Adjoint Functors

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.

In this post we study adjoint functors. They may seem somewhat non-obvious if one learns them only in category theory, but through examples from various fields one can see that they are a very important concept.

Definition of Adjoint Functors

We begin with the definition.

Definition 1 Let two functors \(F: \mathcal{A}\rightarrow \mathcal{B}\), \(G: \mathcal{B}\rightarrow \mathcal{A}\) be given, and suppose there is a natural isomorphism in two variables \(A,B\)

\[\Hom_\mathcal{B}(F(A), B)\cong \Hom_\mathcal{A}(A, G(B))\]

Then we call this pair \((F,G)\) an adjunction, \(F\) the left adjoint of \(G\), and \(G\) the right adjoint of \(F\). We denote this by \(F\dashv G\).

Example 2 Any preordered set \((S,\preceq)\) can always be regarded as a category. (§Categories, ⁋Example 3) Since the morphisms of this category are exactly the preorder relation \(\preceq\), if we view two preordered sets \(S,S'\) as categories then a functor \(F:S \rightarrow S'\) between them is nothing more than an increasing function from the set \(S\) to \(S'\).

Then for two functors \(F: S \rightarrow S'\) and \(G:S' \rightarrow S\), the statement \(F\dashv G\) means that the formula

\[F(x)\preceq y\iff x\preceq G(y)\]

holds. That is, they form a monotone Galois connection. (§Filters, Ideals, and Galois Connections, ⁋Definition 6)

Unit and Counit

Now suppose the following adjunction is given:

Fixing an object \(A\) of \(\mathcal{A}\), by the isomorphism of Definition 1 the following functor

\[\Hom_\mathcal{A}(A, G-): \mathcal{B}\rightarrow\Set\]

is a representable functor, and the natural isomorphism

\[\Hom_\mathcal{A}(A, G-)\cong\Hom_\mathcal{B}(F(A), -)\]

is a representation of this functor. On the other hand, by §Representable Functors, ⁋Theorem 3 (Yoneda), we know that a natural transformation \(\alpha\) from \(\Hom_\mathcal{B}(F(A),-)\) to \(\Hom_\mathcal{A}(A,G-)\) is always determined by the element \(\alpha_{F(A)}(\id_{F(A)})\). That is, in our situation the above natural isomorphism is determined by an element \(\eta_A\) of \(\Hom_\mathcal{A}(A, GF(A))\), and from the naturality of the above isomorphism we know that this family \((\eta_A)_{A\in\obj(\mathcal{A})}\) defines a natural transformation from \(\id_\mathcal{A}\) to \(GF\). We call this the unit of this adjunction. Similarly, fixing an object \(B\) of \(\mathcal{B}\) and proceeding from there, the natural transformation \(\epsilon:FG \rightarrow \id_\mathcal{B}\) obtained is called the counit of this adjunction.

Conversely, if there exist two natural transformations \(\eta:\id_\mathcal{A}\Rightarrow GF\), \(\epsilon: FG\Rightarrow\id_\mathcal{B}\) satisfying the following triangle identities

then it is known that they carry exactly the same information as the adjunction of Definition 1.

Free and Forgetful Functors

To examine the adjunction we will consider in Example 3 (Free \(\dashv\) Forgetful), we first briefly explain what a forgetful functor is. In §Categories, ⁋Example 2 (Concrete categories) we saw that many mathematical structures we already know each form their own category. However, looking at these examples, some mathematical structures are obtained by adding additional data to other mathematical structures. For example, a monoid is obtained by equipping a set \(S\) with a binary operation satisfying associativity and an identity element, and if this binary operation also has inverses then it becomes a group. Conversely, a group \(G\) can always be regarded as a monoid, and a monoid \(M\) can always be regarded as a set. One can check that these examples define functors \(\Grp \rightarrow\Mon\) and \(\Mon \rightarrow \Set\). We call functors of this kind forgetful functors.

Example 3 (Free \(\dashv\) Forgetful) Let two appropriate categories \(\mathcal{A},\mathcal{B}\) and a forgetful functor \(U: \mathcal{A}\rightarrow \mathcal{B}\) be given. Then if the left adjoint \(F:\mathcal{B}\rightarrow \mathcal{A}\) of \(U\) exists, we call it a free functor. This means that the formula

\[\Hom_\mathcal{A}(F(A),B)\cong\Hom_\mathcal{B}(A, U(B))\]

holds.

• The forgetful functor \(U:\Grp \rightarrow \Set\) has a left adjoint \(F:\Set \rightarrow \Grp\). For any set \(X\), we call \(F(X)\) the free group.
• The forgetful functor \(U:\Ab \rightarrow \Set\) has a left adjoint \(F:\Set \rightarrow \Ab\). For any set \(X\), we call \(F(X)\) the free abelian group.

Contravariant Adjoint Functors

Meanwhile, up to now we have only considered adjoint functors for covariant functors, but adjoint functors are also well defined for contravariant functors.

Definition 4 Let contravariant functors \(F:\mathcal{A} \rightarrow \mathcal{B}\) and \(G: \mathcal{B}\rightarrow \mathcal{A}\) be given. Then if there exists a natural isomorphism in two variables \(A,B\)

\[\Hom_\mathcal{B}(F(A), B)\cong\Hom_\mathcal{A}(G(B), A)\]

we say that they are mutually left adjoint. Similarly, if there exists a natural isomorphism in two variables \(A,B\)

\[\Hom_\mathcal{B}(B, F(A))\cong\Hom_\mathcal{A}(A, G(B))\]

we say that they are mutually right adjoint.

Two-Variable Adjoint Functors

Meanwhile, we can extend the notion of adjoint functors to multiple variables.

First consider a bifunctor \(F: \mathcal{A}\times \mathcal{B}\rightarrow \mathcal{C}\). Then fixing an object \(A\) of \(\mathcal{A}\), we obtain a functor

\[F(A,-):\mathcal{B}\rightarrow \mathcal{C}\]

Suppose that \(F(A,-)\) defined in this way has a right adjoint \(G_A: \mathcal{C}\rightarrow \mathcal{B}\). That is, the formula

\[\Hom_\mathcal{B}(B, G_A(C))\cong \Hom_\mathcal{C}(F(A, B), C)\tag{1}\]

holds. A similar argument is obtained starting from \(F(-,B):\mathcal{A}\rightarrow \mathcal{C}\). The following Proposition 5 shows that if such a right adjoint exists for every \(A\in\obj(\mathcal{A})\), then we can combine the \(G_A\)’s into a single bifunctor \(G\).

In this post we have too much content to cover, so we do not prove the following proposition, but one can see that in order to endow the above formula (1) with naturality, \(G\) must be \(\mathcal{A}^\op\times \mathcal{C}\rightarrow \mathcal{B}\). If \(A' \rightarrow A\) is given, then \(F(A',B)\rightarrow F(A,B)\) is given and therefore the right-hand side becomes \(\Hom_\mathcal{C}(F(A',B),C)\), whereas on the left-hand side \(A\) appears in the target of \(\Hom_\mathcal{B}\).

Proposition 5 Fix a bifunctor \(F: \mathcal{A}\times \mathcal{B}\rightarrow \mathcal{C}\).

1. If for every \(A\), \(F(A,-):\mathcal{B}\rightarrow \mathcal{C}\) has a right adjoint \(G_A\), then we can collect them to construct a unique bifunctor \(G:\mathcal{A}^\op\times \mathcal{C}\rightarrow \mathcal{B}\) such that the isomorphism

   \[\Hom_\mathcal{C}(F(A,B),C)\cong\Hom_\mathcal{B}(B, G(A,C))\]

   is natural in all three variables.

2. If for every \(B\), \(F(-,B):\mathcal{A}\rightarrow \mathcal{C}\) has a right adjoint \(H_A\), and therefore by the above result a bifunctor \(H:\mathcal{B}^\op\times \mathcal{C}\rightarrow \mathcal{A}\) is defined, then the two natural isomorphisms

   \[\Hom_\mathcal{C}(F(A,B),C)\cong\Hom_\mathcal{B}(B, G(A,C))\]

   and

   \[\Hom_\mathcal{C}(F(A,B),C)\cong\Hom_\mathcal{A}(A, H(B,C))\]

   imply that the two contravariant functors \(G(-,C): \mathcal{B} \rightarrow \mathcal{C}\) and \(H(-,C):\mathcal{A}\rightarrow \mathcal{C}\) on the right-hand sides are mutually right adjoint.

Through this proposition, one can guess how an adjunction involving a bifunctor should be defined.

Definition 6 Let bifunctors

\[F: \mathcal{A}\times \mathcal{B}\rightarrow \mathcal{C},\quad G: \mathcal{A}^\op\times \mathcal{C}\rightarrow \mathcal{B},\quad H:\mathcal{B}^\op\times \mathcal{C}\rightarrow \mathcal{A}\]

be given. Then if there exists a natural isomorphism in all three variables

\[\Hom_\mathcal{C}(F(A,B), C)\cong \Hom_\mathcal{B}(B, G(A, C))\cong\Hom_\mathcal{A}(A, H(B, C))\]

we call this a two-variable adjunction.

If \(\mathcal{A}=\mathcal{B}=\mathcal{C}\) and \(F:\mathcal{A}\times \mathcal{A}\rightarrow \mathcal{A}\) gives a monoidal product structure, then we call \(G\) and \(H\) the left closure and right closure of \(F\), respectively. Also, if \(G\) and \(H\) are isomorphic, we call \(F\) closed.

Example 7 In particular, consider the cartesian monoidal category \((\Set,\times)\). (§Monoidal Categories, ⁋Example 2) Then the following isomorphism (§Product of Sets, ⁋Proposition 4)

\[\Hom_\Set(A\times B, C)\cong\Hom_\Set(A,\Hom_\Set(B,C))\cong\Hom_\Set(B,\Hom_\Set(A,C))\]

is natural in each component. Moreover, since the left closure and right closure are both the same \(\Hom_\Set(-,-)\), \(\Set\) is a cartesian closed category.

Internal \(\Hom\)

In Example 7 we were able to interpret the bifunctor \(\Hom_\Set\) as a right adjoint to \(\times\). Generalizing this, in a monoidal category we define as follows.

Definition 8 Let a symmetric monoidal category \((\mathcal{A},\otimes)\) be given. Then an internal \(\Hom\) in \(\mathcal{A}\) is a bifunctor \([-,-]:\mathcal{A}^\op\times \mathcal{A}\rightarrow \mathcal{A}\) such that for any object \(A\in\obj(\mathcal{A})\) the following adjunction

\[(-\otimes A)\dashv [A,-]\]

holds. A symmetric monoidal category in which an internal \(\Hom\) exists is called a closed symmetric monoidal category.

We will slowly look at examples of internal \(\Hom\) in the future, but even just looking at the definition one can find differences from the usual \(\Hom\). In general, taking \(\Hom\) in a (locally small) category gives a bifunctor

\[\Hom_\mathcal{A}(-,-):\mathcal{A}^\op\times \mathcal{A}\rightarrow \Set\]

but in the case of internal \(\Hom\), the target category of this bifunctor is not \(\Set\) but \(\mathcal{A}\) itself, so it will contain much richer information. More generally, replacing the target category with an arbitrary closed symmetric monoidal category yields the notion of an enriched category.

Limits and Adjoints

Finally, we introduce the following theorem, again without proof.

Theorem 9 Let an adjunction \(F\dashv G\) be given. Then

1. \(F\) preserves colimits.
2. \(G\) preserves limits.

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References

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

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