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 previously seen that a category of categories exists. Likewise, if we adopt the belief that everything is a category, we can more or less believe that for two categories \(\mathcal{A},\mathcal{B}\), the category \(\Fun(\mathcal{A},\mathcal{B})\) of functors from \(\mathcal{A}\) to \(\mathcal{B}\) also exists. The question we must answer is then: given two functors \(F,G:\mathcal{A}\rightarrow \mathcal{B}\), what is a morphism from \(F\) to \(G\)? This is precisely the natural transformation that we define in this post.

Definition 1 Let two categories \(\mathcal{A},\mathcal{B}\) be 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 \(A_1,A_2\in\obj(\mathcal{A})\),

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, then we call this a natural isomorphism, and say that the two functors \(F,G\) are naturally equivalent. We denote this by \(F\simeq G\).

Based on this, we can define the functor category \(\Fun(\mathcal{A},\mathcal{B})\) from \(\mathcal{A}\) to \(\mathcal{B}\). This is the category consisting of functors from \(\mathcal{A}\) to \(\mathcal{B}\), whose morphisms are natural transformations between functors. Isomorphisms in this category are given by natural isomorphisms.

Equivalent Categories

The notion of equivalence frequently used between categories is not given by isomorphism in \(\Cat\). This is because isomorphism between categories is too strong a condition: two categories that appear sufficiently similar may still be regarded as different.

Definition 2 A functor \(F\) from \(\mathcal{A}\) 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 an equivalence from \(\mathcal{A}\) to \(\mathcal{B}\) exists, then these two categories are said to be equivalent, and we write \(\mathcal{A}\simeq\mathcal{B}\).

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

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

Let \(\mathcal{A}\) be a small category. Then we can take the set \(\obj(\mathcal{A})\), regard isomorphic objects as the same, and pick out only the distinct ones to 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 inclusion functor \(\mathcal{S}\hookrightarrow\mathcal{A}\) defined in the obvious way is a faithful functor. If this functor is also full, then we call \(\mathcal{S}\) a full subcategory.

As in the preceding argument, when we construct a subcategory \(\mathcal{S}\) from a small category \(\mathcal{A}\), it is natural to question 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 no morphism \(A_1'\rightarrow A_2'\) exists when we choose objects \(A_1',A_2'\) isomorphic to \(A_1,A_2\), then we could say that \(\mathcal{S}\) has lost information possessed by \(\mathcal{A}\). However, 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 point of view, the category \(\mathcal{S}\) constructed above can be thought of as essentially containing all the information of \(\mathcal{A}\). Of course, \(\mathcal{S}\) itself may differ depending on which object is chosen from each isomorphism class, but it is easy to prove that a category obtained from a different choice is necessarily isomorphic to \(\mathcal{S}\).

Definition 4 The skeleton of a category \(\mathcal{A}\) is a full subcategory of \(\mathcal{A}\) that is a skeletal category. We write this as \(\sk(\mathcal{A})\).

The proof of the following theorem is long and tedious, so we do not write it out separately. However, a little thought reveals that no new idea is needed for this proof, and it is even quite obvious. In many cases, one adopts this as the definition of equivalence outright.

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 essentially surjective in the following sense:

For each \(B\in\obj(\mathcal{B})\) there exists a suitable \(A\in\mathcal{A}\) such that \(F(A)\cong B\).

Accepting this, the following corollary is also obvious.

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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