U = [U ] ~ E A GLq . n U B ~ there are unique morphisms ~0 ~,,o~/~ ~(:) in XU . ~(i~(~)) ~(le(~))"~ • By the c o n t i n u i t y of (8o4) ]PRO0f . F, g0~ is continuous. Lemma. ~gC~I~C~E A z~ . ~nui~nuy~¢,theneo~any x~u~nui~nu v we have the commutative diagram j~(x) ( ~ ~ u~,,O .....

G. Segal, Classifvin ~ spaces and spectral sequences, Instltut des Hautes Etudes Scientifiques, Publications Matbematlques, No. 34 (1968), pp. 105-112. 8. reference for t h i s Recall the class section the notion of objects as follows. is of C and For each pair a disjoint union. If C. • • We w r i t e is the class X,Y E ~ , C = ~ U • of morphisms of Hom (X,Y) i s a s e t J where ~ is C described and Hom(X,Y), The basic axioms are: X,Y,Z E ~ , there is a map (called composition) HOm(Y,Z) (written [I].

AI~Ocu;A~ r) -~ ~1(u;~r-I) 2r-1 ) 1 E _~t C (U;A~ Define I A 1 kC~ ~aS = ~* (~( Ucx N Us 0 U On 42 2 ))" we work with t h r e e connections and the convex combination of these over to d e f i n e by ( U n Vo • Uy) x C2 • ( u n v~ n u~) x ~ 2 - ~ u E C~2(U;~ r ' 2 ) 2 and we use ~aS~ = by ~A2 - -kO~B Y . ,¢n~0~ E K2r(u). Here each s l = (-1) [ ( I + I ) / 2 ] the r e l a t i o n . 10) • = • i = (-D j ~. j=0 where &p-l(j) i s the j t h f a c e o f ~P . The l a s t e q u a l l t y i s by the 32 combinatorial version o f Stoke's Theorem [I, p.

### A book of curves by E. H. Lockwood

by Jason

4.3