Suppose A is a category with a strict initial object 0. Consider a functor | | from A to the (meta)category of sets (or topological paces). If f: Y --> X is a map in A for simplicity we shall write |f(Y)| for the image |f|(|Y|) of |X|. If f, g: X --> Z is a pair of parallel of maps we denote by V(f, g) the set of points p of |X| such that there is a map h: P --> X with |h(P)| = p and fh = gh; V(f, g) is called a principal algebraic set of |X|. A subset of |X| is called an algebraic set if it is an intersection of some principal algebraic sets.
An object P is called a spot if |P| is a one-point-space. Recall that an object P in A is called simple if any non-initial map to P is epic.
Definition 1. A functor | | is called simple
the following conditions are satisfied:
Definition 2. A category is called simple if the class of simple objects is unidense.
Definition 4. (a) A simple functor | | from A to
the (meta)category of topological spaces is called a Zariski
functor if for any object X a subset U of |X|
is closed iff it is algebraic.
Remark 5. A simple (resp. Zariski) functor (if exists) is uniquely determined by the category up to equivalence.
Example 5.1. Denote by Set, Top, Poset,
HTop, Var, and Ring the categories of sets, topological
spaces, posets, Hausdorff spaces, algebraic varieties (over any field),
and commutative rings.
Example 5.2. Suppose A is an analytic geometry such that the class of simple objects is uni-dense (e.g. if A is a coherent analytic geometry) and any strong mono is regular. Then the analytic topology on A is a Zariski topology.
(to be cont.)
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