Suppose C is a category and X is an object of C. Recall that a set S of maps to an object X is called a sieve on X if S contains any map to X that factors through a map in S. The largest sieve on X is denoted by 1_C/X, called the unit sieve on X, which consists of all the maps to X. A sieve on X not equal to 1_C/X is called a proper sieve on X. If V Í U are two sieves of X we say that V is a subsieve of U. 

A sieve U on X is called active if it is generated by a monomorphism e: Y ® X; since Y is uniquely determined by U up to isomorphism, we often write U for Y (and e_U for e), viewed as a subobject of X (with the inclusion morphism e_U). Sometimes we simply identify a subobject Y of X with the active sieve on X generated by it. If f: Y ® X is a map we simply write f(Y) for the sieve on X generated by f. 

We shall use the following notation: 
(i) C/X for the slice category of all the morphisms with codomain X. 
(ii) If f: Y ® X is a morphism we write C/f: C/Y ® C/X for the induced functor, 
(iii) w_C(X) is the poset of sieves on X. 
(iv) v_C(X) is the poset of subobjects of X. 
(v) C/U for the slice category of all the morphisms in a sieve U of X. 

Remark 0.1. Suppose f: Y ® X is a morphism and W is a sieve on Y. Suppose U and V are two sieves on X. 
(a) f^-1(U) is a sieve on Y and f(W) is a sieve on X. 
(b) The assertions that f Î U, f(1_C/Y) Í U and f^-1(U) = 1_C/Y are equivalent. 
(c) W Í f^-1(U) if and only if f(W) Í U. 
(d) A morphism g: Z ® Y is in f^-1(U) if and only if fg(Z) Í U (by (b) and (c)). 
(e) If {U_i} is a set of sieves on X then Ç {U_i} and È {U_i} are two sieves on X. We have f^-1(Ç {U_i}) = Ç {f^-1(U_i)} and f^-1(È {U_i}) = È {f^-1(U_i)}. 
(f) If f Î U then f^-1(U Ç V ) = 1_C/Y Ç f^-1(V ) = f^-1(V ) (by (b) and (e)). 
(g) Suppose C has pullbacks. If f: Y ® X is a morphism and U is an active sieve on X, then f^-1(U) is an active sieve of Y generated by the monomorphism Y ×_X U ® Y. 

Remark 0.2. If D is a category and F: C ® D is a functor we write F_X: C/X ® D/F(X) for the induced functor. 
(a) If W is a sieve on F(X) then F_X^-1(W) is a sieve on X. 
(b) If {W_i} is a set of sieves on F(X) then F_X^-1(Ç {W_i}) = Ç {F_X^-1(W_i)} and F_X^-1(È {W_i}) = È {F_X^-1(W_i)}. 

Next we introduce a relative version of the notion of a sieve, which will be useful in the study of extensions of framed topologies in º3. If A is a full subcategory of C and X Î C we write A/X for the full subcategory of C/X of all the morphisms f: Y ® X with Y Î A. Suppose A Ê B are two full subcategories of C. By an A/B-sieve on an object X Î C we mean a subset U Í A/X such that a morphism f: Y ® X in A/X is in U if and only if 
for any morphism g: Z ® Y in B/Y, fg is in U. If B = A then we simply say that U is a B-sieve on X. Denote by w_A/B(X) (resp. w_B(X)) the set of A/B-sieves (resp. B-sieves) on X. 

Remark 0.3. (a) There is a natural one-to-one correspondence between w_C/B(X) and w_B(X) preserving orders, sending each C/B-sieve U on X to the B-sieve U Ç B/X on X. 
(b) If f: Y ® X is a morphism and U a B-sieve on X then f_B^-1(U) = {u: Z ® Y | Z Î B and fu Î U} is a B-sieve on Y. 

Remark 0.4. (a) w_C/B(X) is closed under arbitrary intersections. 
(b) If f: Y ® X is a morphism and U is a C/B-sieve on X, then f^-1(U) is a C/B-sieve on Y. 
(c) Suppose U Ê V are two sieves on X, and U is a C/B-sieve. Suppose for any f: Y ® X in U, f^-1(V) is a C/B-sieve on Y, then V is a C/B-sieve. 
 
Example 0.4.1. Suppose B is a small category and C = B^{^} is the category of contravariant functors from B to the category Set of small sets, then a C-sieve U on an object X in C is a C/B-sieve if and only if U is active in C (i.e. U is determined by a subfunctor of X).