xii INTRODUCTION

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

(X,X,ai,T2,z/,/,,e)

where X and X are objects in C and

ax:X-^X, c r

2

: 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

2

o *, = ri

plus the usual inverse axiom. The datum (X, X , ai , cr2) can then be viewed as a

correspondence.

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

2

, /i, 1/, 6, e)

where pr

2

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

correspondences

( X , G x X , p r

2

, / 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

2

, P

3

,... 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