Left (geometric, yin)
Classes
Right (algebraic, yang)
left categories: a category with a strict initial object
(0)
right categories: a category with a strict terminal object
left unitary categories: a left category such that any map with an initial domain is a (generalized) regular mono (1) right unitary categories: a right category such that any map with a terminal codomain is a (generalized) regular epi
left extensive categories: a category with disjoint stable finite sums. (2) right extensive categories: a category with codisjoint costable finite products.
lextensive categories: a left extensive category with finite limits (3) rextensive categories: a right extensive category with finite colimits
left analytic categories: a lextensive category with epi-regular-mono factorizations (4) right analytic categories: a rextensive category with regular-epi-mono factorizations
left analytic geometries: a locally disjunctable reducible perfect left analytic category  (5) right analytic geometries: a locally codisjunctable reducible perfect right analytic category 
left coherent analytic categories: a locally finitely copresentable category whose subcategory of finitely copresentable objects is lextensive. (6) right coherent analytic categories: a locally finitely presentable category whose subcategory of finitely presentable objects is rextensive.
left Stone geometry: a left coherent analytic category in which any strong mono is an intersection of direct monos. (7) right Stone geometry: a right coherent analytic category in which any strong epi is a cointersection of direct epis.
left coherent analytic geometries: a locally disjunctable left coherent analytic category (8) right coherent analytic geometries: a locally codisjunctable right coherent analytic category
left algebraic geometry: a strict left coherent analytic category which is a left algebraic category whose underlying functor is copresented by a disjunctable object. (9) right  algebraic geometry: a strict right coherent analytic category which is a right algebraic category whose underlying functor is presented by a codisjunctable object.
Implications: (9) => (8) => (6) + (5) => (4) => (3) => (2) => (1) => (0) and (7) => (6) + (5)
The only left and right unitary category is the trivial category.
Examples
Left analytic category
Classes
Right analytic category
any elementary topos 
(4)
 
any Grothendieck topos
(5)
 
the category of finite sets
(5)
 
the category of finite topological spaces
(5)
 
the category of finite sober spaces
(5)
 
the category of sets
(5)
the category of complete atomic Boolean algebras
the category of topological spaces
(5)
 
the category of sober spaces (= spatial locales)
(5)
the category of spatial frames
the category of Hausdorff spaces
(5)
 
the category of locales
(5)
the category of frames
the category of coherent spaces
(8)
the category of distributive lattices
Zariski geometry
(8)
Zariski category
the category of Stone spaces
(9)
the category of Boolean algebras
the category of reduced affine schemes
(9)
the category of commutative reduced rings
the category of affine schemes
(9)
the category of commutative rings
 
 
Any left analytic category
The category of affine schemes
(a left algebraic geometry)
The category of commutative rings
(a right algebraic geometry)
Object  
(categorical property)
Affine scheme 
(geometric property)
Commutative ring
(algebraic property)
simple the spectrum of a field field
integral the spectrum of an integral ring = reduced + irreducible integral domain
reduced the spectrum of a reduced ring a ring without non-null nilpotent
pseudo-simple the spectrum has only one point  exactly one prime ideal
quasi-primary ~ ab = 0 => (a or b is nilpotent)
primary ~ any zero divisor is nilpotent
irreducible the spectrum is irreducible the ideal (0) is irreducible with respect to intersection
von Neumann regular ~ von Neumann regular ring
local the spectrum has only one closed point  local ring
Map
Morphism
Homomorphism
coflat map flat flat
normal mono ~ ~
unipotent map surjective morphisms with a nilpotent kernel
analytic mono finitely presentable flat mono finitely presentable flat epi
fraction = flat + normal mono flat mono (local isomorphism) flat epi
Note: The first part of this table was adapted from [Diers 1992, p.45]