Definition 1.2.1. A morphism f: Y ® X in a presite C is called active if for any morphism g: Z ® X with |g(Z)| Í |f(Y)| there is a unique morphism h: Z ® Y such that g = fh. A bicontinuous active morphism is called an effective morphism. 

1.2.2 Using the notion of representable functor we can rephrase the definition of an active morphism as follows: 
An object X of C determines a representable contravariant functor h_X = hom (~, X) from C to Set. Suppose U is a subset of |X|. Then U determines a subfunctor h_U of h_X by 

Any active morphism f is a monomorphism as it represents the contravariant functor h_f(Y). 

1.2.3 Suppose U is a subset of |X|. We say U is active if h_U is represented by a morphism f: Y ® X. If there is an effective morphism f: Y ® X such that |f(Y)| = U, then we say that U is effective. 

Suppose f: Y ® X is an effective morphism with |f(Y)| = U Í |X|. Since Y is uniquely determined by U up to isomorphism, we often write U for Y, and call U an effective subobject of X; f is called the inclusion morphism of U. 

Suppose f: X ® Y is a morphism. If |f(X)| = U is an effective subset of Y, we write f(X) for the effective subobject U of Y. Note that there is a canonical surjective morphism X ® f(X). If V is an effective subset of |Y| and W = f^-1(V) Í |X| is effective, we write f^-1(V) for the effective subobject W of X. 

Remark 1.2.4. An effective morphism has all the familiar properties for an embedding in proper geometry: 
(a) An effective morphism is uniquely determined by its image. 
(b) An effective morphism is a monomorphism. 
(c) An isomorphism is effective. 
(d) A surjective effective morphism is an isomorphism. 
(e) A composite of two effective morphisms is effective. 
(f) Suppose f: X ® Y is an effective morphism. Then for any subset U Í |X|, U is effective if and only if f(U) = V is effective, in which case f_U: U ® V is an isomorphism of subobjects. 

Remark 1.2.5. For active morphisms the situation is more subtle: 
(a) Suppose f: X ® Y and g: Y ® Z are two active morphisms and g is injective. Then gf is quasi-effective. 
(b) The composite of two injective active morphisms is an injective active morphism. 
(c) A surjective active morphism is an isomorphism. 
(d) Suppose fibre products exist in C. If f: Y ® X is a morphism and U an active subset of |X|, then f^-1(U) is an active subset of |Y| as the projection p: W = U ×_X Y ® Y representing the subfunctor h_f-1_(U) of h_Y. Active morphisms are stable under base extensions. 
(e) Suppose f: X ® Y is an active morphism and |f(X)| is effective, then f is effective. 

Definition 1.2.6. A presite C is called effective (resp. everywhere effective) if any open subset (resp. any subset) of the underlying space |X| of any object X e C is effective.  

Suppose C is an everywhere effective presite. A morphism in C is effective if and only if it is active (1.2.5e). If f: X ® Y is a morphism in C then f(X) exists, and f can be factored uniquely (up to isomorphism) as a composition of a surjective morphism X ® f(X) and an effective morphism f(X) ® Y. 

Example 1.2.7. Set is an everywhere effective presite: for any subset U of a set X, the inclusion map U ® X is effective. Any monomorphism (i.e., injective map) in Set is effective. 

Example 1.2.8. Top is an everywhere effective presite: for any subset U of a space X, the inclusion map U ® X from the subspace U to X is effective. A continuous map f: Y ® X in Top is effective if and only if f is bicontinuous.  

Example 1.2.9. The presite w(X) of subspaces of a topological space X is an everywhere effective. The presite W(X) of open subsets of X is an effective presite. Any morphism in w(X) and W(X) is effective. 

Example 1.2.10. RSp is an everywhere effective presite: suppose (X, O_X) is a ringed space and U a subset of X, then the inclusion morphism (U, O_X|_U) ® (X, O_X) is effective, hence U is effective. If X is a local ringed space (resp. geometric space), then (U, O_X|_U) is a local ringed space (resp. geometric space). Thus LSp and GSp are everywhere effective presites.  

Remark 1.2.11. (a) If C is an everywhere effective presite, the presite Dis(C) of discrete objects is an everywhere effective presite.  
(b) Suppose C is a presite such that any point of an object is effective. Then Spot(C) is a coreflective subcategory of the category C_* of pointed objects of C. Spot(C) is the largest generic subpresite of C.  
(c) The subpresite Spot(C) of an everywhere effective presite C is a generic subpresite of C. 

Example 1.2.12. RSet, LSet and GSet are everywhere effective presites.  

Example 1.2.13. If (X, O) is a scheme (resp. reduced scheme) and U an open subset of X, then (U, O|_U) is a scheme (resp. reduced scheme). Hence Sch and GSch are effective presites. Similarly PVar/k is an effective presite. 

Definition 1.2.14. A presite is called locally effective if the following conditions are satisfied: 
(a) For any object X, the open effective subsets together with the empty set form a basis for the underlying space |X| of X. 
(b) If f: Y ® X is a morphism and U an effective open subset of |X|, then  f^-1(U) is effective. 

(1.2.14.b) implies that the intersection of two open effective subobjects of X is an open effective subobject. Any effective or everywhere effective presite is locally effective. 

Similarly we define an active, everywhere active, and locally active presite by replacing "effective" in the definitions by "active". 

If C has fibre products then (1.2.14.b) follows from (1.2.14.a). This is given by the following 

Proposition 1.2.15. Suppose (C, t) is a presite satisfying the condition (1.2.14.a). Then 
(a) Any active open subset of an object is effective. 
(b) (C, t) is a locally effective presite if fibre products exist in C. 

Proof. (a) Suppose U is an active open subset of an object X and f: Y Î X is a morphism representing h_U. For any x Î U let V Í U be an effective open subset containing x. Then the effective morphism V ® X factors through f by a morphism h: V ® Y as f is active. This proves that |f(Y)| = U. If y Î |Y| is any point above x and W Í f^-1(V) is an open effective subset of Y containing y with the inclusion morphism i: W ® Y, then the composition W ® Y ® X has the image in V, hence it factors through the inclusion morphism V ® X by some j: W ® V. Thus we have a composite  W ® V ® Y ® X which is the same as the composite W ® Y ® X. By the uniqueness of factorization in the definition of active morphisms we see that i: W ® Y is the same as hj: W ® V ® Y. Thus we obtain i(y) = y = hj(y) = h(x), indicating that h(x) is the unique point of Y above x, so f is injective. Since the collection of such effective open subsets V covers U = |f(Y)| and each composition V ® Y ® X is bicontinuous, it follows that f is bicontinuous, hence effective. This proves that U is effective. 
(b) Suppose C has fibre products. Suppose f: Y ® X is a morphism and U an open effective subset of |X|. Then f^-1(U) is active (1.2.5.d), hence is effective by (a). Thus (1.2.14.b) holds for C.  

Example 1.2.16. The open sets {X_f = Spec A_f | f Î A} of any affine scheme X = Spec A form a basis for X. Since fibre products exist in ASch (see (4.2.4)), it follows from (1.2.15) that ASch is a locally effective presite. Similarly GASch and AVar/k are locally effective presites. 

Example 1.2.17. In Sch, ASch, GSch, and GASch, the intersection {x}^o of open neighborhoods of a point x of an object X is effective. To see this we may assume that X = Spec A is an affine scheme and x is a prime ideal p of A. Then {p}^o is the image of the canonical morphism Spec A_p ® Spec A corresponding to the localization map A ® A_p. Clearly Spec A_p ® Spec A is effective, hence {p}^o is effective. 

1.2.18 Denote by Emp(C) the full subcategory of empty objects of C. If C is effective, then Emp(C) is a coreflective subpresite of C. 

Example 1.2.19. Any presite C of spots is locally effective, but not effective unless C is empty, because the empty subset of any spot is not effective.  

Example 1.2.20. RPot, LPot and GPot are locally effective.  

Definition 1.2.21. A morphism f: Y ® X in a metric presite C is called a local isomorphism if for any y Î |Y| there exists an effective open neighborhood U of y such that the restriction f_U: U ® X is effective.  

Example 1.2.22. A morphism in Top is a local isomorphism if and only if it is a local homeomorphism. 

Remark 1.2.23. (a) A composite of two local isomorphisms is a local isomorphism. 
(b) Suppose f: X ® S, g: Y ® S are two morphisms such that the fibre product X ×_S Y exists. If f is a local isomorphism, then q: X ×_S Y ® Y is a local isomorphism. Local isomorphism is stable under base extension. 
(c) If f: Y ® X is a local isomorphism, then |f|: |Y| ® |X| is a local homeomorphism in Top.