1.4. Coflat Maps 

Definition 1.4.1. (a) A map f: Y ® X is called precoflat if the pullback of any epi to X along f is epic. 
(b) A map f: Y ® X is called coflat if the pullback of f along any map to X is precoflat. 

Equivalently, a map f: Y ® X is coflat if the pullback functor A/X ® A/Y along f preserves epis. 

Proposition 1.4.2.  (a) Coflat maps are stable, i.e. the pullback of a coflat map is coflat. 
(b) Composite of coflat maps are coflat. 
(c) Finite products of coflat maps are coflat. 
(d) A finite sum of maps is coflat iff each factor is coflat. 

Proof. (a) and (b) are immediate. 
(c) Let f1: Y1 ® X1 and f2: Y2 ® X2 be two coflat maps. Consider the following pullback: 

Since f1: Y1 ® X1 and f2: Y2 ® X2 are coflat and the pullbacks of coflat maps are coflat, Y1 × X2 ® X1 × X2 and X1 × Y2 ® X1 × X2 are coflat. It follows that the cartisian diagram 
consists of coflat maps. Thus (f1 × f2): Y1 × Y2 ® X1 × X2 is coflat by (b). 
(d) Consider two maps u1: X1 ® T1 and u2: X2 ® T2 . If u1 + u2 is coflat, then u1 and u2 are coflat by (a) because they are pullbacks of u1 + u2 by (1.3.4.b). Conversely assume u1 and u2 are coflat. Any map from g to T1 + T2 has the form g = v1 + v2: Y1 + Y2 ® T1 + T2 for two maps v1: Y1 ® T1 and v2: Y2 ® T2. Then by (1.3.4.a) the pullback s of g along u1 + u2 is the sum of X1 × T1 Y1 ® X1 and X2 × T2 Y2 ® X2 . If g is epic then v1 and v2 are epic by (1.3.7), thus X1 × T1 Y1 ® X1 and X2 × T2 Y2 ® X2 are epic because u1 and u2 are coflat. It follows that s is epic by (1.3.7). This shows that u1 + u2 is precoflat. Next the pullback t of u1 + u2 along g is the sum of the coflat map X1 × T1 Y1 ® Y1 and the coflat map X2 × T2 Y2 ® Y2. Thus t is also precoflat. It follows that u1 + u2 is coflat. n 

Proposition 1.4.3. (a) Suppose f: Y ® X is a mono and g: Z ® Y is a map. Then g is coflat if f°g is coflat. 
(b) For any object X, the codiagonal map ÑX: X + X ® X is coflat. 
(c) Suppose {fi: Yi ® X} is a finite family of coflat maps. Then f = å (fi): Y = å Yi ® X is coflat. 

Proof. (a) If f°g is coflat then g is coflat as it is the pullback of f°g along f. 
(b) The pullback of ÑX: X + X ® X along a map Z ® X is the map ÑZ: Z + Z ® Z . Therefore in order to show that ÑX is coflat it suffices to prove that ÑX is precoflat. Assume f: Y ® X is an epi. Then the pullback of f along ÑX is f + f: Y + Y ® X + X, which is an epi by (1.3.7). 
(c) For any pair of coflat maps f: Y ® X and g: Z ® X the induced map (f, g): Y + Z ® X is the composite of the coflat maps f + g: Y + Z ® X + X and ÑX: X + X ® X by (b) and (1.4.2.d). Thus (f, g) is coflat. This result extends immediately to a map of the form f = å(fi): Y = å Yi ® X. n 

Recall that a map in a category is a bimorphism if it is both epic and monic. 

Proposition 1.4.4. (a) Coflat monos are stable. 
{b} Composites of coflat monos (bimorphisms) are coflat monos (bimorphisms). 
(c) Finite intersections of coflat monos (bimorphisms) are coflat mono (bimorphisms). 
(d) A finite sum of monos (bimorphisms) is a coflat mono (bimorphisms) iff each factor is a coflat mono (bimorphisms). 
(e) Suppose f: Y ® X is a coflat bimorphisms. If g: Z ® Y is a map such that f°g is an epi, then g is an epi. 
(f) Suppose f: Y ® X is a coflat mono (bimorphisms) and g: Z ® Y is any map. Then g is a coflat mono (bimorphisms) iff f°g is a coflat mono (bimorphisms). 

Proof. (a) - (d) follows from (1.4.2), (1.3.7) and (1.3.8). 
(e) Since f is a mono and f°g factors through f, (g, 1Z) is the pullback of (f, f°g). If f°g is an epi then g is an epi as f is coflat by assumption. 
(f) follows from (e) and (1.4.3.a). n 

Proposition 1.4.5. Injections of a sum are coflat monos. 

Proof. The injection X ® X + Y is the sum of the coflat mono 1X: X ® X and the coflat mono 0 ® Y . Thus by (1.4.4.d) it is coflat. n 
 
 

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