G}$ such that${\displaystyle \psi \circ \alpha =\varphi }$.${\displaystyle E}$-categories form in a natural manner a2-category, with 1-morphisms being${\displaystyle E}$-functors, and 2-morphisms being natural transformations between${\displaystyle E}$-functors whose components lie in some fibre.

An${\displaystyle E}$-functor between two${\displaystyle E}$-categories is called acartesian functor if it takes cartesian morphisms to cartesian morphisms. Cartesian functors between two${\displaystyle E}$-categories${\displaystyle F,G}$ form a category${\displaystyle {\text{Cart}}_E(F,G)}$, withnatural transformations as morphisms. A special case is provided by considering${\displaystyle E}$ as an${\displaystyle E}$-category via the identity functor: then a cartesian functor from${\displaystyle E}$ to an${\displaystyle E}$-category${\displaystyle F}$ is called acartesian section. Thus a cartesian section consists of a choice of one object${\displaystyle x_S}$ in${\displaystyle F_S}$ for each object${\displaystyle S}$ in${\displaystyle E}$, and for each morphism${\displaystyle f:T\to S}$ a choice of an inverse image${\displaystyle m_f:x_T\to x_S}$. A cartesian section is thus a (strictly) compatible system of inverse images over objects of${\displaystyle E}$. The category of cartesian sections of${\displaystyle F}$ is denoted by

${\displaystyle \underset{⟵}{\mathrm{L}\mathrm{i}\mathrm{m}}(F/E)={\mathrm{C}\mathrm{a}\mathrm{r}\mathrm{t}}_E(E,F).}$

In the important case where${\displaystyle E}$ has aterminal object${\displaystyle e}$ (thus in particular when${\displaystyle E}$ is atopos or the category${\displaystyle E_{/S}}$ ofarrows with target${\displaystyle S}$ in${\displaystyle E}$) the functor

${\displaystyle ϵ:\underset{⟵}{\mathrm{L}\mathrm{i}\mathrm{m}}(F/E)\to F_e,s\mapsto s(e)}$

isfully faithful (Lemma 5.7 of Giraud (1964)).

The technically most flexible and economical definition of fibred categories is based on the concept of cartesian morphisms. It is equivalent to a definition in terms ofcleavages, the latter definition being actually the original one presented in Grothendieck (1959); the definition in terms of cartesian morphisms was introduced in Grothendieck (1971) in 1960–1961.

An${\displaystyle E}$ category${\displaystyle \varphi :F\to E}$ is afibred category (or afibred${\displaystyle E}$-category, or acategory fibred over${\displaystyle E}$) if each morphism${\displaystyle f}$ of${\displaystyle E}$ whosecodomain is in the range of projection has at least one inverse image, and moreover the composition${\displaystyle m\circ n}$ of any two cartesian morphisms${\displaystyle m,n}$ in${\displaystyle F}$ is always cartesian. In other words, an${\displaystyle E}$-category is a fibred category if inverse images always exist (for morphisms whose codomains are in the range of projection) and aretransitive.

If${\displaystyle E}$ has a terminal object${\displaystyle e}$ and if${\displaystyle F}$ is fibred over${\displaystyle E}$, then the functor${\displaystyle ϵ}$ from cartesian sections to${\displaystyle F_e}$ defined at the end of the previous section is anequivalence of categories and moreoversurjective on objects.

If${\displaystyle F}$ is a fibred${\displaystyle E}$-category, it is always possible, for each morphism${\displaystyle f:T\to S}$ in${\displaystyle E}$ and each object${\displaystyle y}$ in${\displaystyle F_S}$, to choose (by using theaxiom of choice) precisely one inverse image${\displaystyle m:x\to y}$. The class of morphisms thus selected is called acleavage and the selected morphisms are called thetransport morphisms (of the cleavage). A fibred category together with a cleavage is called acloven category. A cleavage is callednormalised if the transport morphisms include all identities in${\displaystyle F}$; this means that the inverse images of identity morphisms are chosen to be identity morphisms. Evidently if a cleavage exists, it can be chosen to be normalised; we shall consider only normalised cleavages below.

The choice of a (normalised) cleavage for a fibred${\displaystyle E}$-category${\displaystyle F}$ specifies, for each morphism${\displaystyle f:T\to S}$ in${\displaystyle E}$, afunctor${\displaystyle f^{\ast }:F_S\to F_T}$; on objects${\displaystyle f^{\ast }}$ is simply the inverse image by the corresponding transport morphism, and on morphisms it is defined in a natural manner by the defininguniversal property of cartesian morphisms. The operation which associates to an object${\displaystyle S}$ of${\displaystyle E}$ the fibre category${\displaystyle F_S}$ and to a morphism${\displaystyle f}$ theinverse image functor${\displaystyle f^{\ast }}$ isalmost a contravariant functor from${\displaystyle E}$ to the category of categories. However, in general it fails to commute strictly with composition of morphisms. Instead, if${\displaystyle f:T\to S}$ and${\displaystyle g:U\to T}$ are morphisms in${\displaystyle E}$, then there is an isomorphism of functors

${\displaystyle c_{f,g}:g^{\ast }{f}^{\ast }\to (f\circ g{)}^{\ast }.}$

These isomorphisms satisfy the following two compatibilities:

1. ${\displaystyle c_{f,{\mathrm{i}\mathrm{d}}_T}=c_{{\mathrm{i}\mathrm{d}}_S,f}={\mathrm{i}\mathrm{d}}_{f^{\ast }}}$
2. for three consecutive morphisms${\displaystyle h,g,f:V\to U\to T\to S}$ and object${\displaystyle x\in F_S}$ the following holds:${\displaystyle c_{f,g\circ h}\cdot c_{g,h}(f^{\ast }(x))=c_{f\circ g,h}(x)\cdot h^{\ast }(c_{f,g}(x)).}$

It can be shown (see Grothendieck (1971) section 8) that, inversely, any collection of functors${\displaystyle f^{\ast }:F_S\to F_T}$ together with isomorphisms${\displaystyle c_{f,g}}$ satisfying the compatibilities above, defines a cloven category. These collections of inverse image functors provide a more intuitive view on fibred categories; and indeed, it was in terms of such compatible inverse image functors that fibred categories were introduced in Grothendieck (1959).

The paper by Gray referred to below makes analogies between these ideas and the notion offibration of spaces.

These ideas simplify in the case ofgroupoids, as shown in the paper of Brown referred to below, which obtains a useful family of exact sequences from a fibration of groupoids.

A (normalised) cleavage such that the composition of two transport morphisms is always a transport morphism is called asplitting, and a fibred category with a splitting is called asplit (fibred)category. In terms of inverse image functors the condition of being a splitting means that the composition of inverse image functors corresponding to composable morphisms${\displaystyle f,g}$ in${\displaystyle E}$equals theinverse image functor corresponding to${\displaystyle f\circ g}$. In other words, the compatibility isomorphisms${\displaystyle c_{f,g}}$ of the previous section are all identities for a split category. Thus split${\displaystyle E}$-categories correspond exactly to true functors from${\displaystyle E}$ to the category of categories.

Unlike cleavages, not all fibred categories admit splittings. For an example, seebelow.

One can invert the direction of arrows in the definitions above to arrive at corresponding concepts of co-cartesian morphisms, co-fibred categories and split co-fibred categories (or co-split categories). More precisely, if${\displaystyle \varphi :F\to E}$ is a functor, then a morphism${\displaystyle m:x\to y}$ in${\displaystyle F}$ is calledco-cartesian if it is cartesian for theopposite functor${\displaystyle {\varphi }^{\text{op}}:F^{\text{op}}\to E^{\text{op}}}$. Then${\displaystyle m}$ is also called adirect image and${\displaystyle y}$ a direct image of${\displaystyle x}$ for${\displaystyle f=\varphi (m)}$. Aco-fibred${\displaystyle E}$-category is an${\displaystyle E}$-category such that direct image exists for each morphism in${\displaystyle E}$ and that the composition of direct images is a direct image. Aco-cleavage and aco-splitting are defined similarly, corresponding todirect image functors instead of inverse image functors.

The categories fibred over a fixed category${\displaystyle E}$ form a 2-category${\displaystyle \mathbf{F}\mathbf{i}\mathbf{b}(E)}$, where thecategory of morphisms between two fibred categories${\displaystyle F}$ and${\displaystyle G}$ is defined to be the category${\displaystyle {\text{Cart}}_E(F,G)}$ of cartesian functors from${\displaystyle F}$ to${\displaystyle G}$.

Similarly the split categories over${\displaystyle E}$ form a 2-category${\displaystyle \mathbf{S}\mathbf{c}\mathbf{i}\mathbf{n}(E)}$ (from Frenchcatégorie scindée), where the category of morphisms between two split categories${\displaystyle F}$ and${\displaystyle G}$ is the full sub-category${\displaystyle {\text{Scin}}_E(F,G)}$ of${\displaystyle E}$-functors from${\displaystyle F}$ to${\displaystyle G}$ consisting of those functors that transform each transport morphism of${\displaystyle F}$ into a transport morphism of${\displaystyle G}$. Each suchmorphism of split${\displaystyle E}$-categories is also a morphism of${\displaystyle E}$-fibred categories, i.e.,${\displaystyle {\text{Scin}}_E(F,G)\subset {\text{Cart}}_E(F,G)}$.

There is a natural forgetful 2-functor${\displaystyle i:\mathbf{S}\mathbf{c}\mathbf{i}\mathbf{n}(E)\to \mathbf{F}\mathbf{i}\mathbf{b}(E)}$ that simply forgets the splitting.

While not all fibred categories admit a splitting, each fibred category is in factequivalent to a split category. Indeed, there are two canonical ways to construct an equivalent split category for a given fibred category${\displaystyle F}$ over${\displaystyle E}$. More precisely, the forgetful 2-functor${\displaystyle i:\mathbf{S}\mathbf{c}\mathbf{i}\mathbf{n}(E)\to \mathbf{F}\mathbf{i}\mathbf{b}(E)}$ admits a right 2-adjoint${\displaystyle S}$ and a left 2-adjoint${\displaystyle L}$ (Theorems 2.4.2 and 2.4.4 of Giraud 1971), and${\displaystyle S(F)}$ and${\displaystyle L(F)}$ are the two associated split categories. The adjunction functors${\displaystyle S(F)\to F}$ and${\displaystyle F\to L(F)}$ are both cartesian and equivalences (ibid.). However, while their composition${\displaystyle S(F)\to L(F)}$ is an equivalence (of categories, and indeed of fibred categories), it isnot in general a morphism of split categories. Thus the two constructions differ in general. The two preceding constructions of split categories are used in a critical way in the construction of thestack associated to a fibred category (and in particular stack associated to apre-stack).

There is a related construction to fibered categories called categories fibered in groupoids. These are fibered categories${\displaystyle p:\mathcal{F}\to \mathcal{C}}$ such that any subcategory of${\displaystyle \mathcal{F}}$ given by

1. Fix an object${\displaystyle c\in \text{Ob}(\mathcal{C})}$
2. The objects of the subcategory are${\displaystyle x\in \text{Ob}(\mathcal{F})}$ where${\displaystyle p(x)=c}$
3. The arrows are given by${\displaystyle f:x\to y}$ such that${\displaystyle p(f)={\text{id}}_c}$

is a groupoid denoted${\displaystyle {\mathcal{F}}_c}$. The associated 2-functors from the Grothendieck construction are examples ofstacks. In short, the associated functor${\displaystyle F_p:{\mathcal{C}}^{op}\to \text{Groupoids}}$ sends an object${\displaystyle c}$ to the category${\displaystyle {\mathcal{F}}_c}$, and a morphism${\displaystyle d\to c}$ induces a functor from the fibered category structure. Namely, for an object${\displaystyle x\in \text{Ob}({\mathcal{F}}_c)}$ considered as an object of${\displaystyle \mathcal{F}}$, there is an object${\displaystyle y\in \text{Ob}(\mathcal{F})}$ where${\displaystyle p(y)=d}$. This association gives a functor${\displaystyle {\mathcal{F}}_c\to {\mathcal{F}}_d}$ which is a functor of groupoids.

1. The functor${\displaystyle \text{Ob}:\mathbf{\text{Cat}}\to \mathbf{\text{Set}}}$, sending a category to its set of objects, is a fibration. For a set${\displaystyle S}$, the fiber consists of categories${\displaystyle C}$ with${\displaystyle \text{Ob}(C)=S}$. The cartesian arrows are the fully faithful functors.
2. Categories of arrows: For any category${\displaystyle E}$ thecategory of arrows${\displaystyle A(E)}$ in${\displaystyle E}$ has as objects the morphisms in${\displaystyle E}$, and as morphisms the commutative squares in${\displaystyle E}$ (more precisely, a morphism from${\displaystyle f:X\to T}$ to${\displaystyle g:Y\to S}$ consists of morphisms${\displaystyle a:X\to Y}$ and${\displaystyle b:T\to S}$ such that${\displaystyle bf=ga}$). The functor which takes an arrow to its target makes${\displaystyle A(E)}$ into an${\displaystyle E}$-category; for an object${\displaystyle S}$ of${\displaystyle E}$ the fibre${\displaystyle E_S}$ is the category${\displaystyle E_{/S}}$ of${\displaystyle S}$-objects in${\displaystyle E}$, i.e., arrows in${\displaystyle E}$ with target${\displaystyle S}$. Cartesian morphisms in${\displaystyle A(E)}$ are precisely thecartesian squares in${\displaystyle E}$, and thus${\displaystyle A(E)}$ is fibred over${\displaystyle E}$ precisely whenfibre products exist in${\displaystyle E}$.
3. Fibre bundles: Fibre products exist in the category${\displaystyle \text{Top}}$ oftopological spaces and thus by the previous example${\displaystyle A(\text{Top})}$ is fibred over${\displaystyle \text{Top}}$. If${\displaystyle \text{Fib}}$ is the full subcategory of${\displaystyle A(\text{Top})}$ consisting of arrows that are projection maps offibre bundles, then${\displaystyle {\text{Fib}}_S}$ is the category of fibre bundles on${\displaystyle S}$ and${\displaystyle \text{Fib}}$ is fibred over${\displaystyle \text{Top}}$. A choice of a cleavage amounts to a choice of ordinary inverse image (orpull-back) functors for fibre bundles.
4. Vector bundles: In a manner similar to the previous examples the projections${\displaystyle p:V\to S}$ of real (complex)vector bundles to their base spaces form a category${\displaystyle {\text{Vect}}_{\mathbb{R}}}$ (${\displaystyle {\text{Vect}}_{\mathbb{C}}}$) over${\displaystyle \text{Top}}$ (morphisms of vector bundles respecting thevector space structure of the fibres). This${\displaystyle \text{Top}}$-category is also fibred, and the inverse image functors are the ordinarypull-back functors for vector bundles. These fibred categories are (non-full) subcategories of${\displaystyle \text{Fib}}$.
5. Sheaves on topological spaces: The inverse image functors ofsheaves make the categories${\displaystyle \text{Sh}(S)}$ of sheaves on topological spaces${\displaystyle S}$ into a (cleaved) fibred category${\displaystyle \text{Sh}}$ over${\displaystyle \text{Top}}$. This fibred category can be described as the full sub-category of${\displaystyle A(\text{Top})}$ consisting ofétalé spaces of sheaves. As with vector bundles, the sheaves ofgroups andrings also form fibred categories of${\displaystyle \text{Top}}$.
6. Sheaves on topoi: If${\displaystyle E}$ is atopos and${\displaystyle S}$ is an object in${\displaystyle E}$, the category${\displaystyle E_S}$ of${\displaystyle S}$-objects is also a topos, interpreted as the category of sheaves on${\displaystyle S}$. If${\displaystyle f:T\to S}$ is a morphism in${\displaystyle E}$, the inverse image functor${\displaystyle f^{\ast }}$ can be described as follows: for a sheaf${\displaystyle F}$ on${\displaystyle E_S}$ and an object${\displaystyle p:U\to T}$ in${\displaystyle E_T}$ one has${\displaystyle f^{\ast }F(U)={\text{Hom}}_T(U,f^{\ast }F)}$ equals${\displaystyle {\text{Hom}}_S(f\circ p,F)=F(U)}$. These inverse image make the categories${\displaystyle E_S}$ into asplit fibred category on${\displaystyle E}$. This can be applied in particular to the "large" topos${\displaystyle TOP}$ of topological spaces.
7. Quasi-coherent sheaves on schemes:Quasi-coherent sheaves form a fibred category over the category ofschemes. This is one of the motivating examples for the definition of fibred categories.
8. Fibred category admitting no splitting: A group${\displaystyle G}$ can be considered as a category with one object and the elements of${\displaystyle G}$ as the morphisms, composition of morphisms being given by the group law. A grouphomomorphism${\displaystyle f:G\to H}$ can then be considered as a functor, which makes${\displaystyle G}$ into a${\displaystyle H}$-category. It can be checked that in this set-up all morphisms in${\displaystyle G}$ are cartesian; hence${\displaystyle G}$ is fibred over${\displaystyle H}$ precisely when${\displaystyle f}$ is surjective. A splitting in this setup is a (set-theoretic)section of${\displaystyle f}$ which commutes strictly with composition, or in other words a section of${\displaystyle f}$ which is also a homomorphism. But as is well known ingroup theory, this is not always possible (one can take the projection in a non-splitgroup extension).
9. Co-fibred category of sheaves: Thedirect image functor of sheaves makes the categories of sheaves on topological spaces into a co-fibred category. The transitivity of the direct image shows that this is even naturally co-split.

One of the main examples of categories fibered in groupoids comes fromgroupoid objects internal to a category${\displaystyle \mathcal{C}}$. So given a groupoid object

${\displaystyle x\stackrel{s}{\underset{t}{\rightrightarrows }}y}$

there is an associated groupoid object

${\displaystyle h_x\stackrel{s}{\underset{t}{\rightrightarrows }}{h}_y}$

in the category of contravariant functors${\displaystyle \underset{\_}{\text{Hom}}({\mathcal{C}}^{op},\text{Sets})}$ from theyoneda embedding. Since this diagram applied to an object${\displaystyle z\in \text{Ob}(\mathcal{C})}$ gives a groupoid internal to sets

${\displaystyle h_x(z)\stackrel{s}{\underset{t}{\rightrightarrows }}{h}_y(z)}$

there is an associated small groupoid${\displaystyle \mathcal{G}}$. This gives a contravariant 2-functor${\displaystyle F:{\mathcal{C}}^{op}\to \text{Groupoids}}$, and using theGrothendieck construction, this gives a category fibered in groupoids over${\displaystyle \mathcal{C}}$. Note the fiber category over an object is just the associated groupoid from the original groupoid in sets.

Given a group object${\displaystyle G}$ acting on an object${\displaystyle X}$ from${\displaystyle a:G\to \text{Aut}(X)}$, there is an associated groupoid object

${\displaystyle G\times X\underset{t}{\stackrel{s}{\rightrightarrows }}X}$

where${\displaystyle s:G\times X\to X}$ is the projection on${\displaystyle X}$ and${\displaystyle t:G\times X\to X}$ is the composition map${\displaystyle G\times X\stackrel{\left(a,\text{id}\right)}{\to }\text{Aut}(X)\times X\stackrel{(f,x)\mapsto f(x)}{\to }X}$. This groupoid gives an induced category fibered in groupoids denoted${\displaystyle p:[X/G]\to \mathcal{C}}$.

For anabelian category${\displaystyle \mathcal{A}}$ any two-termcomplex

${\displaystyle {\mathcal{E}}_1\stackrel{d}{\to }{\mathcal{E}}_0}$

has an associated groupoid

${\displaystyle s,t:{\mathcal{E}}_1\oplus {\mathcal{E}}_0\rightrightarrows {\mathcal{E}}_0}$

where

this groupoid can then be used to construct a category fibered in groupoids. One notable example of this is in the study of thecotangent complex for local-complete intersections and in the study ofexalcomm.

• Grothendieck construction
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• SGA 1.VI - Fibered categories and descent - pages 119-153
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