By Joseph Bowden
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Additional info for Elements of the theory of integers
S is a p a r t i a l b. The p r o j e c t i o n s c. 6) Proposition. X Then s: S >S S × S > S are equal. >S are equal. is an i s o m o r p h i s m . Let the k e r n e l Proposition. object Let ~T where S is a p a r t i a l terminal pair. f: X ~ S be c o n s t a n t . T h e n S is a p a r t i a l and S = supp X. As any c o n s t a n t ~ supp X f: S f is an i s o m o r p h i s m . Consider terminal Proof. pl,P2: >S Trivial. object. Proof. object. pl,P2 ~ S × S projections d- T h e d i a g o n a l Proof.
W i l l be c a l l e d a partial terminal object if e v e r y m a p to it is c o n s t a n t . ion. Let S be an o b j e c t of X. Then the following are equivalent. a. S is a p a r t i a l b. The p r o j e c t i o n s c. 6) Proposition. X Then s: S >S S × S > S are equal. >S are equal. is an i s o m o r p h i s m . Let the k e r n e l Proposition. object Let ~T where S is a p a r t i a l terminal pair. f: X ~ S be c o n s t a n t . T h e n S is a p a r t i a l and S = supp X. As any c o n s t a n t ~ supp X f: S f is an i s o m o r p h i s m .
T h e second factor factor supp X , is e x a c t l y a c a r t e s i a n 51 map. 15) Proposition. Let S be a full subcategory of supp X. Then the full subcategory of X consisting of those objects whose support lies in S is regular Proof. Trivial. (and exact when X is). 52 3. 2) the If say that of colimit. means (D,X) is r e p r e s e n t e d )X. We m a y denote In the special there >X and ~: k D~ I commutes. When generated by that relation. as lim(D,Ej), the ---~Ej present (I,D), such that in (D,EJ2).