1.4 Exact Presites
Definition 1.4.1. Suppose
C is a presite and X an object of C. A subset U
of X is called exact if the
following conditions are satisfied:
(a) For any point x Î U
there is a morphism f: Y ®
X such that f(Y) Í
U and x Î f(Y).
(b) For any x Î U the
category C_{*/(x,U)} of pointed objects f:
(y, Y) ® (x, X)
over (x, X) such that f(Y) Í
U is connected.
(c) A subset V of U is open in the subspace U
if for any morphism f: Y ®
X such that f(Y) Í
U, f^{1}(V) is an open subset of Y.
Remark 1.4.2. For any subset U
of X let C_{/U} be the subcategory of C_{/X}
consisting of objects (Y, f) over X such that f(Y)
Í U. We denote by h_{U}
the colimit of the functor tp: C_{/U} ®
C ® Top where t
is the metric functor of C. The maps f: Y ®
U for all (Y, f) Î
C_{/U} determine a continuous map G: h_{U}
® U. Then (1.4.1 a,b,c) are equivalent
to the assertions that G is surjective, injective, and open respectively.
Thus we have
(a) U is active if and only if the map G: h_{U}
® U is a homeomorphism.
(b) If U is active and h_{U} is represented by
a morphism g: Z ® X,
then we may identify h_{U} with the space Z of
Z since (Z, g) is a final object of C_{/U}.
(c) Combining (a) and (b) we see that a subset U of X
is effective if and only if U is active and exact.
Remark 1.4.3. If C is separable,
then (1.4.1b) may be simplified to
(1.4.1b') For any x Î U
and any f: (y, Y) ®
(x, X) and g: (z, Z) ®
(x, X) in C_{*/(x,U)}, there
is a pointed object (w, W) with two morphisms (w,
W) ® (y, Y) and (w,
W) ® (z, Z).
Definition 1.4.4. A presite
C is called exact if any open
subset of the space X of any object X Î
C is exact.
Proposition 1.4.5. Suppose for any
object X Î C the
open exact subsets of X form a basis for X.
Then C is exact.
Proof. Suppose U is an open subset of X. We verify
the conditions of (1.4.1) for U:
(a) (1.4.1.a) follows from the fact that U
has an exact open cover {V_{i}} and (1.4.1a) holds for each
active open subset V_{i}.
(b) Suppose x Î U. We
have to prove that the category C_{*/(x,U)}
defined in (1.4.1a) is connected. Let V be an exact open neighborhood
of x contained in U. Then C_{*/(x,V)}
is a connected subcategory of C_{*/(x,U)}.
It suffices to prove that any object of C_{*/(x,U)}
is connected to an object of C_{*/(x,V)} by
a morphism in C_{*/(x,U)}. Suppose f:
(y, Y) ® (x, X)
is an object of C_{*/(x,U)}. Then f^{1}(V)
is an open subset of Y containing y. By (a) there is a g:
(z, Z) ® (y, Y)
such that g(Z) Í f^{1}(V).
Since fg: (z, Z) ®
(x, X) is an object of C_{*/(x,V)},
we see that the object f Î C_{*/(x,U)}
is connected by g to an object fg e C_{*/(x,V)}.
(c) Suppose V is a subset of U such that (1.4.1.c)
holds for V. Take an exact open cover {V_{i}} of
U. Then (1.4.1c) holds for the subset V Ç
V_{i} of V_{i} for each i. Thus each
V Ç V_{i} is open
in V_{i}. Hence V is open in U.
Corollary 1.4.6. Suppose for any
object X Î C the
open effective subsets of X form a basis for X.
Then C is exact. Any locally effective presite is exact.
Remark 1.4.7. (1.4.6) yields another
proof for (1.2.15) since active, quasieffective
subset is effective.
Suppose (C, t) is a presite and C'
a category containing C as a full subcategory. Since colimits exist
in Top, the metric functor t: C
® Top has a Kan extension t':
C' ® Top. We obtain a presite
(C', t').
Remark 1.4.8. Suppose S is an
object of C'. The underlying space S of S is determined
by the following properties:
(a) For any s Î S there
is a morphism f: (x, X) ®
(s, S) in C'_{*} with X Î
C.
(b) The category C_{*/(s,S)} of pointed
objects f: (x, X) ®
(s, S) over (s, S) with X Î
C is connected.
(c) The topology on S is the finest one such that for each
f: X ® S with X
Î C, the map f: X
® S is continuous. Thus a subset
V of S is open if and only for any X Î
C and f: X ® S,
f^{1}(V) is open in X.
Remark 1.4.9. If (C', t')
is a presite and C a full subcategory of C' such that the
above (a)  (c) hold, then t' is the Kan extension
of the restriction t'_{C} of
t' on C.
Remark 1.4.10. The underlying set of
the space S may be identified with the set of connected components
of the category C_{*/S} of the triples (x,
X, f) where X Î C,
x Î X, and f: X
® S is a morphism in C'; a
morphism from (x, X, f) to (y, Y, g) is a morphism h:
(x, X) ® (y, Y)
such that gh = f.
Remark 1.4.11. If C is separated,
then (1.4.8.b) may be simplified to:
(1.4.8.b') For any s Î S
and any morphisms f: (x, X) ®
(s, S) and g: (y, Y) ®
(s, S) with X, Y Î
C, there exists a pointed object (z, Z) such that
Z Î C, with two morphisms
p: (z, Z) ® (x,
X) and q: (z, Z) ®
(y, Y) such that fp = gq.
Example 1.4.12. (a) If (C, t)
is representable with t* = h'_{T} = hom_{C}
(T, ~) for some object T Î
C, then (C', t') is also representable
with t'* = h'_{T} = hom_{C}'
(T, ~) in C'.
(b) Suppose C has a generic subsite D, then D
is also a generic subsite of (C', t').
Proposition 1.4.13. Suppose (C,
t) is a presite and C' a category containing
C as a full subcategory. Suppose t':
C' ® Top is the Kan extension
of t on C'. Then (C, t)
is separable if and only if (C', t')
is separable.
Proof. Suppose (C, t) is separable.
Suppose f': (x', X') ®
(s, S) and g': (y', Y') ®
(s, S) are two morphisms in C'_{*}. Let f":
(x, X) ® (x', X')
and g": (y, Y) ® (y',
Y') be morphisms in C' with X, Y Î
C (1.4.8.a). We obtain two morphisms f
= f'f": (x, X) ® (s,
S) and g = g'g": (y, Y) ®
(s, S). Applying (1.4.8.b') we find
two morphisms p: (z, Z) ®
(x, X) and q: (z, Z) ®
(y, Y) as required. Thus (C', t') is separable.
Next assume that (C', t') is separable. Suppose f:
(x, X) ® (s, S)
and g: (y, Y) ® (s,
S) are two morphisms in C_{*}. There are two morphisms
p': (z', Z') ® (x,
X), x) and q': (z', Z') ®
(y, Z) in C'_{*}. There is a morphism h:
(z, Z) ® (z', Z')
with Z Î C (1.4.8.a).
Then p'h: (z, Z) ®
(x, X) and q'h: (z, Z) ®
(y, Y) are what we are looking for. This proves that (C,
t) is separable.
1.4.14 Suppose D is a category and
E a full subcategory of D.
(a) We say E is a sieve
of D if any object X Î D
such that there is a morphism from X to an object of E is
in E.
(b) We say E is a final
subcategory of D if for any object X Î
D, the subcategory E_{/X} of C_{/X}
consisting of Xobjects Y Î
E is nonempty and is connected.
Remark 1.4.15. Suppose E is
a final subcategory of D.
(a) If F is a subcategory of E, then F is a final
subcategory of D. (The proof is similar to the following (1.4.15.a).)
(b) If F is a full subcategory of D containing E,
then E is final in F.
Lemma 1.4.16. Suppose D
is a category and E a final subcategory of D.
(a) D is connected if and only if E is
connected.
(b) Suppose F is a sieve of D and
F Ç E is a final subcategory
of E. Then E is connected if and only if
F is connected.
Proof. (a) If E is connected then D is so since
any object of D is connected to an object of E. Conversely
assume D is connected. For any two objects X and Y
in E there are finitely many objects X_{0} = X,
X_{1},...,X_{2n} = Y in C
such that there are morphisms X_{2j2} ®
X_{2j1} and X_{2j }®
X_{2j1} for j = 1,...,n. Since E
is final, there exist morphisms f_{i}: Z_{i}
® X_{i} with Z_{i}
Î E; for i = 0 and n
we let Z_{i} = X_{i}. Now Z_{2j2}
and Z_{2j1} are objects over X_{2j1},
thus they are connected in E because C_{/X2j1
}is connected by (1.4.14b). Similarly Z_{2j}
and Z_{2j1} are connected in E. Thus X
= Z_{0} and Y = Z_{2n} is connected
in E, which shows that E is connected.
(b) Suppose E is connected. Since E is final, for any
X Î F there is a morphism
Y ® X with Y Î
E. But F is a sieve, so Y Î
F, thus Y Î F Ç
E. Hence any object of F is connected to an object of E
Ç F. Since E Ç
F is final in E and E is connected by assumption,
E Ç F is connected by (a).
Conversely, suppose F is connected. Since E is final
in D and F Ç E is
final in E, F Ç E
is final in D by (1.4.15.a), thus is final
in F by (1.4.15.b). Applying (a) we see that
F Ç E and E are
connected because F is so.
Proposition 1.4.17. Suppose
(C, t) is a presite and C'
a category containing C as a full subcategory. Suppose
t': C' ®
Top is the Kan extension of t
on C'. Then (C, t)
is exact if and only if (C', t')
is exact.
Proof. First suppose (C, t) is exact. Suppose U
is an open subset of the space X of an object X Î
C'. We prove that U is exact by verifying the conditions
of (1.4.1).
(a) For any point x Î U
there is a morphism f: (y, Y) ®
(x, X) such that Y Î
C (1.4.8.a). Since C is exact, we can
find an active open neighborhood V of y contained in f^{1}(U).
Let g: (z, Z) ® (y,
Y ) be a morphism with Z Î
C and g(Z) Í V.
Then fg: (z, Z) ® (x,
X) is a morphism such that fg(Z) Í
U. This proves (1.4.1.a) for U.
(b) We need to prove that for any x Î
U, the category F = C'_{*/(x,U)}
of pointed objects f: (y, Y) ®
(x, X) with f(Y) Í
U is connected. Consider the category D = C'_{*/(x,X)}.
Then E = C_{*/(x,X)} is connected and
is final in C'_{*/(x,X)} by (1.4.8),
and F is a sieve in D. Since C is active, F
Ç E = C_{*/(x,U)}
is final in E. Thus we may apply (1.4.16.b)
to see that F = C'_{*/(x,U)} is connected.
(c) Suppose V is a subset of U such that for any morphism
f: Y ® X with f(Y)
Í U, f^{1}(V)
is an open subset of Y. We have to prove that V is open
in U, or equivalently, V is an open subset of X. It
suffices to prove that for any morphism g: Z ®
X with Z Î C, g^{1}(V)
Í Z is open (1.4.8.c).
Since g^{1}(U) is open we only need to show that
g^{1}(V) is open in g^{1}(U).
Since g^{1}(U) is exact, by (1.4.1.c)
it suffices to prove that for any h: W ®
Z with h(W) Í
g^{1}(U), h^{1}(g^{1}(V))
is open. But gh(W) Í
U. Thus (gh)^{1}(V) = h^{1}(g^{1}(V))
is open by assumption.
Conversely suppose (C', t') is exact.
Suppose U is an open subset of the space X of an object
X Î C. We prove that U
is exact in C by verifying the conditions of (1.4.1).
(a') Since U is exact in C', for any point x Î
U there is a morphism f: (y, Y) ®
(x, X) with f(Y) Í
U (1.4.1.a). Let g: (z, Z)
® (y, Y ) be a morphism with
Z Î C. Then fg: (z,
Z) ® (x, X) is a morphism
in C such that fg(Z) Í
U. This proves (1.4.1.a) for U.
(b') We need to prove that for any x Î
U, the category E = C_{*/(x,U)}
of pointed objects f: (y, Y) ®
(x, X) with f(Y) Í
U and Y Î C is connected.
Since U is exact in C', D = C'_{*/(x,U)}
is connected by (1.4.1.b), and E is final in D by
(1.4.8). Thus we may apply (1.4.16.a)
to see that E = C_{*/(x,U)} is connected.
(c') Suppose V is a subset of U such that for any morphism
f: Y ® X with f(Y)
Í U and Y Î
C, f^{1}(V) is an open subset of Y.
We have to prove that V is open in U. Since U is exact
in C', it suffices to prove that for any morphism g: Z
® X with g(Z) Í
U, g^{1}(V) Í
Z is open (1.4.1c). By (1.4.8c) it suffices
to prove that for any h: Y ® Z
with Y Î C, h^{1}(g^{1}(V))
is open. But gh(Y) Í
U. Thus (gh)^{1}(V) = h^{1}(g^{1}(V))
is open by assumption.
