Page 34 - Textos de Matemática Vol. 41

20 Chapter 4. The basic structure of morphisms in Loc
For modelling more special maps of the type, namely embeddings and factorizations, one uses special types of monomorphisms and epimorphisms. In this text we will use the (in our categories coinciding) notions of extremal and strong ones.
A monomorphism m (in a general category) is extremal if in every decom- position m = fe with e an epimorphism, e is an isomorphism.
A monomorphism m is said to be strong if for every epimorphism e, each commutative diagram
· e // · (4.3.1) uv
· m //·
· e //·
u w v

· m //·
with mw = v and we = u.
(This relation of the morphisms e and m is often re-
ferred to as orthogonality.)
Dually, an extremal epimorphism e is an epimorphism such that in every decomposition e = mf with m a monomorphism, m is an isomorphism. A strong epimorphism is such epimorphism that for every monomorphism m in (4.3.1) there is a morphism w such that mw = v and we = u:
· e //·
u w v

· m //·
(Thus, the diagrams in the definition of a strong monomorphism and strong epi- morphism are the same. The difference is that in the former case the monomor- phism is given and started with while in the latter one starts with an epimor- phism.)
It is a well-known fact that
can be completed to a diagram