Page 31 - Textos de Matemática Vol. 46
P. 31
COLIMITS OF MONADS 21
monic in A. If (B,b) lies in C, then so does (A,a):
fA
,,22 TjA a // A Tim Tjm m
✏✏ fB ,,✏✏ //✏✏ TiB 22TjB b B
gB
Since the forgetful functor UTj creates limits, the category ATj is com- plete and wellpowered. Let us prove that it is also cowellpowered. Given a factorization of a homomorphism h : (A, a) ! (B, b) in ATj as a strong epimorphism e : C ! B followed by a monomorphism m : C ! B in A, the diagonal fill-in makes e and m homomorphisms:
TjA a //A
Tje e
✏✏✏✏ c // ✏✏✏✏ TjC C
Tjm m ✏✏ // ✏✏
TjBb B
Thus, if h is a strong epimorphism in ATj then m is an isomorphism (recall that UTj creates limits, thus, reflects isomorphisms), consequently, h is an epimorphism in A. Since A is cowellpowered (see Remark 3.2) we conclude that ATj is cowellpowered.
(b) Every full subcategory of ATj closed under products and subobjects is reflective, see [4], 16.9. Thus, the obvious forgetful functor U : C ! A has a left adjoint.
The theorem now follows from Remark 3.8 and the fact that there exists an isomorphism E of categories such that the triangle
C E //C D
UD U A
TiA
gA
✏✏
