Posetal category
Encyclopedia
In mathematics
, a posetal category is a category
whose homsets each contain at most one morphism. As such a posetal category amounts to a preordered set. As suggested by the name, the further requirement that the category be skeletal is often assumed; in the case of a category that is posetal, being skeletal is equivalent to the requirement that the only isomorphisms are the identity morphisms, equivalently that the preordered set satisfies antisymmetry
and hence is a poset.
All diagrams commute
in a posetal category. When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a posetal category corresponds to an inconsistent theory understood as one satisfying the axiom x = y at all types.
Viewing a 2-category
as an enriched category
whose homobjects are categories, the homobjects of any extension of a posetal category to a 2-category
having the same 1-cells are monoid
s.
Some lattice theoretic structures are definable as posetal categories of a certain kind, usually with the stronger assumption of being skeletal. For example a poset may be defined as a posetal category, a distributive lattice
as a posetal distributive category
, a Heyting algebra
as a posetal finitely cocomplete cartesian closed category
, and a Boolean algebra
as a posetal finitely cocomplete *-autonomous category
. Conversely, categories, distributive categories, finitely cocomplete cartesian closed categories, and finitely cocomplete *-autonomous categories can be considered the respective categorification
s of posets, distributive lattices, Heyting algebras, and Boolean algebras.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a posetal category is a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
whose homsets each contain at most one morphism. As such a posetal category amounts to a preordered set. As suggested by the name, the further requirement that the category be skeletal is often assumed; in the case of a category that is posetal, being skeletal is equivalent to the requirement that the only isomorphisms are the identity morphisms, equivalently that the preordered set satisfies antisymmetry
Antisymmetry
In linguistics, antisymmetry is a theory of syntactic linearization presented in Richard Kayne's 1994 monograph The Antisymmetry of Syntax. The crux of this theory is that hierarchical structure in natural language maps universally onto a particular surface linearization, namely...
and hence is a poset.
All diagrams commute
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
in a posetal category. When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a posetal category corresponds to an inconsistent theory understood as one satisfying the axiom x = y at all types.
Viewing a 2-category
2-category
In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category...
as an enriched category
Enriched category
In category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely...
whose homobjects are categories, the homobjects of any extension of a posetal category to a 2-category
2-category
In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category...
having the same 1-cells are monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
s.
Some lattice theoretic structures are definable as posetal categories of a certain kind, usually with the stronger assumption of being skeletal. For example a poset may be defined as a posetal category, a distributive lattice
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
as a posetal distributive category
Distributive category
In mathematics, a category is distributive if it has finite products and finite coproducts such that for every choice of objects A,B,C, the canonical map...
, a Heyting algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
as a posetal finitely cocomplete cartesian closed category
Cartesian closed category
In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in...
, and a Boolean algebra
Boolean algebra
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets...
as a posetal finitely cocomplete *-autonomous category
*-autonomous category
In mathematics, a *-autonomous category C is a symmetric monoidal closed category equipped with a dualizing object \bot.-Definition:Let C be a symmetric monoidal closed category...
. Conversely, categories, distributive categories, finitely cocomplete cartesian closed categories, and finitely cocomplete *-autonomous categories can be considered the respective categorification
Categorification
In mathematics, categorification refers to the process of replacing set-theoretic theorems by category-theoretic analogues. Categorification, when done successfully, replaces sets by categories, functions with functors, and equations by natural isomorphisms of functors satisfying additional...
s of posets, distributive lattices, Heyting algebras, and Boolean algebras.