1.7. Disjunctable Objects 

Definition 1.7.1. (a) An analytic category is disjunctable if any strong mono is disjunctable. 
(b) An analytic category is locally disjunctable if any strong mono is an intersection of disjunctable strong monos
(c) A mono is called locally disjunctable if it is an intersection of disjunctable strong monos. 
(d) An object in an analytic category is called disjunctable (resp. locally disjunctable) if its diagonal map is a disjunctable (resp. locally disjunctable) strong mono. 
 
Proposition 1.7.2.  (a) An object X is disjunctable (resp. locally disjunctable) iff the equalizer of any pair of maps to X is a disjunctable regular (resp. locally disjunctable) mono. 
(b) Any subobject of a disjunctable (resp. locally disjunctable) object is disjunctable (resp. locally disjunctable). 
(c) Suppose any map in A has an epi-reg-mono factorization. Then A is disjunctable (resp. locally disjunctable) iff any object is disjunctable (resp. locally disjunctable). 

Proof. (a) The diagonal map is the equalizer of the two projections from X to X. Thus the condition is sufficient. For any pair (r1, r2): T --> X of maps to X, the equalizer of (r1, r2) is the pullback of the diagonal map X: X --> X  X along the map T --> X  X determined by (r1, r2). If X is disjunctable (resp. locally disjunctable) then its pullback is disjunctable (resp. locally disjunctable). Thus the condition is necessary. 
(b) follows from (a). 
(c) Assume any map has an epi-reg-mono factorization. Then by (1.1.5.d) any strong mono in A is regular, thus is the equalizer of a pair of maps. From the proof of (a) we know that any regular mono is the pullback of a diagonal map. If any object is disjunctable (resp. locally disjunctable), then any diagonal is a disjunctable (resp. locally disjunctable) regular mono, then so is any of its pullback. This shows that A is disjunctable (resp. locally disjunctable) if any object is disjunctable (resp. locally disjunctable). The other direction is trivial.   
 
Proposition 1.7.3. (a) Any intersection of locally disjunctable monos is locally disjunctable. 
(b) Any product of locally disjunctable objects is locally disjunctable. 

Proof. (a) is obvious by definition. 
(b) Suppose X is a product of locally disjunctable objects {Xi} with the projections pi: X --> Xi. Consider a pair (r, s): Y --> X of maps to X . The equalizer V of (r, s) is the intersection of the equalizer Vi of (pir, pis): Y --> Xi. Since each Xi is locally disjunctable, by (1.2.2.a) the equalizer Vi is locally disjunctable. Thus by (a) the intersection V of these Vi is locally isjunctable.   

An injection of a sum is simply called a direct mono 

Definition 1.7.4. (a) An analytic category is decidable if any strong mono is a direct mono. 
(b) An analytic category is locally decidable if any strong mono is an intersection of direct strong monos. 
(c) A mono is called locally direct if it is an intersection of direct monos. 
(d) An object in an analytic category is called decidable (resp. locally decidable) if its diagonal map is a direct regular (resp. locally direct) mono. 

Note that (1.7.2) and (1.7.3) also holds for decidable and locally decidable objects. 

Recall that a set  of objects in a category A is called a cogenerating set provided that for a pair of maps (r1, r2): X --> T such that r1  r2 there exists an object   and a map g: T --> R such that gr1  gr2. If  is a small set we say that  is a small cogenerating set. If  consists of a single object T then T is a cogenerator for A
 
Remark 1.7.5. Suppose A is complete. Then a small set  of objects is a cogenerating set iff every object is a subobject of a product of objects in 

Proposition 1.7.6. Suppose  is a set of cogenerators. Then any regular mono to an object X is an intersection of regular monos which are equalizers of pairs of maps from X to objects in 

Proof. Suppose v: V --> X is the equalizer of a pair of maps (r1, r2): X --> T . Consider a map f: Y --> X not factor through V --> X . Since V --> X is the equalizer of (r1, r2) , we have r1 r2f. Since  is a set of cogenerators, we can find an object W in  and a map g: T --> W such that gr1 gr2f . The equalizer z: Z --> X of (gr1, gr2) contains v but not t . This shows that v is an intersection of regular monos which are equalizers of pairs of maps from X to objects in .  
 
Proposition 1.7.7. Suppose an analytic category A has a disjunctable (resp. decidable) cogenerating set. Then 
(a) Any regular mono is locally disjunctable (resp. decidable). 
(b) If A has epi-reg-mono factorizations then it is locally disjunctable (resp. decidable}. 

Proof. (a) follows from (1.7.6) in view of (1.7.2.a); if A has epi-reg-mono factorizations then any strong mono is regular by (1.1.5.d), thus (b) follows from (a).  
 
 

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