Definition 4.2.1. A category is coherent
if it salsifies the following conditions:
A category is a coherent analytic category iff its opposite is a locally indecomposable category in the sense of Diers [1986]. A locally finitely copresentable category is a coherent analytic category iff its full subcategory of finitely copresentable objects is lextensive. This was proved by Diers in [Diers 1983] in the dual situation. Example 4.2.1.1. (see [Johnstone, 1982]) Recall that an open subset is quasi-compactif every open cover has a finite subcover (note that in [Johnstone 1982] quasi-compact open subset is simply called compact). If X is a topological space we denote by ^{c}(X) the family of quasi-compact open subsets of X, then ^{c}(X) is closed under finite unions. A sober topological space X is coherentif ^{c}(X) is closed under finite intersection (this is equivalent to that ^{c}(X) is a distributive lattice), and forms a base for the topology on the space. A continuous mapping f: X --> Y of two coherent spaces is called coherent if f^{-1}(U) is quasi-compact for any quasi-compact open subset U of Y. A coherent space X is uniquely determined by the distributive lattice ^{c}(X). In fact, according to Stone's representation theorem for distributive lattices, the category of coherent spaces CohSp is dual to the category of distributive lattices DLat. Since DLat is locally finitely presentable (in fact, finitary algebraic), CohSp is locally finitely copresentable. Clearly finite sums in CohSp are stable disjoint and the space of two points is finitely copresentable, thus CohSp = DLat^{op} is a coherent analytic category. Suppose A is a coherent analytic category. Proposition 4.2.2. (a) A
is a spatial analytic category.
Proof. (a) A coherent analytic category is a perfect analytic
category by (4.1.2). It is spatial by the
assertion (d) and (2.6.5.a).
Proposition 4.2.3. (a) Cofiltered limits
and products of coflat maps are coflat.
Proof. (a) Consider a cofiltered diagram of coflat maps (f_{i}:
X_{i} --> Y_{i }| i
I}, whose limit is f: X --> Y with the projections (s_{i}:
X --> X_{i}, t_{i}: Y --> Y_{i}}. For
any i I let
f_{i}':
X_{i}' --> Y be the pullback of f_{i}
along t_{i}. Then f is a cofiltered limit of {f_{i}'}.
Let m: (U, u) --> (V, v) be an epi in
A/Y. Denote the pullback of m
along f_{i}' by m_{i}':(U_{i},
x_{i}) --> (V_{i}, v_{i}) and
the pullback of m along f by m':(U', u')
--> (Y', y'). As pullbacks commute with limits, m
is the limit of {m'}. As f_{i}' is coflat and m
is epic, each m_{i}' is epic. Since any cofiltered limit
of epis is epic in a locally finitely copresentable category, m'
is epic (cf. [Adamek and Rosicky 1994]). As a
result, f is coflat. The second assertion follows from the fact
that a product of coflat maps is a cofiltered limit of finite products
of coflat maps, therefore is coflat.
Definition 4.2.4. (a) An object is rationally
closed if any epic fraction to it is an isomorphism.
Proposition 4.2.5. (a) Any object X
has a unique rationally closed epic fraction R(X).
Proof. (a) The intersection of epic fractions of an object X
is an epic fraction R(X) by (4.2.3.d).
Since any composition of epic fractions is an epic fraction, clearly R(X)
is rationally closed. Any rationally closed epic fraction of X must
be the intersection of all the epic fractions of X, thus is unique.
Definition 4.2.6. (a) The unique rationally
closed epic fraction R(X) of an object X is called
the rational
hull of X.
Recall that a map to an object X is called prelocal if it does not factor through any proper analytic mono to X. Definition 4.2.7. A preanalytic mono is a map such that any of its pullbacks is not proper prelocal. Proposition 4.2.8. (a) Any preanalytic
mono is a mono; any analytic mono is preanalytic.
Proof. Similar to (1.1.2) and (1.1.3). Proposition 4.2.9. (a) The class of
preanalytic mono is the smallest class of monos containing the class of
analytic monos which is closed under composition and arbitrary intersection.
Proof. (a) Consider a class S of monos which is closed
under composition and intersection. Assume S contains the class
of analytic monos. We prove that S contains the class of preanalytic
monos. Consider a preanalytic mono u: U --> X. Let v:
V --> X be the intersection of all the monos to X in S
through which u factors. Then v is a mono in S through
which u factors in the form u = ve therough a mono e:
U
--> V. If w is an analytic mono to V through which
e
factors, then vw is a mono in S through which u factors.
Since by assumption v is the intersection of such monos,
vw
factors through v. Thus w is an isomorphism, which means
that e is a prelocal map. But u: U --> X is preanalytic
implies that e: U --> V is preanalytic by (4.2.8.c),
thus e is an isomorphism by (4.2.8.b). Since
v
is in S, the relation u = ve implies that u
is in S.
Remark 4.2.9. Suppose f: Y --> X is a map. If f = gl with l: Y --> Z a quasi-local (resp. prelocal) map and g: Z --> X a fraction (resp. analytic mono), then we say that (l, g) is a quasi-local-fraction (resp. prelocal-preanalytic) factorization of f, and Z is the quasi-local (resp. prelocal) image of Y in X. It is easy to see that g is the intersection of all the fractions (resp. presingular monos) to X such that f factors, thus such a factorization is unique. As in the case of epi-strong-mono factorization, quasi-local-fraction factorizations and prelocal-presingular factorizations exist because these classes of monos are closed under composition and intersection. |