Natural Transformations

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Definition of Natural Transformations

We have seen earlier that a category of categories exists. Keeping faith with the conviction that everything is a category, we can also believe, to some extent, that for 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 \(G\)? This is precisely what we define in this post as a natural transformation.

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 the 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 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 precisely the natural isomorphisms.

Equivalent Categories

The notion of being equivalent, which is 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: two categories that appear sufficiently similar may still be treated as different.

Definition 2 A functor \(F\) from a category \(\mathcal{A}\) to a category \(\mathcal{B}\) 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 furnishes a sufficiently good notion of sameness. To do so, we first need the following definition.

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

Let \(\mathcal{A}\) be a small category. Then from the set \(\obj(\mathcal{A})\), we may identify isomorphic objects as the same, pick out only the distinct ones, and form a subset \(\mathcal{S}\) of \(\obj(\mathcal{A})\). 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 obviously 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 we construct a subcategory \(\mathcal{S}\) from a small category \(\mathcal{A}\) as above, it is natural to ask whether \(\mathcal{S}\) retains enough information to describe \(\mathcal{A}\). For instance, if a morphism \(f:A_1\rightarrow A_2\) exists in \(\mathcal{A}\), but choosing objects \(A_1',A_2'\) isomorphic to \(A_1,A_2\) yields no morphism \(A_1'\rightarrow A_2'\), then one might say that \(\mathcal{S}\) has lost information present in \(\mathcal{A}\). But a moment’s thought shows that this can never happen: whenever a morphism \(f:A_1\rightarrow A_2\) is given, we can compose it with isomorphisms \(A_1'\rightarrow A_1\) and \(A_2\rightarrow A_2'\) to produce a morphism \(A_1'\rightarrow A_2'\).

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 representative we choose from each isomorphism class, but it is easy to prove that any category obtained from a different choice is necessarily isomorphic to \(\mathcal{S}\).

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

We omit the proof of the following theorem, as it is long and tedious. However, a moment’s thought reveals that no new ideas are needed for this proof, and it is in fact quite obvious. In many cases, one simply takes this 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 fully faithful and 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. Thus, 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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