3.1. Reduced Objects Consider an analytic category A.  Definition 3.1.1. (a) An object is reduced if any unipotent map to it is epic.  (b) An analytic category is reduced if any unipotent map is epic (i.e. any object is reduced).  (c) An analytic category is reducible if any non-initial object has a non-initial reduced subobject.  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).  (b) A unipotent strong subobject contains any reduced subobject.  (c) A reduced unipotent strong subobject is the largest reduced and the smallest unipotent strong subobject (therefore is unique).  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 epi-strong-mono 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.  (b) If V is a unipotent strong subobject of X and U a reduced subobject, then by (1.1.2) and (2.2.3) V U is a unipotent strong subobject of the reduced object U, thus V U = U by (a), i.e. V U.  (c) follows from (b). n  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.  (b) If f: Y --> X is a map and U is a reduced strong subobject of Y then its strong image f+1(U) in X is reduced.  (c) Any reduced subobject is contained in a reduced strong subobject.  (d) The join of a set of reduced strong subobjects is reduced.  (e) Any analytic subobject of a reduced object is reduced.  (f) An analytic category is reducible iff the class of reduced objects is uni-dense.  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).  (b) The induced map U --> f+1(U) is epic, thus f+1(U) is reduced if U is reduced by (b).  (c) The strong image of any reduced subobject is a reduced strong subobject by (b).  (d) Suppose V is the join of a set {Vi} of reduced strong subobjects of an object X. Then each Vi is also a strong reduced subobject of V. Any unipotent strong subobject W of V contains each reduced subobject Vi by (3.1.2.b), so it contains their join V, hence V = W, and V is reduced by (3.1.2.a).  (e) Suppose uc: Uc --> X is an analytic subobject of a reduced object X, which is the complement of a strong subobject u: U --> X. Consider any unipotent strong subobject v: V --> Uc of Uc. Since uc is coflat, by (1.5.3) there is a strong subobject w: W --> X such that W Uc = V. Since {Uc, U} is a unipotent cover of X and v: V --> Uc is unipotent, {V, U} is a unipotent cover of X. But V W, {W, U} is also a unipotent cover of X. Thus W U is a unipotent strong subobject of X, and X = W U by (3.1.2.a) as X is reduced. Since uc: Uc --> X is coflat, we have  Uc = Uc X = Uc (W U) = (Uc W) (Uc U) = Uc W = V by (1.5.3). This shows that Uc is reduced.  (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 non-isomorphic epic (because epic strong mono is isomorphic), thus not non-isomorphism 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).  (b) The reduced model of an object X is the largest reduced (strong) subobject of X, denoted by red(X).  (c) A reduced unipotent strong subobject of X is called the radical of X, denoted by rad(X).  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.  (b) If any object in A has the reduced model then the full subcategory of reduced subobjects is a coreflective subcategory of A.  (c) The radical an object X is the reduced model of 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.  (b) follows from (a) and (c) from (3.1.2.c).   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 non-initial object X. Consider a non-initial map t: T --> X. Since the category is reducible, there is a non-initial reduced subobject v: V --> T. Then tv: V --> X is a non-initial 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.  (a) If V is a unipotent strong subobject of U, then U = V (cf. 2.1).  (b) If U has a radical rad(U) then U = rad(V).  (c) If U and V are reduced then U = V iff U = V.  (d) Suppose U and V have radicals rad(U) and rad(V) respectively. Then U = V iff rad(U) = rad(V) as strong subobjects of 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.  (b) follows from (a).  (c) Assume that U is not smaller than V. Then U V is a proper strong subobject of U. Since U is reduced, U V is not a unipotent subobject of U. There is a non-initial map t to U which is disjoint with U V. The composite of t with the mono U --> X is disjoint with V, but not with U, so u  v as desired.  (d) If rad(U) = rad(V) then by (b) we have U = rad(U) = rad(V) = V. Conversely assume U = V. Then rad(U) = U = V = rad(V) by (b). But rad(U) and rad(V) are reduced, so rad(U) = rad(V) by (c).   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 {vi: Vi --> X} of disjunctable strong subobjects. Then  {(Vi)c} = V.  Proof. Since V {(Vi)c}, we have V =   V   {(Vi)c}. To see the other direction, it suffices to prove that for any non-initial map t: T --> X in V, there is a non-initial map to X which factors through t and some analytic mono (ui)c. Replacing T by a non-initial reduced subobject if necessary we may assume that T is reduced. Since v is the intersection of {vi}, t does not factors through at least one such vi. It follows that the pullback z of vi along t is a proper strong subobject of T. Since T is reduced, z is not unipotent. There is a non-initial map u to T which is disjoint with z. Then t°u is disjoint with vi. Thus tu factors through (vi)c, and also through t.     [Next Section][Content][References][Notations][Home] 