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 epiregularmono factorizations 
(4) 
right analytic categories:
a rextensive category with regularepimono 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 





