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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.
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Examples
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Left analytic category
|
Classes
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Right analytic category
|
| any elementary topos |
(4)
|
|
| any Grothendieck topos |
(5)
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| the category of finite sets |
(5)
|
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| 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)
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| 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 |
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