4
MICHAEL MAKKAI
Z3. The regular factorization of an arrow of the form Gf in S, with any arrow f in
A , is the surjective/injective factorization.
1. Proposition. Suppose a Zawadowski setup (A,...). Then the coregular factorization in
A is the surjective/injective factorization.
i
Proof. Note, first of all, that Z2 applied to GA GA , gives that e, is surjective;
hence, by Zl, it is an isomorphism (the forgetful functor on A reflects iso's). This means that
the Zawadowski setup presents the category A
o p
as a full reflective subcategory of 5 , among
others. Let's write for the actions of both F and G. Start with
A
B
in A;
construct
A - B JK , (1)
the cokernel pair of f. Apply G:
f-
A B ^ K
we get the kernel pair of f , since G as a right adjoints preserves limits (takes colimits in A
to limits in S). Take the regular factorization in s:
A. B ) K . (2)
Apply F:
**
A
reflection, this i
A -
re precisely, we
*
i
^ S * - " e
**
^B
is essentially the same as
f
have
—B
\ * * Xc
c
u
'
(3)
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