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In mathematics, given a
category 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) *C ...
''C'', a quotient of an
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...
''X'' by an equivalence relation f: R \to X \times X is a
coequalizer In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition A coequalizer is a ...
for the pair of maps :R \ \overset\ X \times X \ \overset\ X,\ \ i = 1,2, where ''R'' is an object in ''C'' and "''f'' is an equivalence relation" means that, for any object ''T'' in ''C'', the image (which is a set) of f: R(T) = \operatorname(T, R) \to X(T) \times X(T) is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
; that is, a reflexive,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
and transitive relation. The basic case in practice is when ''C'' is the category of all schemes over some scheme ''S''. But the notion is flexible and one can also take ''C'' to be the category of sheaves.


Examples

*Let ''X'' be a set and consider some equivalence relation on it. Let ''Q'' be the set of all
equivalence classes In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
in ''X''. Then the map q: X \to Q that sends an element ''x'' to the equivalence class to which ''x'' belongs is a quotient. *In the above example, ''Q'' is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is p ...
''H'' of ''X''. 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 ...
, one might replace ''H'' by a
Hilbert scheme In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is ...
or disjoint union of Hilbert schemes. In fact, Grothendieck constructed a relative
Picard scheme In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
of a flat projective scheme ''X''One also needs to assume the geometric fibers are integral schemes; Mumford's example shows the "integral" cannot be omitted. as a quotient ''Q'' (of the scheme ''Z'' parametrizing
relative effective divisor In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumfo ...
s on ''X'') that is a closed scheme of a Hilbert scheme ''H''. The quotient map q: Z \to Q can then be thought of as a relative version of the
Abel map Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd wh ...
.


See also

*
Categorical quotient In algebraic geometry, given a category ''C'', a categorical quotient of an object ''X'' with action of a group ''G'' is a morphism \pi: X \to Y that :(i) is invariant; i.e., \pi \circ \sigma = \pi \circ p_2 where \sigma: G \times X \to X is the ...
, a special case


Notes

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References

*Nitsure, N. ''Construction of Hilbert and Quot schemes.'' Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, American Mathematical Society 2005, 105–137. Binary relations Scheme theory