Natural Transformations

Definition of Natural Transformations

We have seen earlier that a category of categories exists. If we likewise hold the belief that everything is a category, we can also believe, to some extent, that given two categories \(\mathcal{A},\mathcal{B}\), there exists a category \(\Fun(\mathcal{A},\mathcal{B})\) of functors from \(\mathcal{A}\) to \(\mathcal{B}\). The question we must answer is: given two functors \(F,G:\mathcal{A}\rightarrow \mathcal{B}\), what is a morphism from \(F\) to \(G\)? That is precisely the natural transformation we define in this post.

Definition 1 Suppose two categories \(\mathcal{A},\mathcal{B}\) are given, and let \(F,G\) be two functors from \(\mathcal{A}\) to \(\mathcal{B}\). If a family of morphisms indexed by \(\obj(\mathcal{A})\)

\[\bigl(\alpha_A:F(A)\rightarrow G(A)\bigr)_{A\in\obj(\mathcal{A})}\]

makes the following diagram commute for each \(A_1,A_2\in\obj(\mathcal{A})\) and every \(f\in\Hom_\mathcal{A}(A_1,A_2)\)

then we call \(\alpha=(\alpha_A)_{A\in\obj(\mathcal{A})}\) a natural transformation and denote it by \(\alpha:F\Rightarrow G\).

If each \(\alpha_A\) is an isomorphism, we call this a natural isomorphism, and we say that the two functors \(F,G\) are naturally equivalent. We write this as \(F\simeq G\).

Based on this, we can define the functor category \(\Fun(\mathcal{A},\mathcal{B})\) from category \(\mathcal{A}\) to \(\mathcal{B}\). This is the category consisting of functors from \(\mathcal{A}\) to \(\mathcal{B}\), whose morphisms are natural transformations between functors. Composition is given componentwise by \((\beta\circ\alpha)_A=\beta_A\circ\alpha_A\), and the identity morphism of a functor \(F\) is given by \((\id_F)_A=\id_{F(A)}\). The isomorphisms in this category are given by natural isomorphisms.

Equivalent Categories

The notion of being equivalent, frequently used between categories, is not given by isomorphism in \(\Cat\). (§Functor, ⁋Definition 9) This is because isomorphism between categories is too strong a condition, so that even two categories that appear sufficiently similar are treated as different.

Definition 2 A functor \(F\) from a category \(\mathcal{A}\) to \(\mathcal{B}\) is called an equivalence of categories if there exists a functor \(G:\mathcal{B}\rightarrow \mathcal{A}\) such that \(\id_\mathcal{A}\simeq G\circ F\) and \(\id_\mathcal{B}\simeq F\circ G\). If there exists an equivalence from \(\mathcal{A}\) to \(\mathcal{B}\), we say that these two categories are equivalent and write \(\mathcal{A}\simeq\mathcal{B}\).

Let us examine in what sense this notion of equivalence between categories gives a sufficiently good notion of sameness. For this, we must first define the following.

Definition 3 A category \(\mathcal{A}\) is a skeletal category if, for any \(A\in\obj(\mathcal{A})\), the only object isomorphic to \(A\) in \(\mathcal{A}\) is itself.

Let \(\mathcal{A}\) be a small category. Then, after viewing mutually isomorphic objects in the set \(\obj(\mathcal{A})\) as the same, we can pick out only distinct ones to form a subset of \(\obj(\mathcal{A})\), namely \(\mathcal{S}\). For any \(S_1,S_2\in\mathcal{S}\), set \(\Hom_\mathcal{S}(S_1,S_2)=\Hom_\mathcal{A}(S_1,S_2)\). By definition, \(\mathcal{S}\) is a subcategory of \(\mathcal{A}\), and the trivially defined inclusion functor \(\mathcal{S}\hookrightarrow\mathcal{A}\) is a faithful functor. (§Category, ⁋Definition 5) If this functor is also full, we call \(\mathcal{S}\) a full subcategory. (§Functor, ⁋Definition 10)

When constructing from a small category \(\mathcal{A}\) a subcategory \(\mathcal{S}\) as in the preceding argument, it is a natural question whether \(\mathcal{S}\) has enough information to describe \(\mathcal{A}\). For instance, if in \(\mathcal{A}\) a morphism \(f:A_1\rightarrow A_2\) exists, but when choosing objects isomorphic to \(A_1,A_2\), say \(A_1',A_2'\), no morphism \(A_1'\rightarrow A_2'\) exists, then one could say that \(\mathcal{S}\) has lost the information that \(\mathcal{A}\) holds. However, a moment’s thought shows that this never happens. This is because whenever a morphism \(f:A_1\rightarrow A_2\) is given, we can compose the isomorphisms \(A_1'\rightarrow A_1\), \(A_2\rightarrow A_2'\) and \(f\) to produce \(A_1'\rightarrow A_2'\).

From this perspective, the category \(\mathcal{S}\) constructed above can be thought of as essentially containing all the information of \(\mathcal{A}\). Of course, \(\mathcal{S}\) itself will vary depending on which of the isomorphic objects we choose, but it is easy to prove that a category obtained from a different choice is also necessarily isomorphic to \(\mathcal{S}\).

Definition 4 A skeleton of a category \(\mathcal{A}\) is a full subcategory of \(\mathcal{A}\) that is a skeletal category such that every \(A\in\obj(\mathcal{A})\) is isomorphic to some object of that subcategory. We denote this by \(\sk(\mathcal{A})\).

The proof of the following theorem is long and tedious, so we do not write it down separately. However, a moment’s thought shows that no special idea is needed for this proof, and it is even quite obvious. In many cases, one takes this itself as the definition of an equivalence.

Theorem 5 A functor \(F:\mathcal{A}\rightarrow\mathcal{B}\) is an equivalence between categories if and only if \(F\) is a fully faithful functor and is essentially surjective in the following sense.

For every \(B\in\obj(\mathcal{B})\), there exists some \(A\in\obj(\mathcal{A})\) such that \(F(A)\cong B\).

Considering a skeleton of \(\mathcal{A}\), the inclusion functor \(\sk(\mathcal{A})\hookrightarrow\mathcal{A}\) is fully faithful because it is the inclusion of a full subcategory, and the last condition of Definition 4 is precisely that this functor is essentially surjective. Therefore, by Theorem 5, this inclusion is an equivalence, and \(\mathcal{A}\simeq\sk(\mathcal{A})\) holds. From this we obtain the following.

Corollary 6 Two small categories \(\mathcal{A}\) and \(\mathcal{B}\) are equivalent if and only if their skeletal subcategories \(\sk(\mathcal{A})\) and \(\sk(\mathcal{B})\) are isomorphic.

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References

[Rie] Emily Riehl. Category Theory in Context. Dover Publications, 2016.

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