Adjoint Functors

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

First, we begin with the definition.

Definition 1 Given two functors \(F: \mathcal{A}\rightarrow \mathcal{B}\), \(G: \mathcal{B}\rightarrow \mathcal{A}\), if the following isomorphism

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

natural in the two variables \(A,B\) is given, 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 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, 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 \(G:S' \rightarrow S\), \(F\dashv G\) means that the following equivalence

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

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

Unit and Counit

Now suppose that the following adjunction between two locally small categories \(\mathcal{A},\mathcal{B}\)

is given. Once we fix 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 \(\alpha_{F(A)}(\id_{F(A)})\) 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 starting with fixing an object \(B\) of \(\mathcal{B}\) is called the counit of this adjunction.

Conversely, it is known that if there exist 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, however, 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 two suitable categories \(\mathcal{A},\mathcal{B}\) and a forgetful functor \(U: \mathcal{A}\rightarrow \mathcal{B}\) are given. Then, if \(U\) has a left adjoint \(F:\mathcal{B}\rightarrow \mathcal{A}\), we call it a free functor. This means that the following equation

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

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 Suppose contravariant functors \(F:\mathcal{A} \rightarrow \mathcal{B}\) and \(G: \mathcal{B}\rightarrow \mathcal{A}\) are given. Then, if there exists the following isomorphism natural in the two variables \(A,B\)

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

they are said to be mutually left adjoint. Similarly, if there exists the following isomorphism natural in the two variables \(A,B\)

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

they are said to be mutually right adjoint.

Two-Variable Adjoint Functors

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

First, consider a bifunctor \(F: \mathcal{A}\times \mathcal{B}\rightarrow \mathcal{C}\). Then, fixing in \(\mathcal{A}\) an object \(A\), 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 following equation

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

holds. A similar argument can be obtained starting from \(F(-,B):\mathcal{A}\rightarrow \mathcal{C}\). The following Proposition 5 shows that if such a right adjoint exists for all \(A\in\obj(\mathcal{A})\), then we can combine the \(G_A\) 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 equation (1) above, \(G\) must be \(\mathcal{A}^\op\times \mathcal{C}\rightarrow \mathcal{B}\). This is because if \(A' \rightarrow A\) is given, then \(F(A',B)\rightarrow F(A,B)\) is given and thus the right-hand side becomes \(\Hom_\mathcal{C}(F(A',B),C)\), whereas on the left-hand side the term containing \(A\) is 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 form a unique bifunctor \(G:\mathcal{A}^\op\times \mathcal{C}\rightarrow \mathcal{B}\) such that the following 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 in 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 the following isomorphism

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

natural in all three variables, 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 by \(F(A,B)\cong F(B,A)\) given by the braiding, there exists a natural isomorphism between \(G\) and \(H\), but in general these two may differ.

Example 7 Consider in particular the cartesian monoidal category \((\Set,\times)\). (§Monoidal Categories, ⁋Example 2) Then one can see that 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 in this way 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 \(\times\). Generalizing this, we define in a monoidal category as follows.

Definition 8 Let a symmetric monoidal category \((\mathcal{A},\otimes)\) be given. Then, in \(\mathcal{A}\), an internal \(\Hom\) 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 explore examples of internal \(\Hom\) in due course, but even from the definition alone we 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 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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