Fibred categories (or fibered categories) are abstract entities in
mathematics used to provide a general framework for
descent theory. They formalise the various situations in
geometry and
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
in which ''inverse images'' (or ''pull-backs'') of objects such as
vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
from a topological space ''X'' to another topological space ''Y'' is associated the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
functor taking bundles on ''Y'' to bundles on ''X''. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, which is the context in which fibred categories originally appeared. Fibered categories are used to define
stack
Stack may refer to:
Places
* Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group
* Blue Stack Mountains, in Co. Donegal, Ireland
People
* Stack (surname) (including a list of people ...
s, which are fibered categories (over a site) with "descent". Fibrations also play an important role in categorical semantics of
type theory, and in particular that of
dependent type
In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifier ...
theories.
Fibred categories were introduced by , and developed in more detail by .
Background and motivations
There are many examples in
topology and
geometry where some types of objects are considered to exist ''on'' or ''above'' or ''over'' some underlying ''base space''. The classical examples include vector bundles,
principal bundles, and
sheaves over topological spaces. Another example is given by "families" of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
parametrised by another variety. Typical to these situations is that to a suitable type of a
map between base spaces, there is a corresponding ''inverse image'' (also called ''pull-back'') operation
taking the considered objects defined on
to the same type of objects on
. This is indeed the case in the examples above: for example, the inverse image of a vector bundle
on
is a vector bundle
on
.
Moreover, it is often the case that the considered "objects on a base space" form a category, or in other words have maps (
morphisms
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, 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 a
functor. Again, this is the case in examples listed above.
However, it is often the case that if
is another map, the inverse image functors are not ''strictly'' compatible with composed maps: if
is an object ''over''
(a vector bundle, say), it may well be that
:
Instead, these inverse images are only
naturally isomorphic. 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 in
descent 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.
Formal definitions
There are two essentially equivalent technical definitions of fibred categories, both of which will be described below. All discussion in this section ignores the
set-theoretical issues related to "large" categories. The discussion can be made completely rigorous by, for example, restricting attention to small categories or by using
universes.
Cartesian morphisms and functors
If
is a
functor between two
categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
*Categories (Peirce)
* ...
and
is an object of
, then the
subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
of
consisting of those objects
for which
and those morphisms
satisfying
, is called the ''fibre category'' (or ''fibre'') ''over''
, and is denoted
. The morphisms of
are called
''-morphisms'', and for
objects of
, the set of
-morphisms is denoted by
. The image by
of an object or a morphism in
is called its ''projection'' (by
). If
is a morphism of
, then those morphisms of
that project to
are called
''-morphisms'', and the set of
-morphisms between objects
and
in
is denoted by
.
A morphism
in
is called
''-cartesian'' (or simply ''cartesian'') if it satisfies the following condition:
: if
is the projection of
, and if
is an
-morphism, then there is ''precisely one''
-morphism
such that
.
A cartesian morphism
is called an ''inverse image'' of its projection
; the object
is called an ''inverse image'' of
''by
''.
The cartesian morphisms of a fibre category
are precisely the isomorphisms of
. There can in general be more than one cartesian morphism projecting to a given morphism
, possibly having different sources; thus there can be more than one inverse image of a given object
in
by
. However, it is a direct consequence of the definition that two such inverse images are isomorphic in
.
A functor
is also called an
''-category'', or said to make
into an
-category or a category ''over''
. An
-functor from an
-category
to an
-category
is a functor
such that
.
-categories form in a natural manner a
2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of catego ...
, with 1-morphisms being
-functors, and 2-morphisms being natural transformations between
-functors whose components lie in some fibre.
An
-functor between two
-categories is called a ''cartesian functor'' if it takes cartesian morphisms to cartesian morphisms. Cartesian functors between two
-categories
form a category
, with
natural transformations as morphisms. A special case is provided by considering
as an
-category via the identity functor: then a cartesian functor from
to an
-category
is called a ''cartesian section''. Thus a cartesian section consists of a choice of one object
in
for each object
in
, and for each morphism
a choice of an inverse image
. A cartesian section is thus a (strictly) compatible system of inverse images over objects of
. The category of cartesian sections of
is denoted by
:
In the important case where
has a
terminal object (thus in particular when
is a
topos or the category
of
arrows with target
in
) the functor
:
is
fully faithful (Lemma 5.7 of Giraud (1964)).
Fibred categories and cloven categories
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 of ''
cleavages'', 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
category
is a ''fibred category'' (or a ''fibred
-category'', or a ''category fibred over
'') if each morphism
of
whose codomain is in the range of projection has at least one inverse image, and moreover the composition
of any two cartesian morphisms
in
is always cartesian. In other words, an
-category is a fibred category if inverse images always exist (for morphisms whose codomains are in the range of projection) and are ''transitive''.
If
has a terminal object
and if
is fibred over
, then the functor
from cartesian sections to
defined at the end of the previous section is an
equivalence of categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fr ...
and moreover
surjective on objects.
If
is a fibred
-category, it is always possible, for each morphism
in
and each object
in
, to choose (by using the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
) precisely one inverse image
. The class of morphisms thus selected is called a ''cleavage'' and the selected morphisms are called the ''transport morphisms'' (of the cleavage). A fibred category together with a cleavage is called a ''cloven category''. A cleavage is called ''normalised'' if the transport morphisms include all identities in
; 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
-category
specifies, for each morphism
in
, a ''functor''
; on objects
is simply the inverse image by the corresponding transport morphism, and on morphisms it is defined in a natural manner by the defining universal property of cartesian morphisms. The operation which associates to an object
of
the fibre category
and to a morphism
the ''inverse image functor''
is ''almost'' a contravariant functor from
to the category of categories. However, in general it fails to commute strictly with composition of morphisms. Instead, if
and
are morphisms in
, then there is an isomorphism of functors
:
These isomorphisms satisfy the following two compatibilities:
#
# for three consecutive morphisms
and object
the following holds:
It can be shown (see Grothendieck (1971) section 8) that, inversely, any collection of functors
together with isomorphisms
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 of
fibration of spaces.
These ideas simplify in the case of
groupoids
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*'' Group'' with a partial funct ...
, as shown in the paper of Brown referred to below, which obtains a useful family of exact sequences from a fibration of groupoids.
Splittings and split fibred categories
A (normalised) cleavage such that the composition of two transport morphisms is always a transport morphism is called a ''splitting'', and a fibred category with a splitting is called a ''split'' (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
in
''equals'' the inverse image functor corresponding to
. In other words, the compatibility isomorphisms
of the previous section are all identities for a split category. Thus split
-categories correspond exactly to true functors from
to the category of categories.
Unlike cleavages, not all fibred categories admit splittings. For an example, see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
.
Co-cartesian morphisms and co-fibred categories
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
is a functor, then a morphism
in
is called ''co-cartesian'' if it is cartesian for the
opposite functor . Then
is also called a ''direct image'' and
a direct image of
for
. A ''co-fibred''
-category is an
-category such that direct image exists for each morphism in
and that the composition of direct images is a direct image. A ''co-cleavage'' and a ''co-splitting'' are defined similarly, corresponding to ''direct image functors'' instead of inverse image functors.
Properties
The 2-categories of fibred categories and split categories
The categories fibred over a fixed category
form a 2-category
, where the ''category'' of morphisms between two fibred categories
and
is defined to be the category
of cartesian functors from
to
.
Similarly the split categories over
form a 2-category
(from French ''catégorie scindée''), where the category of morphisms between two split categories
and
is the full sub-category
of
-functors from
to
consisting of those functors that transform each transport morphism of
into a transport morphism of
. Each such ''morphism of split
-categories'' is also a morphism of
-fibred categories, i.e.,
.
There is a natural forgetful 2-functor
that simply forgets the splitting.
Existence of equivalent split categories
While not all fibred categories admit a splitting, each fibred category is in fact ''equivalent'' to a split category. Indeed, there are two canonical ways to construct an equivalent split category for a given fibred category
over
. More precisely, the forgetful 2-functor
admits a right 2-adjoint
and a left 2-adjoint
(Theorems 2.4.2 and 2.4.4 of Giraud 1971), and
and
are the two associated split categories. The adjunction functors
and
are both cartesian and equivalences (''ibid''.). However, while their composition
is an equivalence (of categories, and indeed of fibred categories), it is ''not'' 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 the
stack
Stack may refer to:
Places
* Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group
* Blue Stack Mountains, in Co. Donegal, Ireland
People
* Stack (surname) (including a list of people ...
associated to a fibred category (and in particular stack associated to a
pre-stack).
Categories fibered in groupoids
There is a related construction to fibered categories called categories fibered in groupoids. These are fibered categories
such that any subcategory of
given by
# Fix an object
# The objects of the subcategory are
where
# The arrows are given by
such that
is a groupoid denoted
. The associated 2-functors from the Grothendieck construction are examples of
stacks. In short, the associated functor
sends an object
to the category
, and a morphism
induces a functor from the fibered category structure. Namely, for an object
considered as an object of
, there is an object
where
. This association gives a functor
which is a functor of groupoids.
Examples
Fibered categories
#The functor
, sending a category to its set of objects, is a fibration. For a set
, the fiber consists of categories
with
. The cartesian arrows are the fully faithful functors.
#Categories of arrows: For any category
the ''category of arrows''
in
has as objects the morphisms in
, and as morphisms the commutative squares in
(more precisely, a morphism from
to
consists of morphisms
and
such that
). The functor which takes an arrow to its target makes
into an
-category; for an object
of
the fibre
is the category
of
-objects in
, i.e., arrows in
with target
. Cartesian morphisms in
are precisely the
cartesian square
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A ...
s in
, and thus
is fibred over
precisely when
fibre product
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often ...
s exist in
.
#Fibre bundles: Fibre products exist in the category
of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s and thus by the previous example
is fibred over
. If
is the full subcategory of
consisting of arrows that are projection maps of
fibre bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E an ...
s, then
is the category of fibre bundles on
and
is fibred over
. A choice of a cleavage amounts to a choice of ordinary inverse image (or ''pull-back'') functors for fibre bundles.
#Vector bundles: In a manner similar to the previous examples the projections
of real (complex)
vector bundles
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
to their base spaces form a category
(
) over
(morphisms of vector bundles respecting the
vector space structure of the fibres). This
-category is also fibred, and the inverse image functors are the ordinary ''pull-back'' functors for vector bundles. These fibred categories are (non-full) subcategories of
.
#Sheaves on topological spaces: The inverse image functors of
sheaves make the categories
of sheaves on topological spaces
into a (cleaved) fibred category
over
. This fibred category can be described as the full sub-category of
consisting of
étalé spaces of sheaves. As with vector bundles, the sheaves of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
and
rings also form fibred categories of
.
#Sheaves on topoi: If
is a
topos and
is an object in
, the category
of
-objects is also a topos, interpreted as the category of sheaves on
. If
is a morphism in
, the inverse image functor
can be described as follows: for a sheaf
on
and an object
in
one has
equals
. These inverse image make the categories
into a ''split'' fibred category on
. This can be applied in particular to the "large" topos
of topological spaces.
#Quasi-coherent sheaves on schemes:
Quasi-coherent sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
form a fibred category over the category of
schemes. This is one of the motivating examples for the definition of fibred categories.
#Fibred category admitting no splitting: A group
can be considered as a category with one object and the elements of
as the morphisms, composition of morphisms being given by the group law. A group
homomorphism can then be considered as a functor, which makes
into a
-category. It can be checked that in this set-up all morphisms in
are cartesian; hence
is fibred over
precisely when
is surjective. A splitting in this setup is a (set-theoretic)
section of
which commutes strictly with composition, or in other words a section of
which is also a homomorphism. But as is well known in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, this is not always possible (one can take the projection in a non-split
group extension).
#Co-fibred category of sheaves: The
direct image In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topol ...
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.
Category fibered in groupoids
One of the main examples of categories fibered in groupoids comes from
groupoid objects internal to a category
. So given a groupoid object
:
there is an associated groupoid object
:
in the category of contravariant functors
from the
yoneda embedding
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (view ...
. Since this diagram applied to an object
gives a groupoid internal to sets
:
there is an associated small groupoid
. This gives a contravariant 2-functor
, and using the
Grothendieck construction, this gives a category fibered in groupoids over
. Note the fiber category over an object is just the associated groupoid from the original groupoid in sets.
Group quotient
Given a group object
acting on an object
from
, there is an associated groupoid object
:
where
is the projection on
and
is the composition map
. This groupoid gives an induced category fibered in groupoids denoted
.
Two-term chain complex
For an abelian category
any two-term
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
:
has an associated groupoid
:
where
:
this groupoid can then be used to construct a category fibered in groupoids. One notable example of this is in the study of the
cotangent complex In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic ob ...
for local-complete intersections and in the study of
exalcomm.
See also
*
Grothendieck construction
*
Stack (mathematics)
*
Artin's criterion In mathematics, Artin's criteria are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces or as Algebraic stacks. In particular, these cond ...
*
Fibration of simplicial sets In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions \Lambda^n_i \subset \Delta^n, 0 \le i < n. A right ...
References
*
*
*
*
*
*
*
*
*.
*
*
*
*
External links
*
SGA 1.VI - Fibered categories and descent - pages 119-153
*
{{Category theory
Category theory