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

Definition of Adjoint Functors

We begin with the definition.

Definition 1 Given two functors \(F: \mathcal{A}\rightarrow \mathcal{B}\), \(G: \mathcal{B}\rightarrow \mathcal{A}\), if for variables \(A,B\) there is a natural isomorphism

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

then we call the pair \((F,G)\) an adjunction, \(F\) the left adjoint of \(G\), and \(G\) the right adjoint of \(F\). We write this as \(F\dashv G\).

Example 2 Any preordered set \((S,\preceq)\) can always be viewed as a category. (§Category, ⁋Example 3) Since the morphisms of this category are exactly the preorder \(\preceq\), 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 \(S'\).

Now for two functors \(F: S \rightarrow S'\) and \(G:S' \rightarrow S\), the statement \(F\dashv G\) means that the following identity

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

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

Unit and Counit

Now consider an adjunction between two locally small categories \(\mathcal{A},\mathcal{B}\)

Fix an object \(A\) of \(\mathcal{A}\). Then 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 \(\alpha_{F(A)}(\id_{F(A)})\) of \(\Hom_\mathcal{A}(A, GF(A))\). 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, the natural transformation \(\epsilon:FG \rightarrow \id_\mathcal{B}\) obtained by fixing an object \(B\) of \(\mathcal{B}\) is called the counit of this adjunction.

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

then this carries exactly the same information as the adjunction of Definition 1.

Free Functor and Forgetful Functor

To examine the adjunction we will look at in Example 3 (Free \(\dashv\) Forgetful), we first briefly explain what a forgetful functor is. In §Category, ⁋Example 2 (Concrete categories) we saw that many mathematical structures we already knew each form their own category. Looking at these examples, we see that some mathematical structures are obtained by adding extra data to other mathematical structures. For instance, a monoid is obtained from a set \(S\) by adding information about 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 viewed as a monoid, and a monoid \(M\) can always be viewed as a set. We can verify 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) Suppose we are given suitable categories \(\mathcal{A},\mathcal{B}\) and a forgetful functor \(U: \mathcal{A}\rightarrow \mathcal{B}\). Then if the left adjoint \(F:\mathcal{B}\rightarrow \mathcal{A}\) of \(U\) exists, we call it a free functor. This means that the following identity

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

holds.

• The forgetful functor \(U:\Grp \rightarrow \Set\) has a left adjoint \(F:\Set \rightarrow \Grp\). For any set \(X\), we call \(F(X)\) a free group.
• The forgetful functor \(U:\Ab \rightarrow \Set\) has a left adjoint \(F:\Set \rightarrow \Ab\). For any set \(X\), we call \(F(X)\) a 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 for 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 for 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 this \(F(A,-)\) has a right adjoint \(G_A: \mathcal{C}\rightarrow \mathcal{B}\). That is, the following identity

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

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

Since there is much to cover in this post, we will not prove the following proposition, but we can see that in order to give naturality to the above equation (1), \(G\) 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, so the right-hand side becomes \(\Hom_\mathcal{C}(F(A',B),C)\), whereas on the left-hand side the term containing \(A\) appears in the target part of \(\Hom_\mathcal{B}\).

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

1. If for all \(A\), \(F(A,-):\mathcal{B}\rightarrow \mathcal{C}\) has a right adjoint \(G_A\), then we can combine them to make 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 all \(B\), \(F(-,B):\mathcal{A}\rightarrow \mathcal{C}\) has a right adjoint \(H_B\), and therefore by the above result a bifunctor \(H:\mathcal{B}^\op\times \mathcal{C}\rightarrow \mathcal{A}\) is defined, then from the two natural isomorphisms

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

   and

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

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

Through this proposition, we can guess how to define an adjunction involving a bifunctor.

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 \(G,H\) the left, right closure of \(F\) respectively, and when both closures exist we call \(F\) closed. If \(F\) gives a symmetric monoidal structure, then the braiding gives a natural isomorphism between \(G\) and \(H\) via \(F(A,B)\cong F(B,A)\), but in general these two may differ.

Example 7 Consider in particular the cartesian monoidal category \((\Set,\times)\). (§Monoidal Categories, ⁋Example 2) Then the following isomorphism ([Set Theory] §Product of Sets, ⁋Proposition 4)

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

is natural in each component. This means that \(\times\) forms a two-variable adjunction as in Definition 6 with left closure and right closure given by \(G(A,C)=\Hom_\Set(A,C)\) and \(H(B,C)=\Hom_\Set(B,C)\) respectively, so \((\Set,\times)\) is closed. A cartesian monoidal category whose monoidal product is closed is called a cartesian closed category. Here, because of the symmetry of the product, both closures coincide as \(\Hom_\Set(-,-)\).

Internal \(\Hom\)

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

Definition 8 Let a symmetric monoidal category \((\mathcal{A},\otimes)\) be given. Then the 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 from the definition we can find points that differ 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 gives 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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