# Dual (category theory)

Last updated

In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category Cop. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about C, then its dual statement is true about Cop. Also, if a statement is false about C, then its dual has to be false about Cop.

## Contents

Given a concrete category C, it is often the case that the opposite category Cop per se is abstract. Cop need not be a category that arises from mathematical practice. In this case, another category D is also termed to be in duality with C if D and Cop are equivalent as categories.

In the case when C and its opposite Cop are equivalent, such a category is self-dual. [1]

## Formal definition

We define the elementary language of category theory as the two-sorted first order language with objects and morphisms as distinct sorts, together with the relations of an object being the source or target of a morphism and a symbol for composing two morphisms.

Let σ be any statement in this language. We form the dual σop as follows:

1. Interchange each occurrence of "source" in σ with "target".
2. Interchange the order of composing morphisms. That is, replace each occurrence of ${\displaystyle g\circ f}$ with ${\displaystyle f\circ g}$

Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions.

Duality is the observation that σ is true for some category C if and only if σop is true for Cop. [2] [3]

## Examples

• A morphism ${\displaystyle f\colon A\to B}$ is a monomorphism if ${\displaystyle f\circ g=f\circ h}$ implies ${\displaystyle g=h}$. Performing the dual operation, we get the statement that ${\displaystyle g\circ f=h\circ f}$ implies ${\displaystyle g=h.}$ For a morphism ${\displaystyle f\colon B\to A}$, this is precisely what it means for f to be an epimorphism. In short, the property of being a monomorphism is dual to the property of being an epimorphism.

Applying duality, this means that a morphism in some category C is a monomorphism if and only if the reverse morphism in the opposite category Cop is an epimorphism.

• An example comes from reversing the direction of inequalities in a partial order. So if X is a set and ≤ a partial order relation, we can define a new partial order relation ≤new by
xnewy if and only if yx.

This example on orders is a special case, since partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). For example, if we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices.

## Related Research Articles

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototyical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.

In mathematics, a category is a collection of "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 objects are sets and whose arrows are functions.

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X ↪ Y.

In category theory, an epimorphism is a morphism f : XY that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: YZ,

In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra.

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In category theory, a branch of mathematics, the opposite category or dual categoryCop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, .

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory, and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of elements.

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category of topological spaces or of chain complexes, via the acyclic model theorem. The concept was introduced by Daniel G. Quillen (1967).

In category theory, a regular category is a category with finite limits and coequalizers of a pair of morphisms called kernel pairs, satisfying certain exactness conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of images, without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic.

In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f : XY and g : YX are morphisms whose composition fog : YY is the identity morphism on Y, then g is a section of f, and f is a retraction of g.

In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category. This idea often allows restating of definitions or properties of morphisms given by a universal property in more familiar terms, by stating their relation to elements. Some very general theorems, such as Yoneda's lemma and the Mitchell embedding theorem, are of great utility for this, by allowing one to work in a context where these translations are valid. This approach to category theory, in particular the use of the Yoneda lemma in this way, is due to Grothendieck, and is often called the method of the functor of points.

In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism is a representation of as a product , where is a strong epimorphism, a bimorphism, and a strong monomorphism.

## References

1. Jiří Adámek; J. Rosicky (1994). Locally Presentable and Accessible Categories. Cambridge University Press. p. 62. ISBN   978-0-521-42261-1.
2. Mac Lane 1978, p. 33.
3. Awodey 2010, p. 53-55.