In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
PreOrd has
preordered sets as
objects and
order-preserving functions as
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s.
This is a category because the
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
* Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...
of two order-preserving functions is order preserving and the identity map is order preserving.
The
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphis ...
s in PreOrd are the
injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
order-preserving functions.
The
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
(considered as a preordered set) is the
initial object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element) ...
of PreOrd, and the
terminal objects are precisely the
singleton preordered sets. There are thus no
zero object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
s in PreOrd.
The categorical
product in PreOrd is given by the
product order
In mathematics, given partial orders \preceq and \sqsubseteq on sets A and B, respectively, the product order (also called the coordinatewise order or componentwise order) is a partial order \leq on the Cartesian product A \times B. Given two pa ...
on the
cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
.
We have a
forgetful functor
In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure or properties mapping to the output. For an algebraic structure of ...
PreOrd →
Set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
that assigns to each preordered set the underlying
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, and to each order-preserving function the underlying
function. This functor is
faithful, and therefore PreOrd is a
concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects of the category as sets with additional ...
. 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).
While PreOrd is a category with different properties, the category of preordered groups, denoted PreOrdGrp, presents a more complex picture, nonetheless both imply preordered connections.
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 PreOrd a
2-category
In category theory in mathematics, a 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transformation between functors.
...
(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 (mathematics), category whose Category (mathematics)#Small and large categories, homsets each contain at most one morphism. As such, a posetal catego ...
).
With this 2-category structure, a
pseudofunctor F from a category ''C'' to PreOrd 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
Notes
References
*
{{DEFAULTSORT:Category Of Preordered Sets
Preordered sets