**Dinatural Transformation**
**Definition**
A dinatural transformation is a generalization of a natural transformation between functors, defined in the context of category theory. It relaxes the naturality condition by requiring a weaker form of commutativity, called dinaturality, which involves a family of morphisms indexed by objects of a category and satisfying a specific coherence condition.
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## Introduction
In category theory, natural transformations provide a fundamental way to compare functors, capturing the idea of a „morphism between functors” that respects the structure of the categories involved. However, certain constructions and concepts require a more flexible notion than natural transformations. Dinatural transformations arise as such a generalization, playing a crucial role in the study of ends, coends, and other categorical limits and colimits.
Dinatural transformations were introduced to handle situations where the strict naturality condition is too restrictive, particularly in the context of bifunctors and profunctors. They are essential in defining ends and coends, which are universal constructions that generalize limits and colimits for functors of two variables.
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## Background: Natural Transformations
Before exploring dinatural transformations, it is helpful to recall the concept of a natural transformation. Given two functors ( F, G : mathcal{C} to mathcal{D} ) between categories (mathcal{C}) and (mathcal{D}), a natural transformation (eta : F Rightarrow G) consists of a family of morphisms (eta_X : F(X) to G(X)) in (mathcal{D}), one for each object (X) in (mathcal{C}), such that for every morphism (f : X to Y) in (mathcal{C}), the following naturality square commutes:
[
begin{array}{ccc}
F(X) & xrightarrow{eta_X} & G(X) \
F(f) downarrow & & downarrow G(f) \
F(Y) & xrightarrow{eta_Y} & G(Y)
end{array}
]
This condition ensures that the transformation (eta) respects the morphisms in (mathcal{C}) in a coherent way.
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## Definition of Dinatural Transformation
Let (mathcal{C}) and (mathcal{D}) be categories, and consider two functors (S, T : mathcal{C}^{op} times mathcal{C} to mathcal{D}). A **dinatural transformation** (alpha : S Rightarrow T) is a family of morphisms
[
alpha_X : S(X, X) to T(X, X)
]
in (mathcal{D}), indexed by objects (X) of (mathcal{C}), satisfying the **dinaturality condition**: for every morphism (f : X to Y) in (mathcal{C}), the following diagram commutes:
[
T(f, 1_Y) circ alpha_Y circ S(1_Y, f) = alpha_X = T(1_X, f) circ alpha_X circ S(f, 1_X)
]
More explicitly, the dinaturality condition requires that the two compositions
[
S(Y, X) xrightarrow{S(1_Y, f)} S(Y, Y) xrightarrow{alpha_Y} T(Y, Y) xrightarrow{T(f, 1_Y)} T(X, Y)
]
and
[
S(Y, X) xrightarrow{S(f, 1_X)} S(X, X) xrightarrow{alpha_X} T(X, X) xrightarrow{T(1_X, f)} T(X, Y)
]
are equal for every morphism (f : X to Y).
This condition is weaker than the naturality condition for natural transformations, which requires commutativity of squares for morphisms between objects, whereas dinaturality involves a „mixed” condition relating morphisms in both variables of the bifunctors.
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## Motivation and Intuition
Dinatural transformations arise naturally when dealing with bifunctors (S, T : mathcal{C}^{op} times mathcal{C} to mathcal{D}), where the two variables are contravariant and covariant respectively. The classical naturality condition is too strong in this setting because it would require commutativity for morphisms acting simultaneously on both variables in a strict manner.
Instead, dinaturality captures a form of coherence that allows morphisms to „slide” along the diagonal of the bifunctor’s domain, reflecting the interplay between the contravariant and covariant actions. This is particularly important in the theory of ends and coends, which are defined as universal dinatural transformations.
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## Ends and Coends
Ends and coends are universal constructions in category theory that generalize limits and colimits for bifunctors. They are defined using dinatural transformations.
– **End:** Given a bifunctor (S : mathcal{C}^{op} times mathcal{C} to mathcal{D}), an **end** of (S) is an object (int_{X} S(X, X)) of (mathcal{D}) equipped with a universal dinatural transformation
[
pi_X : int_{X} S(X, X) to S(X, X)
]
such that for any other object (N) with a dinatural transformation (alpha_X : N to S(X, X)), there exists a unique morphism (u : N to int_X S(X, X)) making the relevant diagrams commute.
– **Coend:** Dually, a **coend** of (S) is an object (int^{X} S(X, X)) equipped with a universal dinatural transformation
[
iota_X : S(X, X) to int^{X} S(X, X)
]
satisfying a dual universal property.
Dinatural transformations are the morphisms that mediate these universal properties, making them central to the theory of ends and coends.
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## Formal Properties
### Relation to Natural Transformations
Every natural transformation between functors (F, G : mathcal{C} to mathcal{D}) can be viewed as a dinatural transformation when the functors are considered as bifunctors constant in one variable. However, the converse is not true: dinatural transformations are strictly more general.
### Composition
Dinatural transformations do not compose in the same straightforward way as natural transformations. While natural transformations form the morphisms of functor categories, dinatural transformations do not generally form morphisms in a category of bifunctors. Instead, they are better understood as morphisms in a bicategorical or enriched setting.
### Examples
– **Trace in Monoidal Categories:** The trace of an endomorphism in a traced monoidal category can be expressed using coends and dinatural transformations.
– **Profunctors:** Dinatural transformations appear naturally in the theory of profunctors (also called distributors or bimodules), which are functors (mathcal{C}^{op} times mathcal{D} to mathbf{Set}). Morphisms between profunctors are given by dinatural transformations.
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## Applications
### Enriched Category Theory
In enriched category theory, where hom-objects take values in a monoidal category rather than (mathbf{Set}), dinatural transformations generalize to enriched dinatural transformations. These play a role in defining enriched ends and coends, which are fundamental in the study of enriched limits and colimits.
### Homological Algebra and Topology
Ends and coends, and thus dinatural transformations, are used in homological algebra and algebraic topology to define various constructions such as tensor products, hom complexes, and derived functors. The flexibility of dinaturality allows these constructions to be expressed categorically.
### Computer Science
In theoretical computer science, particularly in semantics of programming languages and type theory, dinatural transformations appear in the study of parametric polymorphism and in the semantics of recursive types.
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## Historical Context
The concept of dinatural transformations was introduced by Saunders Mac Lane in the 1960s as part of the development of category theory. Mac Lane’s work on natural transformations, ends, and coends laid the foundation for much of modern category theory, and dinatural transformations emerged as a natural extension to handle more complex functorial constructions.
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## Summary
Dinatural transformations generalize natural transformations by relaxing the naturality condition to a dinaturality condition, which is suited for bifunctors with contravariant and covariant arguments. They are essential in defining ends and coends, universal constructions that generalize limits and colimits. Dinatural transformations have broad applications across mathematics and theoretical computer science, particularly in enriched category theory, homological algebra, and semantics.
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## See Also
– Natural transformation
– End (category theory)
– Coend
– Profunctor
– Enriched category theory
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**Meta Description:**
Dinatural transformations generalize natural transformations by relaxing naturality conditions, playing a key role in defining ends and coends in category theory. They are fundamental in various mathematical and theoretical computer science contexts.