This post is a continuation of a mini-series on category theory, starting with this post. As before, knowing the background for every single example is not required for understanding the overall concepts.
Previously, we have defined categories and the morphisms between them. In the case that one has two parallel functors , between two categories and , it is frequently the case that for any object , one has a morphism . In this situation, the notion of a natural transformation conceptualizes what it means for this collection of morphisms to vary “naturally” with the object , leading to a concept essential to all of category theory. Using this concept, we can define a coarser notion of equivalences than the obvious one of isomorphic categories.
Definition C3.1 Let and be categories and let be functors. Then a natural transformation consists of the following data:
- For every object a morphism in , called the component of at .
Such that the following condition holds:
- For every objects and every morphism , the following diagram commutes:
which means that
One can think of the naturality condition as saying that “interpolates” between and .
Definition C3.2 If all the components of a natural transformation are isomorphisms, is called a natural isomorphism and and are called naturally isomorphic.
Example C3.3 Let be a group and and be -sets, considered as functors . Then a natural transformation has only one component, as has only one object, so it is given by a map of the underlying sets satisfying a naturality condition. This condition requires that we have for any (i.e. for any morphism in ) , (by abuse of notation, we write for its action on and ), so that natural transformations correspond precisely to -equivariant maps.
Example C3.4 Reasoning as in the last example, one concludes that if we have a group and a field , then natural transformations between two representations, considered as functors correspond to morphisms of representations.
Example C3.5 Let be the category of commutative and unital rings with ring homomorphisms and be the category of monoids. Fix . Let be the functor that sends a commutative ring to the multiplicative monoid of the matrix ring: . Every ring homomorphism induces a monoid homomorphism of multiplicative monoids by applying to every component (this is in fact a ring homomorphism of the matrix rings). Let be the “forgetful” functor that forgets about addition and sends a commutative ring to its multiplicative monoid. Every ring homomorphism respects multiplication and the unit element, so that it induces a monoid homomorphism on the multiplicative monoids. The determinant is defined by a polynomial with coefficients in . Hence it doesn’t matter if we apply the ring homomorphism componentwise to a square matrix and then take the determinant or if we take the determinant first and then apply to the result. We hence get a commutative diagram: stating that the determinant is natural in .
Natural Transformations as Homotopies
(Image source: https://commons.wikimedia.org/wiki/File:HomotopySmall.gif)
Reminder/Definition Consider two topological spaces and and let and be continuous maps, then a homotopy between and is a continuous map such that for all and for all . If there exists such a homotopy, and are called homotopic, denoted by .
One thinks of a homotopy as a continuous deformation or an interpolation of to and of the first parameter from as a time parameter, as in the picture above. The definition states that behaves on like and on like . Continuity implies that it is a continuous deformation of to .
If we think of categories as the objects of study in category theory and functors as the morphisms between them, then natural transformations interpolate between morphisms between the objects. This situation might be reminiscent of the situation in topology: objects in topology are topological spaces, the morphisms between them are continuous functions and homotopies interpolate between continuous functions. We will see in this section that this analogy can be made even closer than this vague similarity by defining the analogue of products and the unit interval for categories.
Definition C3.6 The arrow category or interval category is the following category: we have two objects and three morphisms: the two identities and a unique morphism . One easily observes that there is only one possible way to define composition which makes this into a category.
This category plays a role for natural transformations analogous to the unit interval
Definition C3.7 Let and be categories, then the product category consists of the following data:
- For and , we have
- Composition is defined componentwise.
Using these definitions, we can define the analogue of a homotopy: let and be categories and let and be functors. Then a “categorical homotopy” from to is a functor such that for all and for all and for all morphisms in , we have and . This means that on the subcategory , behaves like and on the subcategory , behaves like .
Proposition C3.8 Let be categories and let be functors, then natural transformations correspond to categorical homotopies from to .
Proof Let be a natural transformation, then we define a categorical homotopy as follows:
- For all , let for and .
- For all morphisms in , let and let and let (the last equality holds by naturality of .)
Let us check that this is indeed a functor. It clearly respects identities. The only non-obvious cases for compatibility with composition involve in the first component. We have for morphisms and in ,
In the other case we have
Such that is indeed a functor. It is clear from the construction that is a categorical homotopy from to , completing one half of the correspondence. For the other direction, let be a categorial homotopy from to . To obtain a natural transformation, set for all . Naturality now follows from functoriality of ,
as for any morphism in , we have:
Using the same computations, one checks that those two constructions are inverse to each other.
Equivalences of Categories and Related Properties
There’s an obvious notion of isomorphisms for categories: a functor is an isomorphism if there is a functor such that and are the identity functors on and , respectively. For many purposes, however, this notion is too strict and a coarser notion is much more useful. We can let the analogy between natural transformations and homotopies guide us.
Reminder/Definition A continuous map between topological spaces and is called a homotopy equivalence if there is a continuous map such that and are homotopic to the identities on and , respectively.
Example The -dimensional sphere is homotopy equivalent to .
This leads “naturally” to the following definition:
Definition C3.9 A functor between two categories and is called an equivalence of categories if there is a functor such that and are naturally isomorphic to the identity functor on and on , respectively.
It is sometimes easier to verify some sufficient (and necessary) conditions for a functor to be an equivalence than to use the definition directly by constructing a functor in the other direction. We shall now define those properties, which are important in their own right.
Definitions C3.10 A functor between two categories and is called
- faithful, if for all , the induced map is injective.
- full, if for all , the induced map is surjective.
- fully faithful, if for all , the induced map is bijective.
We state and prove a useful property of fully faithful functors:
Lemma C3.11 Let be a fully faithful functor, then reflects isomorphisms in the following sense: if are two objects such that and are isomorphic in , then and are isomorphic in .
Proof Take an isomorphism with inverse , then there are some such that and as the induced maps on -sets is surjective. We get , so as is faithful. By symmetry, , so is an isomorphism with inverse .
Lemma C3.12 A natural isomorphism of functors preserves fullness, faithfulness and full faithfulness
Proof Let be functors and let be a natural isomorphism. One immediately sees that for any objects the map is a bijection with inverse given by .
By naturality, we have for any morphism in , which implies that From this equation, we conclude that the maps induced by and , respectively, on Hom-sets are related by the bijection , hence one is injective, surjective or bijective if and only if the other one is.
Lemma C3.13 If and are functors such that is faithful, then is faithful. If is full, then is full. If is fully faithful, then is full/faithful/fully faithful if and only is.
Proof These assertions follow immediately from the corresponding elementary statements about injective, surjectiv and bijective maps. (E.g. if is surjective, then is surjective.)
Corollary C3.14 Any equivalence of categories is fully faithful.
Proof Let be a functor and be a functor such that is naturally isomorphic to the identity functor on and is naturally isomorphic to the identity functor on . By using lemma C3.12, we can conclude that and are fully faithful, as the identity functor is evidently fully faithful. In virtue of lemma C3.13, this implies that is both full and faithful, hence fully faithful.
Definition C3.15 A functor is called essentially surjective if for every object , there is an object such that is isomorphic to .
Lemma C3.16 An equivalence of categories is essentially surjective.
Proof Let be a functor and be a functor such that is naturally isomorphic via to the identity functor on . Then for any object , we have an isomorphism , showing that is essentially surjective.
At this point, we have collected enough necessary properties of equivalences of categories to obtain a characterization that doesn’t rely on the existence of another functor, which consitutes the main result of this post.
Theorem C3.17 A functor is an equivalence of categories if and only if it is fully faithful and essentially surjective.
Proof Lemma C3.16 and corollary C3.14 furnish one half of the proof.
For the other half, let be a fully faithful and essentially surjective functor. For any object , choose an object such that is isomorphic to (which is possible, as is essentially surjective.) Choose an isomorphism . Let be a morphism in . Then is a morphism . By full faithfulness of , there is a unique morphism such that .
We need to check that these assignments make into a functor: let and be morphisms in , then from the definition of , we obtain , such that by faithfulness of , . As for identities, we get for any object , so that by faithfulness, .
We now need to show that and are naturally isomorphic to the respective identities: The defining equation yields so that is a natural isomorphism from the identity functor on to .
To finish the proof, we have to show that is naturally isomorphic the identity functor on . Note that since is naturally isomorphic to the identity functor on we get that is naturally isomorphic to , because we have for any object an isomorphism such that for any morphism in , we have . Now as is fully faithful, there exist for each pair of objects a unique morphism such that . To show that is natural, insert into so that the definition of , so that we get which implies by functionariality so that by faithfulness, we get , which means precisely that is a natural transformation from the identity functor to . As is an isomorphism for each and is fully faithful, one concludes that is an isomorphism. (If is a fully faithful functor and is an isomorphism, then is an isomorphism, cf. the proof of C3.11.) This shows that is naturally isomorphic to the identity functor on , which concludes the proof and also this post.
(Exercises and remarks for the stray logician or set theorist:
The above proof doesn’t work in ZFC, why? Give an example of an underlying set theory as a meta-theory such that the proof does indeed work.
Prove that the above proof does work in ZFC if one restricts the statement to small categories and that the statement for small categories is equivalent to choice over ZF.
In practice, set-theoretic issues are often ignored by those doing category theory, or one uses Tarski-Grothendieck set theory)