Introduction

Introduction

(yin and yang in category theory)

A category is left unitary if it has a strict initial object and any object is unitary. The opposite of a left unitary category is a right unitary category. A left unitary category is never right unitary unless it is trivial. These are the yin (left or geometric) and yang (right or algebraic) in category theory. Categorical geometry studies the geometric properties of unitary categories. The categorical approach to algebraic geometry was initiated by Yves Diers in his pioneer book Categories of Commutative Algebras (Oxford University Press, 1992). This site contains the papers and notes I wrote on this subject since 1992. Many of the new concepts and results described below are due to (or under the influences of) Diers's book. The reader is referred to Diers's book, and the online book Categorical Geometry for details.

Strict Initial Object: An object 0 in a category is an initial object if for any object X there is a unique map 0 --> X; an initial object 0 is strict if any map X --> 0 is an isomorphism. Geometrically a strict initial object plays the role of the "empty set". A map f: Y --> X is initial if its domain Y is an initial object. The dual notions are terminal object 1, strict terminal object, and terminal map.

Unitary Object: An object X is (left) unitary if for any non-initial map f: Y --> X there is a distinct pair of maps u, v: X --> Z such that uf ¹ vf. One can show that any object with a map to a unitary object is unitary. Thus any object in a category with a unitary terminal object is unitary. A pair of parallel maps f, g: X --> Z is a unitary (or disjointed) condition on X if any map t: T --> X satisfying the condition tf = tg is initial. Any object with a unit condition is (left) unitary.

Unitary Category: A category with a strict initial object is called (left) unitary if any object is unitary. The dual notions are (right) unitary object and (right) unitary category. Since any left unitary terminal object in a non-trivial category is not strict, a left unitary category is never right unitary unless it is trivial (i.e. any object is initial) (see Unitary Categories and Left and Right Unitary Categories).

In the following we assume A is a left unitary category with a strict initial object 0. A class of objects is called unidense if any non-initial object is the codomain of a map with a non-initial domain in the class.

Reduced Category: A pair of parallel maps f, g: X --> Z is nilpotent if any map t: T --> X such that (tf, tg) is a unitary condition is initial. An object X is reduced if any pair of distinct parallel maps with domain X is not nilpotent. A category is reduced if any object is reduced; it is reducible if the class of reduced objects is unidense. One can show that any epic quotient of a reduced object is reduced. A non-initial object is simple if any non-initial map to it is epic. Any simple object is reduced (see Reduced Categories and Categorical Geometry Chapter 3).

Atomic Functor: A functor E from A to the (meta)category of sets is called atomic (or aunifunctor) if the following three conditions are satisfied:
(a) E(X) is empty iff X is an initial object.
(b) For any element p of E(X) there is a map h: P --> X in A such that E(P) has only one element and E(h)(E(P)) = p.
(c) For any two non-initial maps f: P --> X and g: Q --> X in A such thatE(f)(E(P)) = E(g)(E(Q)) there are non-initial maps u: O --> P and v: O --> Q such that fu = gv.
An atomic unifunctor is uniquely determined by the category up to equivalence.

Atomic Category: A non-initial object P is called atomic (or unisimple) if for any two non-initial maps f: X --> P andg: Y --> P there are two non-initial maps r: R --> X ands: R --> Y such that fr = gs. A category is calledatomic if the class of atomic objects is unidense. Any reduced atomic object is simple (thus any reduced atomic category is reducible). A category is atomic iff it has an atomic functor. In an atomic category a map is unipotent iff its image under the atomic functor is surjective. (see Atomic Categories and Uniform functors).

Zariski Functor: Consider a functor T from A to the (meta)category of topological spaces. If f, g: X --> Z is a pair of parallel of maps we denote by V(f, g) the set of points p of T(X) such that there is a map h: P --> X with T(h)(T(P)) = p and fh = gh; V(f, g) is called a principal algebraic set of T(X). A subset of T(X) is called an algebraic set if it is an intersection of some principal algebraic sets.
We say T is a Zariski functor if its set-theoretic underlying functor is atomic and for any object X a subset U of T(X) is closed iff it is algebraic. The Zariski functor (if exists) is uniquely determined by A up to equivalence. For instance, if A is the category of affine schemes or affine varieties then it carries a Zariski functor which coincides with the classical Zariski topology (see Zariski Topology).

Unipotent Map: Two maps u: U ® X and v: V ® X are disjoint if 0 is the pullback of (u, v). A map f: Y --> X is called unipotent if any non-initial map to X is not disjoint with f. Unipotent maps are closed under composition and stable under pullbacks. The notion of a unipotent map plays the role of a "surjective" map in categorical geometry. (see Categorical Geometry Chapter 2).

Normal Mono: A mono u: U --> X is called normal if for any map v: V --> X not factored through u there is a non-initial map s: S --> V such that vs is disjoint with u. If A has pullbacks then a mono is normal iff any of its pullbacks is not proper (i.e. non-isomorphic) unipotent. Normal monos are closed under composition and stable under pullbacks. The notion of a normal mono plays the role of an "embedding" in categorical geometry. If A has finite limits then an object X is reduced iff any unipotent map to it is epic, and A is reduced iff any strong (or regular) mono is normal (see Categorical Geometry Chapter 2).

Coflat Map: A map f: Y --> X is called coflat (resp. faithful) if the pullback of f along any map to X exists and the pullback functor F: A/X --> A/Y preserves (resp. reflects) epis. A faithful and coflat map is called a faithfully coflat map (the dual of these notions are due to Diers [1992]).A normal coflat mono is called a fraction. Coflat (resp. faithful, resp. fraction) maps are closed under composition and stable under pullback. In a left unitary category a map is faithful iff it is universal epic (i.e. any of its pullback is epic), and any faithful map is unipotent. (see Categorical Geometry Chapter 1 and 5).

Analytic Category: Suppose X + Y is the sum of two objects with the injections x: X --> X + Y and y: Y --> X + Y. Then X + Y is disjoint if the injections x and y are disjoint and monic. The sum X + Y is stable if for any map f: Z --> X + Y, the pullbacks Z_X --> Z and Z_Y --> Z of x and y along f exist, and the induced map Z_X + Z_Y --> Z is an isomorphism. Assume the category has pullbacks. A strong mono is a map (in fact, a mono) such that any of its pullbacks is not proper (i.e. non-isomorphic) epic. If a map f is the composite me of an epi e followed by a strong mono m then the pair (e, m) is called an epi-strong-mono factorization of f.
An analytic category is a category satisfying the following axioms:
     (Axiom A1) Finite limits and finite sums exist.
     (Axiom A2) Finite sums are disjoint and stable.
     (Axiom A3) Any map has an epi-strong-mono factorization.
Any analytic category is left unitary.
(see The Language of Analytic Categories and Categorical Geometry Chapter 1).

In the following we assume A is an analytic category.

Analytic Mono: A mono u^c: U^c --> X is a complement of a mono u: U --> X if u and u^c are disjoint, and any map v: T --> X such that u and v are disjoint factors through u^c (uniquely). An analytic mono is a coflat complement of a strong mono. A mono is disjunctable if it has a coflat complement. An analytic category is disjunctable if any strong mono is disjunctable; it is locally disjunctable if any strong mono is an intersection of disjunctable strong monos. Analytic monos are closed under composition and stable under pullbacks; (see Categorical Geometry Chapter 1).

Integral Object: A non-initial object is primary if any non-initial analytic subobject is epic. An integral object is a reduced primary object. A prime of an object is an integral strong subobject. A non-initial object is irreducible if it is not the join of two proper strong subobjects. Any simple object is integral. Any quotient of a primary (resp. integral) object is primary (resp. integral). Any non-initial analytic subobject of a primary object is primary; any non-initial analytic subobject of an integral object is integral. If A is locally disjunctable then an object is integral iff it is reduced and irreducible. (see Categorical Geometry Chapter 3).

Spectrum: For any object X denote by Spec(X) the set of primes of X. If U is any analytic subobject of X we denote by X(U) the set of primes of X which is not disjoint with U, called an affine subset of X. One can show that the class of affine subsets is closed under intersection. Thus affine subsets form a base for a topology on Spec(X). The resulting topological space Spec(X) is called the spectrum of X. Since the pullback of an analytic mono is analytic, and any quotient of an integral object is integral, Spec is naturally a functor from A to the (meta)category of topological spaces. For instance, if A is the category of affine schemes then Spec coincides with the classical Zariski topology. (see Categorical Geometry Chapter 3).
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