3.2. Integral Objects

Definition 3.2.1. (a) A non-initial object is primary if any non-initial analytic mono to it is epic. 
(b) A non-initial object is quasi-primary if any two non-initial analytic monos are not disjoint with each other. 
(c) An integral object is a reduced primary object. 
(d) A prime of an object is an integral strong subobject. 
(e) A non-initial object is irreducible if it is not the join of two proper strong subobjects. 

Proposition 3.2.2. (a) Any quotient of a primary object is primary. 
(b) If f: Y ® X is a map and U a primary subobject of Y, then the strong image f+1(U) of f in X is a primary subobject of X. 
(c) Any primary object is quasi-primary. 

Proof. (a) Let f: Y ® X be an epi with Y primary. Consider any non-initial analytic mono t: T ® X. Let (g: W ® Y, h: W ® T) be the pullback of (t, f). Then h is non-initial epic as it is the pullback of an epi along the non-initial  coflat map t, and g as the pullback of an analytic mono is a non-initial analytic mono, so g is epic as Y is primary. It follows that f°g = t°h is epic, and so t is epic. 

(b) follows from (a) 
(c) Suppose U and V are two non-initial analytic subobject of a primary object X. Since U and V are coflat and epic, U Ç V is an epic analytic mono of U and V (cf. (1.6.6.b)), thus is non-initial. n 

Corollary 3.2.3. (a) Any quotient of an integral object is integral. 
(b) If f: Y ® X is a map and U a prime of Y, then f+1(U) is a prime of X. 

Proof. (a) follows from (3.2.2.a) and (3.1.3.a). 
(b) follows from (3.2.2.b) and (3.1.3.b). n 

Proposition 3.2.4. (a) Any non-initial analytic subobject of a primary object is primary. 
(b}Any non-initial analytic subobject of an integral object is integral. 

Proof. (a) Suppose u: U ® X is a non-initial analytic subobject of a primary object X. Consider a non-initial analytic subobject t: T ® U. Since u°t is analytic and X is primary, u°t is epic, so t is epic (as the pullback of the epi u°t along the coflat mono u), thus U is primary as desired. 
(b) follows from (a) and (3.1.3.e). n 

Proposition 3.2.5. Suppose X = U Ú V is the join of two strong subobjects U and V of X
(a) If t: T ® X is a coflat map which is disjoint with V then it factors through U
(b) If U and V are disjunctable then Uc and Vc are disjoint. 

Proof. (a) By (1.5.3) we have 

T = t-1(X) = t-1(U Ú V) = t-1(U) Ú t-1(V) = t-1(U).
So t factors through U
(b) Uc Ç Vc = (U Ú V)c = Xc is initial by (1.6.4). n 

Proposition 3.2.6. Suppose A is locally disjunctable. The following are equivalent for a non-initial reduced object X
(a) Any non-initial coflat map to X is epic. 
(b) X is primary (i.e. X is integral). 
(c) X is quasi-primary. 
(d) X is irreducible. 

Proof. Clearly (a) implies (b), and (b) implies (c) by (3.2.2.c). We prove that (c) implies (d). Assume X is not irreducible. Then X = U Ú V is the join of two proper strong subobjects U and V. Since A is locally disjunctable, we may assume U and V are disjunctable. Since X is reduced, Uc and Vc are non-initial, and they are disjoint by (3.2.5). Thus X is not quasi-primary. Hence (c) implies (d). 
Next we show that (d) implies (b). Assume X is irreducible. Consider a non-initial analytic subobject U, which is the complement of a proper strong object V of X . We prove that U is epic. Consider any strong subobject W of X containing U. Since W Ú V contains U = Vc and V, it is unipotent, thus X = W Ú V as X is reduced. Since V is proper, by (d) we have W = X. This shows that U is an epic subobject by (1.1.3.e). This shows that (d) implies (b). 
Finally we prove that (b) implies (a). Consider a non-initial coflat map f: Y ® X. As A is locally disjunctable, any proper strong subobject is contained in a proper disjunctable strong subobject. To see that f is epic by (1.1.3.e)  it suffices to show that it does not factor through any proper disjunctable strong subobject V of X. Since X is reduced, Vc is a non-initial analytic mono, therefore epic by (b). Since f is non-initial and coflat, f is not disjoint with Vc. Since Vc is disjoint with V, this implies that f: Y ® X does not factor through any proper disjunctable strong subobject V, so it is epic. This shows that (b) implies (a). n 

Corollary 3.2.7. Suppose A is locally disjunctable. 
(a) An object is integral iff it is reduced and quasi-primary. 
(b) An object is integral iff it is reduced and irreducible. 

Proof. By (3.2.6). n 

Later we shall prove in Section 5 that if A is an analytic geometry then a non-initial object X is quasi-primary iff its radical is integral. 
 
 

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