Page 279 - Textos de Matemática Vol. 46

ON THE LOCALNESS OF THE EMBEDDING OF ALGEBRAS 269
Thus, EY is contained in X; let i : EY ! X be the inclusion map. And let a : X ! Y be the morphism such that sa = ⌘X. Then we have that UFj·UFi·nY ·a = UFdY ·nY ·a = UFj·s·a = UFj·⌘X. Since j is a monomorphism, so is UFj, and, thus, UFi·nY ·a = ⌘X. By (b) of Lemma 3.1, the square part of the commutative diagram
X
t nYa
!!E ))// UFE
idX Y ⌘EY ⇠⇠ ✏✏
Y
i UFi
// ✏✏
X ⌘X UFX
isapullback.Hence,thereisamapt:X!EY withit=idX.Sinceiisan
inclusion, we conclude that X = EY . Theorem 3.3. Let
N
C
sumptions of Section 2. Moreover, assume that:
(H0) U is nontrivial, preserves directed colimits, and has an intersection preserving left adjoint F;
(H1) U creates U-separated epimorphisms;
(H2) U0 locally detects B-morphisms;
(H3) A contains all objects in C with local A-behaviour.
Then, in B, an object B is a subobject of some object NA with A 2 A, whenever every finitely generated subobject of B is so.
Proof. Let B 2 B be such that every finitely generated subobject of B is a subobject of some object NA with A 2 A.
1. Involving an inverse limit of nonempty finite sets. Let mX :X,!UFU0B, X2F,
A ✏ o
be a commutative diagram of categories and functors satisfying General As-
// ??B
M !!
⇤
L
U0
U // Set