Definition 2.2.1. (a) A set S
of maps to an object X is called a unipotent
cover on X if 0 --> X is the only map that is
disjoint with S.
The term " unipotent " is suggested by the following (2.2.2) and (2.2.2.1): Remark 2.2.2. Suppose S
is a set of maps to X. Then by (2.1.1)
we have
Example 2.2.2.1. Consider a homomorphism of commutative rings : A --> B. The induced morphism Spec(): Spec (B) --> Spec (A) is unipotent in the category of affine schemes iff the kernel of is a nilpotent ideal. This suggests that the dual of unipotent map is called a nilpotent map. Unipotent covers (resp. unipotent maps) play the important role of surjective covers (resp. maps) in categorical geometry. For instance, a map in the categories of sets, topological spaces and affine schemes is unipotent iff it is surjective, and this is true for most of natural metric sites. Proposition 2.2.3. (a) Isomorphisms
are unipotent maps.
Proof. (a) is obvious.
Proposition 2.2.4. Suppose f:
Y --> X is a map. Suppose S and
T are two sets of maps to Y.
Write f_{*}(S) for the
set {fu | u S}.
Proof. (a) Assume T S. Consider a map t: T --> Y in T. Since t S, for any non-initial map s: S --> T we can find a non-initial map w: W --> S such that tsw factors through some map v: V --> Y in S. Then the non-initial map ftsw factors through the map fv in f_{*}(S). Thus f_{°}t f_{*}(S). Since this is true for any t in T, we obtain f_{*}(T) f_{*}(S). Definition 2.2.5. A mono u: U --> X is normal if u generates the normal sieve {u} (i.e. sieve(u) = {u}). Equivalently, u is normal iff t {u} implies that t factors through u. Proposition 2.2.6. (a) Isomorphisms
and initial maps are normal monos.
Proof. (a) is obvious.
(e) follows from (2.1.2.b). (f) follows from (2.1.2.a). Corollary 2.2.7. If A is an analytic category then any singular or analytic mono is normal. Proposition 2.2.8. If A has pullbacks, then a mono is normal iff its pullback along any map is not proper unipotent. Proof. Suppose u: U --> X is a mono. If u is normal then any pullback of u is normal by (2.2.6.d), thus not proper unipotent by (2.2.6.b). Conversely, assume the condition is satisfied. If t: T --> X is a map in {u}, then its pullback along u is normal unipotent, thus an isomorphism by (2.2.6.b), which implies that t factors through u. This shows that u is normal. Definition 2.2.9. A class of objects is called uni-dense if any non-initial object is the codomain of a map start from a non-initial object in the class. Proposition 2.2.10. Suppose C is a uni-dense class of objects. A map f: Y --> X is unipotent iff any map from a non-initial object in C to X is not disjoint with f. Proof. The condiiton is clearly necessary. Conversely, assume the condition is satisfied. If t: T --> X is a non-initial map, then there is a non-initial map r: R --> T with the domain T in C. Now tr is not disjoint with f, so t is not disjoint with f. |