Just as a functor is a morphism between categories, a natural transformation is a 2-morphism between two functors.
Natural transformations are the 2-morphisms in the 2-category Cat.
(natural transformations)
Given categories and and functors a natural transformation between them, denoted
is an assignment to every object in of a morphism in (called the component of at ) such that for any morphism in , the following diagram commutes in :
(composition of natural transformations) Natural transformations between functors and compose in the obvious way to natural transformations (this is their vertical composition in the 2-category Cat) and functors with natural transformations between them form the functor category
The notation alludes to the fact that this makes Cat a closed monoidal category. (Since is in fact a cartesian closed category, another common notation is , cf. exponential objects.)
In fact, if we want to be cartesian closed, the definition of natural transformation is forced (since an adjoint functor is unique). This is discussed in a section below.
There is also a horizontal composition of natural transformations, which makes Cat a 2-category: the Godement product. See there for details.
In fact, Cat is a 2-category (a -enriched category) because it is (cartesian) closed: closed monoidal categories are automatically enriched over themselves, via their internal hom.
(categories of presheaves)
The functor categories (1) are also called categories of presheaves, in particular if they are of the form
hence if they are categories whose
objects are functors out of the opposite category of a given category into the category Set of sets and functions
morphisms are natural transformations between these.
Similarly, functor categories of the form
are also called categories of copresheaves.
An alternative but ultimately equivalent way to define a natural transformation is as an assignment to every morphism in of a morphism , in such a way as that for every binary composition in (or equivalently for every ternary composition in ).
The relation of this to the previous definition is that the commutative squares in the previous definition for any morphism give the value , and the identity morphisms for any object give the component .
Vertical composition of natural transformations can be specified directly in terms of this account as well: specifically, an -ary composition of natural transformations is uniquely determined by the property that , for every -ary composition in .
Horizontal composition is even easier, as the horizontal composite of is just .
The definition of the functor category with morphisms being natural transformations is precisely the one that makes a cartesian closed monoidal category.
The category Cat of all categories (regarded for the moment just as an ordinary 1-category) is a cartesian monoidal category: for every two categories and there is the cartesian product category , whose objects and morphisms are simply pairs of objects and morphisms in and : .
It therefore makes sense to ask if there is for each category an internal hom functor that would make Cat into a closed monoidal category in that for we have natural isomorphisms of sets of functors
This is precisely the case for being the functor category with functors as objects and natural transformations, as defined above, as morphisms.
Since here is cartesian closed, one often uses the exponential notation for the functor category.
To derive from this the definition of natural transformations above, it is sufficient to consider the interval category . For any category , a functor is precisely a choice of morphism in . This means that we can check what a morphism in the internal hom category is by checking what functors are. But by the defining property of as an internal hom, such functors are in natural bijection to functors .
But, as mentioned above, we know what the category is like: its morphisms are pairs of morphisms in and , subject to the obvious composition law, which says in particular that for any morphism in we have
Here the right side is more conveniently depicted as a commuting square
So a natural transformation between functors is given by the images of such squares in . By tracing back the way the hom-isomorphism works, one finds that the image of such a square in for a natural transformation is the naturality square from above:
There is a nice way of describing these structures due to Charles Ehresmann. For a category let be the double category of commutative squares in . Then the class of natural transformations of functors can be described as . But then induces a category structure on this and so we get .
An advantage of this approach is that it applies to the case of topological categories and groupoids (working in a convenient category of spaces).
An analogous approach works for strict cubical -categories with connections, using the good properties of cubes, so leading to a monoidal closed structure for these objects. This yields by an equivalence of categories a monoidal closed structure on strict globular omega-categories, where the tensor product is the Crans-Gray tensor product.
(unit of double-dualization)
For any ground field , the canonical linear map from a vector space to its double dual
is a natural transformation
from the identity functor on to the composition of the linear dual-endofunctor with itself.
This example was the motivating example of Eilenberg & MacLane 1945 (right on the first pages) for introducing the notion of natural transformation (and with it category theory) in the first place.
The conceptual subtlety that these authors sought to resolve here is that for any finite-dimensional vector space there exists also an isomorphism from to its single-dual vector space:
But these linear maps are conceptually different from (2) in that they are not natural in the technical sense that they do not form the components of a natural transformation between the evident functors. Instead they involve an arbitrary choice equivalent to that of an (possibly indefinite) inner product on , which is not preserved by general linear isomorphisms (but just by the corresponding isometries).
The determinant is a natural transformation from the general linear group to the group of units of a ring, which are both functors from Ring to Grp.
The Frobenius homomorphism is a natural transformation from the identity functor on the full subcategory of Ring containing all rings with characteristic to itself.
The Hurewicz homomorphism is a natural transformation from the homotopy group to singular homology, which are both functors from Top to Grp.
The inversion for every group yields a natural transformation from the identity functor on Grp to the opposite group-assigning functor.
The coprojection for every group yields a natural transformation from the identity functor on to the abelianization functor.
(homomorphisms of diagrams)
By Remark , every category identified with a category of presheaves or copresheaves has its morphisms identified with natural transformations.
For instance, the category of directed graphs (digraphs) may be identified with the category of copresheaves on the diagram shape , and under this identification the natural transformations between functors are identified with digraph homomorphisms.
For functors between higher categories, see lax natural transformation etc.
A transformation which is natural only relative to isomorphisms may be called a canonical transformation.
For functors with more complicated shapes than , see extranatural transformation and dinatural transformation.
See natural transformation (discussion) for an informal discussion about natural transformations.
The notion of natural transformations between functors is due to
where it served as the motivation for the definition of categories and functors in the first place.
Textbook accounts:
Saunders MacLane, §I.4 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
Francis Borceux, Section 1.3 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory [doi:10.1017/CBO9780511525858]
See also category theory - references.
Last revised on June 13, 2024 at 09:11:25. See the history of this page for a list of all contributions to it.