morphism / : X —• X*; cf. Equation 1.4. Categorical quotients of correspondences
of the form X*, or more generally of the form X, should be viewed as (categorical)
spaces of orbits of the "dynamical system defined by s".
Another example of correspondences appearing in nature are groupoids. To
review this we shall tacitly assume that all products intervening (explicitly or im-
plicitly) in the discussion below exist in C A groupoid in C is a tuple
where X and X are objects in C and
ax:X-^X, c r
: X - X , z / i X x ^ x ^ X ^ X , ^ : X - X , e : X - X
are morphisms such that v satisfies the usual associativity axiom, e satisfies o\ oe =
a2 o e = idx plus the usual unit axiom, and 1 satisfies o\ o t = cr2, c
o *, = ri
plus the usual inverse axiom. The datum (X, X , ai , cr2) can then be viewed as a
In their turn groupoids often arise from group actions. Again we tacitly assume
that all products intervening (explicitly or implicitly) in the discussion below exist
in C. Let G be a group object in the category C and /i : G x X » X an action on
an object X in C. Then one can consider the groupoid
(X, G x X,pr
, /i, 1/, 6, e)
where pr
is the second projection, and v, 1, e are naturally induced by the group
operations and the action. So in particular group actions fi : G x X ^ X define
( X , G x X , p r
, / i ) .
Many of the examples of correspondences we shall be interested in will not
come, however, from dynamical systems or groupoids.
0.1.2. Basic pathology. Typically it turns out that, in many classical geo-
metric situations, one has interesting correspondences X = (X, a) = (X, X , cri, cr2)
whose categorical quotient X/a is trivial. This "basic pathology" manifests itself,
for instance, in the case when C is the category of algebraic varieties over an alge-
braically closed field and the trivial objects are the points: if X and X above are
algebraic varieties and the smallest equivalence relation (a) C X x X containing
the image of o\ x a2 ' X X x X is Zariski dense in X x X then the categorical
quotient X/a is trivial. Examples of this kind are very common. For instance
assume that dim X = dim X = 1, and a has an infinite orbit by which we mean
that there exists an infinite sequence of points Pi , P
, P
,... G X and a sequence of
points Pi,P2,p3,... X such that ai(P,) = Pu a2(Pt) = Pz+i for all i 1. Then
(a) C X x X is dense in X x X and hence X/a reduces to a point in the category
of varieties. Note, by the way, that in the example above it is impossible to embed
X into a correspondence X ; = (X ; , X',a[, a'2) in the category of schemes of finite
type such that X ' admits a groupoid structure and dim Xf dim X' = 1; here,
by an embedding we mean a morphism (L,L) : X —• X ' with 1 and I embeddings.
Intuitively, X does not "generate" a groupoid "of finite type" and, in some sense,
this is what prevents usual algebraic geometry from "controlling" the quotient.
With the above "basic pathology" in mind one is tempted to "enlarge" usual
algebraic geometry so as to make X/a non-trivial; one can then pursue the study
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