Back to Categorical Geometry Homepage

 Zariski Topologies   Zhaohua Luo (9/29/98) Revised(10/18/98) 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 if the following conditions are satisfied:  (a) |X| is empty iff X is an initial object of A.  (b) For any point p of |X| there is a map h: P --> X in A such that |h(P)| = p and P is a simple spot.  (c) Suppose P and Q are two simple spots and u: P --> X and v: Q --> X are two maps in A. Then |u(P)} = |v(Q)| iff for any pair of parallel of maps f, g: X --> Z, fu = gu iff fv = gv.  Definition 2. A category is called simple if the class of simple objects is unidense.  Remark 3. Since any reduced atomic object is simple, any reduced atomic category is simple. One can show that a category A is simple iff it has a simple functor E.  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.  (b) A Zariski functor is called a (strict) Zariski topology if it is a (strict) metric topology.  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.  (a) The identity functor Set --> Set is a (strict) Zariski topology.  (b) The underlying functor Top --> Set is a Zariski topology.  (c) The underlying functor Poset --> Set is a Zariski topology.  (d) The identity functor HTop --> HTop is a (strict) Zariski topology.  (e) The Zariski topology Var --> Top is a (strict) Zariski topology.  (f) The spectrum functor Ringop --> Top is a (strict) Zariski topology.  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.) 