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0 an initial object in an analytic category. 
1 a terminal object in an analytic category. 
D(G) the class of open effective monos for a framed topology G.
Loc(X) the dual of the poset of reduced strong subobjects of an object X in an analytic geometry.
R(X) the set of strong subobjects of an object X in an analytic category. 
rad(X) the radical of an object X in an analytic category, which is the unipotent reduced strong subobject of X. 
red(X) the reduced model of an object X in an analytic category, which is the largest reduced strong subobject of X. 
Spec(X) the set of prime subobjects of an object X in an analytic category.
S(X) the set of reduced strong subobjects of an object X in an analytic category.  
ØS the set of maps to an object X which is disjoint with a set S of maps to X.
T(G) The collection of open effective covers for a framed topology G.
sie(S) the sieve on an object X generated by a set S of maps to X.
(X) the set of normal sieves on an object X
FD(X) (or simply F(X)) the set of D-sieves on an object X for a divisor D.