2.5. Framed TopologiesSuppose A is a category with a strict
initial object 0. Consider a functor G
from A to the category of locales.
If
A framed topology
is
Suppose is a framed
topology on X such that (X)
is the join of {(U)},
then we say that {_{i}U} (resp. {(_{i}U)})
is an _{i}open
effective cover on X (resp. (X)).
U)
= 0).
(b) If u is a normal mono then t is dominated by u
iff (t) factors
through (u).
U)),
so s must be initial if (t)((^{-1}U))
= 0 by (2.5.2.a). Conversely if the open sublocale (t)((^{-1}U)
0 then it is a join of non-initial open effective sublocales {v:_{i}
V} by (2.5.2.b), and each _{i} --> Ttv
factors through both _{i}u and t. Thus t is not disjoint
with u.
(b) If u is normal then t is dominated by u iff
t factors through u, which is equivalent to that (t)
factors through (u)
by (2.5.1.b) as u is effective.
U} is a set of open effective monos and _{i} --> Xt:
T --> X is a map.
(a) t is disjoint with each u iff (_{i}t)
is disjoint with the join of (u).
_{i}(b) t is dominated by {u:_{i} U} if (_{i}
--> Xt)
factors through the join of (U).
_{i}
t)((^{-1}U))_{i}
= 0 for each U by (2.5.3.a).
Thus
_{i}t)(((^{-1}U))
= ((_{i}t)((^{-1}U))
= 0,_{i}t)
is disjoint with the join of (u).
The other direction is obvious.
_{i}(b) Suppose ( t)
factors through the join of (u).
Consider a map _{i}s: S --> T such that ts is disjoint
with each u. Then (_{i}ts)
is disjoint with the join of (u)
by (a). But (_{i}ts)
also factors through the join of (u)
because (_{i}t) is so.
This means that (ts)
is the initial locale. Hence S is initial by (2.5.2.a).
This shows that t is dominated by {u}.
_{i}Denote by
u: U --> X and v:
V --> U are two open effective monos. Then (uv)
= (u)(v)
as a composite of open embeddings of locales is an open embedding. If t:
T --> X is a map in such that (At)
factors through (u)(v),
then t factors through u uniquely in a map s: T
--> U because u is open effective and (t)
factors through (u).
It follows that s factors through v in a map r:
T --> V as v is open effective and (s)
factors through (v).
This shows that t factors through uv. Thus uv is open
effective.
().
Suppose Df: Y --> X is a map and u: U --> X
is an open effective map. By (2.5.2.b) the open sublocale (f)((^{-1}U))
of (Y) is the join
of a set of open effective sublocales (V),
where each _{i}v:_{i} V is an open
effective subobject of _{i} --> YY. Then fv factors through
_{i}u for each v. Suppose _{i}t: T --> Y
is a map to Y such that ft factors through u. Then (t)
factors through the open sublocale (f)((^{-1}U))
of (Y). Applying
(2.5.4.b) we see that t is dominated by {v}.
_{i}The monic divisor open
divisor of .
X. Then (X)
is the join of {(U)}.
Applying (2.5.4.b) we see that the identity map _{i}X
® X is dominated by {u:_{i}
U®_{i} X}. Thus {u}
is a unipotent cover on _{i}X.
U} is an
open cover on _{i} --> XX and f: Y --> X is any map. Then each (f)((^{-1}U)
is a join of open effective sublocales {(_{i}v):_{ij} (V)_{ij}
® (Y)}.
Since (Y) is the
join of {(f)((^{-1}u)},
it is the join of {(_{i}v)}.
Thus {_{ij}v:_{ij} V} is an open effective
cover on _{ij} --> YY such that for each ij, fv
factors through _{ij}u.
_{i}u:_{i} U} is an open
effective cover on _{i} --> XX. Suppose for each i one has a open effective
cover {u}, then (_{ij}: U_{ij} --> U_{i}X)
is the join of {(u)}.
Thus the collection {_{ij}u:_{i}u_{ij} U} is an open effective cover on _{ij}
--> XX.
These together with (2.5.6) show that T()
is a unipotent Grothendieck topology on A.
Suppose is a framed
topology on |