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 epiregmono
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 X to X.
Thus the condition is sufficient. For any pair (r_{1}, r_{2}):
T > X of maps to X, the equalizer of (r_{1}, r_{2})
is the pullback of the diagonal map _{X}:
X > X X along the map
T > X X determined
by (r_{1}, r_{2}). 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 epiregmono 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 {X_{i}}
with the projections p_{i}: X > X_{i}.
Consider a pair (r, s): Y > X of maps to X . The
equalizer V of (r, s) is the intersection of the equalizer
V_{i} of (p_{i}r, p_{i}s): Y >
X_{i}. Since each X_{i} is locally disjunctable,
by (1.2.2.a) the equalizer V_{i}
is locally disjunctable. Thus by (a) the intersection V of these
V_{i} 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 (r_{1}, r_{2}):
X > T such that r_{1}
r_{2} there exists an object R
and a map g: T > R such that gr_{1}
gr_{2}. 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 (r_{1}, r_{2}): X > T . Consider
a map f: Y > X not factor through V > X . Since
V > X is the equalizer of (r_{1}, r_{2})
, we have r_{1}f
r_{2}f. Since
is a set of cogenerators, we can find an object W in
and a map g: T > W such that gr_{1}f
gr_{2}f . The equalizer z: Z > X of (gr_{1},
gr_{2}) 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 epiregmono 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
epiregmono factorizations then any strong mono is regular by (1.1.5.d),
thus (b) follows from (a). ––
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