By Donald Knutson

ISBN-10: 3540054960

ISBN-13: 9783540054962

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59 Relations ........................... 72 and D e s c e n t (Grothendieck) of a c a t e g o r y covering C = Cat the Theory ToDoloqv T and in C a t ~ on a c a t e - a set C o v ~ called r a n g e U of the m a p s ~ of coverings ~i is fixed) satisfying i) If ~ is an i s o m o r p h i s m 2) If {U i ~ U} each i then position 3) If 6 e Cov the then T and family {~] £ Cov ~. [Vij + U I ] c C o v {Vij ~ U} obtained T for by com- is in C o y T. [U i + U] 29 58 {Ui ----~ U ] i E I of m a p s in e a c h Theory ............

X.. ~ X appearing 13 13 set associated to 6, the map T h e n ~ is e f f e c t i v e (and hence the map { (X) ~ X is in D). ). 4 and c o n s i d e r for a given c a t e g o r y C. fiber products. First definition. 15: effectively A family {U i ~ U}iEI epimorphic objects W of C, and m a p s W + U, family of maps of C is (UEEF) if for all and for all o b j e c t s V e C, the following d i a g r a m of sets is exact: HOmc(W,V) ~--~Hom (W X Ui,V) i C U ~ ~-~HOmc(W. × z,3 U A single map f:V ~ U in C is a u n i v e r s a l (UEE) family if t h e [f] is a UEEF.

Effective (U i × Uj),V) U epimorphism I. 16: Let B be a closed on C, ~B (also called subcategory the t o p o l o g y of C. on C asso- to B) has Cat T B = C C o v rB = A l l families and in w h i c h (One can check that {Ui-----~U } which each m a p ~i this d e f i n i t i o n are U E E F is in B. i and the axiom A0). 17: To get an i n t e r e s t i n g on B m u s t be satisfied. l - - w r i t t e n if X is the c a t e g o r i c a l disjoint definition, exactly one element. [Xi}i¢ I of o b j e c t s X.

### Algebraic Spaces by Donald Knutson

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