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 ...

, specifically

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, a posetal category, or thin category, is a

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 ( ...

whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class (or a

preordered set In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name is meant to suggest that preorders are ''almost'' partial orders, but not quite, as they are not necessar ...

, if its objects form a

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 ...

). As suggested by the name, the further requirement that the category be

skeletal A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fram ...

is often assumed for the definition of "posetal"; 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 class satisfies

antisymmetry In linguistics, antisymmetry, is a theory of syntax described in Richard S. Kayne's 1994 book ''The Antisymmetry of Syntax''. Building upon X-bar theory, it proposes a universal, fundamental word order for phrases (Branching (linguistics), branchin ...

and hence, if a set, is a

poset In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...

. All

diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age o ...

s 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 codiscrete posetal category corresponds to an inconsistent theory understood as one satisfying the axiom ''x'' = ''y'' at all types. Viewing 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. ...

as an

enriched category In category theory, a branch of mathematics, an enriched category generalizes the idea of a category (mathematics), category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many pract ...

whose hom-objects are categories, the hom-objects of any extension of a posetal category to a

having the same 1-cells are

monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...

s. Some lattice-theoretic structures are definable as posetal categories of a certain kind, usually with the stronger assumption of being skeletal. For example, under this assumption, a poset may be defined as a small posetal category, a

distributive lattice In mathematics, a distributive lattice is a lattice (order), lattice in which the operations of join and meet distributivity, distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice o ...

as a small posetal

distributive category In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects A,B,C, the canonical map : mathit_A \times\iota_1, \mathit_A \times\iota_2: A\!\times\!B \,+ A\!\times\!C \to ...

, a

Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' call ...

as a small posetal finitely

cocomplete In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in ...

cartesian closed category In category theory, a Category (mathematics), category is Cartesian closed if, roughly speaking, any morphism defined on a product (category theory), product of two Object (category theory), objects can be naturally identified with a morphism defin ...

, and a

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...

as a small 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 In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural ...

s of posets, distributive lattices, Heyting algebras, and Boolean algebras.

References

{{reflist Category theory