Consider an analytic category A. Definition 3.1.1. (a) An object is reduced
if any unipotent map to it is epic.
Proposition 3.1.2. (a) An object is
reduced iff any unipotent strong mono to it is an isomorphism (i.e. it
has no proper unipotent strong subobject).
Proof. (a) Any unipotent strong mono to a reduced object is epic,
thus must be an isomorphism by (1.1.2.b).
Conversely assume that any unipotent strong mono to an object X
is an isomorphism. To see that X is reduced we have to prove that
any unipotent map f: Y > X is epic. Write (e, m)
for the epistrongmono
factorization of f. Then m: e(Y) > X
is a unipotent strong mono by (2.2.3.d),
thus must be an isomorphism. Therefore f = me is epic as required.
If f: Y > X is an epi we simply say that X is a quotient of Y. Proposition 3.1.3. (a) Any quotient
of a reduced object is reduced.
Proof. (a) Suppose f: Y > X is an epi and Y is
reduced. If U is a unipotent strong subobject of X, f^{1}(U)
is a unipotent strong subobject of the reduced object Y by (1.1.2.c)
and (2.2.3.b), thus f^{1}(U)
= Y by (3.1.2.a). Thus f factors through
the strong mono U > X. But f is epic implies that U
is epic, so U > X as an epic strong mono is an isomorphism,
and X is reduced by (3.1.2.a).
(f) follows from (b). Proposition 3.1.4. An analytic category is reduced iff any strong mono is normal. Proof. A mono is normal (resp. strong) iff any of its pullback is not proper unipotent (resp. epic). If f is a strong mono in a reduced category, any of its pullback is not nonisomorphic epic (because epic strong mono is isomorphic), thus not nonisomorphism unipotent (because any unipotent map is epic in a reduced category), so f is normal. Conversely, suppose any strong mono is normal. Then any unipotent strong mono f: U > X is normal, thus is an isomorphism as any normal unipotent mono is an isomorphism. By (3.1.2.a) X is reduced. Definition 3.1.5. (a) An analytic category
is perfect
if any intersection of strong monos exists (or equivalent, the lattice
of strong subobjects of any object is complete).
Proposition 3.1.6. Any object in a perfect analytic category has a reduced model. Proof. The join of all the reduced strong subobjects of X is the largest reduced strong subobject by (3.1.3.d), thus is the reduced model of X. Proposition 3.1.7. (a) If red(X)
is the reduced model of X then any map from a reduced object to
X factors uniquely through the mono red(X) >
X.
Proof. (a) If f: Y > X is a map with a reduced
domain Y, then its strong image f^{+1}(Y)
in X is a reduced strong subobject of X by (3.1.3.b),
so f^{+1}(Y)
red(X) because by definition red(X) is the maximal
reduced subobject. It follows that f factors through the mono red(X)
> X.
Proposition 3.1.8. The reduced model of any object in a reducible analytic category is unipotent (thus is the radical). Proof. The assertion is trivial for an initial object. Suppose red(X) is the reduced model of a noninitial object X. Consider a noninitial map t: T > X. Since the category is reducible, there is a noninitial reduced subobject v: V > T. Then tv: V > X is a noninitial map, which factors through red(X) > X by (3.1.7.a) as V is reduced. Thus t is not disjoint with red(X), and red(X) is unipotent. Proposition 3.1.9. Suppose U
and V are two strong subobjects of an object X.
Proof. (a) Consider a map t: Z > X which is disjoint
with V. The pullback of t along the inclusion U > X
is disjoint with V > U, thus is an initial map as V is
a unipotent strong subobject of U, so t is disjoint with
U.
Proposition 3.1.10. Suppose X is an object in a reducible analytic category. Suppose v: V > X is a strong subobject which is an intersection of a set {v_{i}: V_{i} > X} of disjunctable strong subobjects. Then {(V_{i})^{c}} = V. Proof. Since V
{(V_{i})^{c}}, we have V
= V {(V_{i})^{c}}.
To see the other direction, it suffices to prove that for any noninitial
map t: T > X in V,
there is a noninitial map to X which factors through t and
some analytic mono (u_{i})^{c}. Replacing
T by a noninitial reduced subobject if necessary we may assume
that T is reduced. Since v is the intersection of {v_{i}},
t does not factors through at least one such v_{i}.
It follows that the pullback z of v_{i} along t
is a proper strong subobject of T. Since T is reduced, z
is not unipotent. There is a noninitial map u to T which
is disjoint with z. Then t_{°}u
is disjoint with v_{i}. Thus tu factors through (v_{i})^{c},
and also through t.
