In mathematics

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...

, categorification refers to the process of replacing set-theoretic

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

 theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s by category-theoretic

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

 analogues. Categorification, when done successfully, replaces sets by categories, function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s with functor

Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

s, and equation

Equation

An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

s by natural isomorphisms

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

 of functors satisfying additional properties. The term was coined by Louis Crane.

Categorification is the reverse process of decategorification. Decategorification is a systematic process by which isomorphic

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

 objects in a category are identified as equal. Whereas decategorification is a straightforward process, categorification is usually much less straightforward, and requires insight into individual situations.

Examples of categorification

One form of categorification takes a structure described in terms of sets, and interprets the sets as isomorphism classes of objects in a category. For example, the set of natural numbers can be seen as the set of cardinalities of finite sets (and any two sets with the same cardinality are isomorphic). In this case, operations on the set of natural numbers, such as addition and multiplication, can be seen as carrying information about products and coproducts of the category of finite sets. Less abstractly, the idea here is that manipulating sets of actual objects, and taking coproducts (combining two sets in a union) or products (building arrays of things to keep track of large numbers of them) came first. Later, the concrete structure of sets was abstracted away - taken "only up to isomorphism", to produce the abstract theory of arithmetic. This is a "decategorification" - categorification reverses this step.

Other examples include homology theories

Homology theory

In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...

 in topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

. See also Khovanov homology

Khovanov homology

In mathematics, Khovanov homology is an invariant of oriented knots and links that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial....

 as a knot invariant

Knot invariant

In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...

 in knot theory

Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

.

See also

• Combinatorial proof
  Combinatorial proof
  In mathematics, the term combinatorial proof is often used to mean either of two types of proof of an identity in enumerative combinatorics that either states that two sets of combinatorial configurations, depending on one or more parameters, have the same number of elements , or gives a formula...
  
  , the process of replacing number theoretic
  Number theory
  Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
  
  theorems by set-theoretic analogues.