In
mathematics, the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
Ord has
preordered sets as
objects and
order-preserving functions as
morphisms. This is a category because the
composition of two order-preserving functions is order preserving and the identity map is order preserving.
The
monomorphisms in Ord are the
injective order-preserving functions.
The
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...
(considered as a preordered set) is the
initial object of Ord, and the
terminal objects are precisely the
singleton preordered sets. There are thus no
zero objects in Ord.
The categorical
product in Ord is given by the
product order on the
cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ ...
.
We have a
forgetful functor Ord →
Set that assigns to each preordered set the underlying
set, and to each order-preserving function the underlying
function. This functor is
faithful, and therefore Ord is a
concrete category. This functor has a left
adjoint (sending every set to that set equipped with the equality relation) and a right adjoint (sending every set to that set equipped with the total relation).
2-category structure
The set of morphisms (order-preserving functions) between two preorders actually has more structure than that of a set. It can be made into a preordered set itself by the pointwise relation:
: (''f'' ≤ ''g'') ⇔ (∀''x'' ''f''(''x'') ≤ ''g''(''x''))
This preordered set can in turn be considered as a category, which makes Ord a
2-category (the additional axioms of a 2-category trivially hold because any equation of parallel morphisms is true in a
posetal category
In mathematics, specifically category theory, a posetal category, or thin category, is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class (or a preordered set, if its objects form ...
).
With this 2-category structure, a
pseudofunctor F from a category ''C'' to Ord is given by the same data as a 2-functor, but has the relaxed properties:
: ∀''x'' ∈ F(''A''), F(''id''
''A'')(''x'') ≃ ''x'',
: ∀''x'' ∈ F(''A''), F(''g''∘''f'')(''x'') ≃ F(''g'')(F(''f'')(''x'')),
where ''x'' ≃ ''y'' means ''x'' ≤ ''y'' and ''y'' ≤ ''x''.
See also
*
FinOrd
*
Simplex category
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects.
Formal definit ...
{{DEFAULTSORT:Category Of Preordered Sets
Preordered sets