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 f_{1}: Y_{1} > X_{1}
and f_{2}: Y_{2} > X_{2} be two
coflat maps. Consider the following pullback:
Since f_{1}: Y_{1} > X_{1} and
f_{2}: Y_{2} > X_{2} are coflat
and the pullbacks of coflat maps are coflat, Y_{1}
X_{2} > X_{1}
X_{2} and X_{1}
Y_{2} > X_{1}
X_{2} are coflat. It follows that the cartisian diagram
consists of coflat maps. Thus (f_{1}
f_{2}): Y_{1}
Y_{2} > X_{1}
X_{2} is coflat by (b).
(d) Consider two maps u_{1}: X_{1} > T_{1}
and u_{2}: X_{2} > T_{2} . If u_{1}
+ u_{2} is coflat, then u_{1} and u_{2}
are coflat by (a) because they are pullbacks of u_{1} + u_{2}
by (1.3.4.b).
Conversely assume u_{1} and u_{2} are coflat.
Any map from g to T_{1} + T_{2} has the form
g = v_{1} + v_{2}: Y_{1} + Y_{2}
> T_{1} + T_{2} for two maps v_{1}:
Y_{1} > T_{1} and v_{2}: Y_{2}
> T_{2}. Then by (1.3.4.a)
the pullback s of g along u_{1} + u_{2}
is the sum of X_{1} _{T1}
Y_{1} > X_{1} and X_{2} _{T2}
Y_{2} > X_{2} . If g is epic then
v_{1} and v_{2} are epic by (1.3.7),
thus X_{1} _{T1}
Y_{1} > X_{1} and X_{2} _{T2}
Y_{2} > X_{2} are epic because u_{1}
and u_{2} are coflat. It follows that s is epic by
(1.3.7). This shows that u_{1}
+ u_{2} is precoflat. Next the pullback t of u_{1}
+ u_{2} along g is the sum of the coflat map X_{1} _{T1}
Y_{1} > Y_{1} and the coflat map X_{2} _{T2}
Y_{2} > Y_{2}. Thus t is also precoflat.
It follows that u_{1} + u_{2} is coflat.
Proposition 1.4.3. (a) Suppose f:
Y > X is a mono and g: Z > Y is a map. Then g
is coflat if fg is coflat.
(b) For any object X, the codiagonal map _{X}:
X + X > X is coflat.
(c) Suppose {f_{i}: Y_{i} > X} is a
finite family of coflat maps. Then f =
(f_{i}): Y =
Y_{i} > X is coflat.
Proof. (a) If fg is coflat then g is coflat as
it is the pullback of fg along f.
(b) The pullback of _{X}:
X + X > X along a map Z > X is the map _{Z}:
Z + Z > Z . Therefore in order to show that _{X}
is coflat it suffices to prove that _{X}
is precoflat. Assume f: Y > X is an epi. Then the pullback
of f along _{X}
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 _{X}:
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 = (f_{i}):
Y = Y_{i} > X.
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 fg 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 fg 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 fg factors through f,
(g, 1_{Z}) is the pullback of (f, fg).
If fg is an epi then g is an epi as f is coflat by
assumption.
(f) follows from (e) and (1.4.3.a).
Proposition 1.4.5. Injections of a sum
are coflat monos.
Proof. The injection X > X + Y is the sum of the coflat
mono 1_{X}: X > X and the coflat mono 0 >
Y . Thus by (1.4.4.d) it is coflat.
[Next Section][Content][References][Notations][Home]
