nLab adjoint functor theorem

Idea

Adjoint functor theorems are theorems stating that under certain conditions a functor that preserves limits is a right adjoint, and that a functor that preserves colimits is a left adjoint.

A basic result of category theory is that right adjoint functors preserve all limits that exist in their domain, and, dually, left adjoints preserve all colimits. An adjoint functor theorem is a statement that (under certain conditions) the converse holds: a functor which preserves limits is a right adjoint.

The basic idea of an adjoint functor theorem is that if we could assume that a large category $D$ had all limits over small and large diagrams, then for $R : D \to C$ a functor that preserves all these limits we might define its would-be left adjoint $L$ by taking $L c$ to be the limit

$$L c \coloneqq \lim_{c\to R d} d$$

This notation stands for a limit over the comma category $c/R$ (whose objects are pairs $(d,f:c\to R d)$ and whose morphisms are arrows $d\to d'$ in $D$ making the obvious triangle commute in $C$) of the projection functor $\pi: c/R \to D$ that forgets the morphism $f$:

$$L c = \lim_{c\to R d} \pi \,.$$

Because with this definition there would be, for every $d$, an obvious morphism

$$L R d \stackrel{=}{\to} \lim_{R d \to R d'} d' \to d$$

(the component map over $d$ of the limiting cone). Moreover, because $R$ preserves limits, we would have an isomorphism

$$R L c \simeq \lim_{c\to R d} R d$$

(the limit of the functor $R \pi$), and hence an obvious morphism of cone tips

$$c \to R L c \,.$$

It is easy to check that these would be the unit and counit of an adjunction $L\dashv R$. See adjoint functor for more.

The problem with this would-be argument is that in general the comma category $(c/G)$ may not be small category. But one can generally not expect a large category to have all large limits: even if we pass to a universe in which $(c/G)$ is considered small, a classical theorem of Freyd says that any complete small category is a preorder (see complete small category for the proof, which is valid in classical logic and also holds classically in any Grothendieck topos). Thus, the argument we gave above is necessarily only an adjoint functor theorem for preorders:

Theorem

If $G:D\to C$ is any functor between (small) preorders such that $D$ has, and $G$ preserves, all small meets, then $G$ has a left adjoint.

(This theorem holds in constructive mathematics, although not in predicative mathematics; the classical reasoning before this explains why the theorem is not more general, but the proof itself is already constructive.)

To obtain adjoint functor theorems for categories that are not preorders, one must therefore impose various additional “size conditions” on the category $D$ and/or the functor $G$.

Statement

Theorem

Sufficient conditions for a limit-preserving functor $R : C \to D$ to be a right adjoint include:

• $C$ is complete and locally small, and $R$ satisfies the solution set condition.

  This is Freyd’s original version, sometimes called the “General Adjoint Functor Theorem”.

• $C$ is complete, locally small, well-powered, and has a small cogenerating set, and $D$ is locally small.

  This is sometimes called the “Special Adjoint Functor Theorem”, and abbreviated to SAFT.

• $C$ is locally small and cototal, and $D$ is locally small.

In the first two cases, which work by replacing large limits by small ones, it suffices to assume that $R$ preserves small limits (that it preserves all limits will follow). The third case works by assuming that $C$ has, while not all large limits, enough so that the theorem goes through; thus is this case $R$ must be already known to preserve large limits as well.

Proof

Here is a proof of the General Adjoint Functor Theorem: that a functor $R : C \to D$ out of a locally small category $C$ with all small limits has a left adjoint if it preserves these limits and satisfies the solution set condition.

From the discussion at adjoint functors – In terms of universal arrows we have that the existence of the adjoint is equivalent to the existence for each $d \in D$ of an initial object $i_d : d \to R L d$ in the comma category $(d \downarrow R)$: an object such that for each $f : d \to R d'$ there is a unique $\tilde f$ such that

$$\array{ && d \\ & {}^{\mathllap{i_d}}\swarrow && \searrow^{\mathrlap{f}} \\ R L d &&\underset{R \tilde f}{\to}&& R d' \\ \\ L d &&\underset{\tilde f}{\to}&& d' }$$

commutes. Now an initial object is the limit of the identity functor, but this is generally a large limit; we replace this with some small limit conditions that guarantee existence of an initial object.

1. Let $Y$ be a category. Call a small family of objects $F$ weakly initial if for every object $y$ of $Y$ there exists $x \in F$ and a morphism $f: x \to y$.

2. Suppose $Y$ has small products. If $F$ is a weakly initial family, then $\prod_{x \in F} x$ is a weakly initial object.

3. Claim: Suppose $Y$ is locally small and has joint equalizers of small families. If $x$ is a weakly initial object, then the domain $e$ of the joint equalizer $i: e \to x$ of all arrows $x \to x$ is an initial object. Proof: clearly $e$ is weakly initial. Suppose given an object $y$ and arrows $f, g: e \to y$; we must show $f = g$. Let $j: d \to e$ be the equalizer of $f$ and $g$. There exists an arrow $k: x \to d$. The arrow $i: e \to x$ equalizes $1_x$ and $i j k: x \to x$, so $i j k i = i$. Since $i$ is monic, $j (k i) = 1_e$. Thus $j$ is an epi, and $f = g$ follows.

If $C$ is locally small and small-complete and $R: C \to D$ preserves limits, then $d \downarrow R$ is locally small and small-complete for every object $d$ of $D$.

If in addition each $d \downarrow R$ has a weakly initial family (solution set condition), then by 2. and 3. each $d \downarrow R$ has an initial object. This restates the condition that $R$ has a left adjoint.

In fact, it suffices for $C$ to be Cauchy complete and for $R$ to be $\alpha$-flat for every $\alpha$ (i.e. for its Yoneda extension to be continuous). See Borceux.

Proof of SAFT

As before, the proof proceeds by constructing initial objects of comma categories. We assume that $C$ is locally small, small-complete, well-powered, has a cogenerating set $\{c_\alpha: \alpha \in A\}$, and that $R: C \to D$ is a small-continuous functor into a locally small category $D$.

As before, for each object $d$ of $D$, the comma category $d \downarrow R$ is locally small and small-complete. Moreover, it is easy to check that it is well-powered, and that the set of all objects of the form $d \to R c_\alpha$ is a cogenerating set for $d \downarrow R$.

It then remains to prove that any locally small, small-complete, well-powered category $X$ with a cogenerating set $\{k_s: s \in S\}$ has an initial object. The initial object $0$ is constructed as the intersection = pullback of all subobjects of $\prod_s k_s$, i.e., the minimal subobject. Then, given $f, g: 0 \to x$, the equalizer $Eq(f, g)$ is isomorphic to $0$ because $0$ is minimal, and so $f = g$: there is at most one arrow $0 \to x$ for each $x$.

On the other hand, for each $x$ the canonical map

$$i: x \to \prod_{s \in S} k_{s}^{\hom(x, k_s)}$$

is monic since the $k_s$ cogenerate. The following pullback of $i$,

$$\array{ k & \to & x \\ \downarrow & & \downarrow \mathrlap{i} \\ \prod_s k_{s}^1 & \stackrel{\prod_s k_{s}^!}{\to} & \prod_s k_{s}^{\hom(x, k_s)} },$$

gives a subobject $k$ of $\prod_s k_s$ that maps to $x$, and into which $0$ embeds. Thus there exists a map $0 \to x$, and we conclude $0$ is initial.

In locally presentable categories

In practice an important special case is that of functors between locally presentable categories. For these there is the following version of an adjoint functor theorem.

Theorem

Let $F \colon C \to D$ be a functor between locally presentable categories. Then

• $F$ has a right adjoint if and only if it preserves all small colimits.

• $F$ has a left adjoint if and only if it is an accessible functor and preserves all small limits.

The second statement, characterizing when $F$ has a left adjoint, is (AdamekRosicky, theorem 1.66). In the “if” direction, this is an application of the general adjoint functor theorem: any accessible functor satisfies the solution set condition. The “only if”, particularly that having a left adjoint forces accessibility, takes a little work. But in any case there are easy examples that show that continuity alone is insufficient, i.e., examples of continuous functors between locally presentable categories that do not have left adjoints. See below in the section In locally presentable categories.

The first statement, characterizing when $F$ has a right adjoint, can be proven using the special adjoint functor theorem: by a non-trivial theorem (AdamekRosicky, theorem 1.58), any locally presentable category is co-wellpowered.

Thus, the first statement can be strengthened by removing the assumption that $D$ is locally presentable: it is enough that $D$ be locally small. For example, if $C$ is locally presentable, then every continuous functor

$$C^{op} \to Set$$

has a left adjoint (is representable), because its opposite $C \to Set^{op}$ is cocontinuous and therefore has a right adjoint, even though $Set^{op}$ is not locally presentable.

A right adjoint to any cocontinuous functor $F \colon C \to D$ between locally presentable categories can also be constructed directly. If $C$ is locally $\lambda$-presentable and $P_\lambda$ is the subcategory of $\lambda$-small objects, then $C$ is equivalent to the full subcategory of $[P_\lambda^{op},Set]$ of presheaves that preserve $\lambda$-small limits (AdamekRosicky, theorem 1.46). The presheaves in the image of the functor $D \to [P_\lambda^{op},Set]$ defined by $d \mapsto \hom(F-,d)$ preserve $\lambda$-small limits because $F$ is cocontinuous. So this functor factors through the subcategory $C$. The functor $D \to C$ so-constructed is a right adjoint to $F$.

Examples

In locally presentable categories

The following is a counter-example, indicating the need for something more than just continuity to force a functor between locally presentable categories to be a right adjoint; as stated in theorem , the missing extra condition is precisely accessibility.

Example

For every infinite cardinal number $\kappa$, let $G_\kappa$ be a simple group of cardinality $\kappa$. Define the functor $ML:$ Group $\to$ Set to be the product of all the representable functors $Hom(G_\kappa,-)$. Since no group can admit a nontrivial homomorphism from proper-class-many of the $G_\kappa$, this functor does indeed land (or can be redefined to land) in Set. Since it is a product of representables, it is continuous (and of course Group and Set are locally presentable categories), but it is not itself representable (hence has no left adjoint).

André Joyal has been attributing this example to Saunders MacLane; it appears in print for instance right at the beginning of (AdámekKoubekTrnková01).

In cocomplete categories

Suppose $C$ and $D$ are categories that admit small colimits (i.e., are cocomplete) and $F\colon C\to D$ is a functor that preserves small colimits (i.e., is cocontinuous). The $F$ has a right adjoint if and only if for any object $d\in D$ the functor

$$C^{op} \to Set$$

that sends

$$c\mapsto D(F(c),d)$$

is a small presheaf on $C$.

See the MathOverflow answer by Ivan Di Liberti.

In toposes

Proposition

Every sheaf topos is a total category and a cototal category.

See the discussion at Grothendieck topos.

It follows that

Corollary

Let $F : C \to D$ be a functor between sheaf toposes. Then

• $F$ has a right adjoint precisely if it preserves all small colimits;

• $F$ has a left adjoint precisely if it preserves all small limits.

In presheaf categories

It is instructive to spell out the construction of the right adjoint from a colimit preserving functor $L$ in the simple case where all categories are categories of presheaves. This is a particularly simple case, but is useful in itself and serves as a template for the general case.

So let now $C$ and $D$ be small categories and $L$ a colimit-preserving functor between their categories of presheaves (which we abbreviate $\widehat C \;\coloneqq\; [C^{op}, Set]$, etc):

$$L \;\colon\; \widehat{C} \longrightarrow \widehat{D}$$

Then its right adjoint $R \;\colon\; \widehat{D} \longrightarrow \widehat{C}$ is given (with $y_c \coloneqq C(-, c)$ denoting the Yoneda embedding) by

(1)$$R \;\colon\; A \;\mapsto\; R(A) \;\coloneqq\; \widehat{D} \left( L(y_{(-)}), A \right) \,,$$

as we shall check in a moment. But first notice that using the co-Yoneda lemma this may be rewritten as

$$\cdots \;\simeq\; \int^{c \in C} \widehat{D} \left( L(y_c), A \right) \cdot y_c \,,$$

where the coend is equivalently given by the colimit

$$\cdots \;=\; \lim_{\underset{L c \to A}{\to}} c \,.$$

This is the formula for the would-be right adjoint from the general discussion above, only that here the colimit is only over the representables, hence over a small category.

Now we check that the functor $R$ thus obtained is indeed right adjoint to $L$, by explicitly checking the hom-isomorphism (here) of the pair of adjoint functors:

We compute $\widehat{D}(L(X),A)$. In the first step

$$\widehat{D}(L(X), A) \simeq \widehat{D} \left( L \left( \int^{c \in C} X(c) \cdot y_c \right), A \right)$$

we use the co-Yoneda lemma for $X$. Then, because $L$ preserves colimits, this is

$$\cdots \simeq \widehat{D} \left( \int^c X(c) \cdot L(y_c), A \right) \,.$$

Since the hom-functor preserves limits in both arguments, we can take the coend out to get an end

$$\cdots \simeq \int_{c \in C} \widehat{D}(X(c) \cdot L(y_c), A) \,.$$

Then we use the standard tensoring of our categories over Set to get

$$\cdots \simeq \int_{c \in C} Set\left( X(c), \widehat{D}(L(y_c),A) \right) \,.$$

And finally this is recognized as the formula for the hom-set of presheaves (see the discussion at functor category):

$$\cdots \simeq \widehat{C} \left( X, \widehat{D}(L(y_{(-)}),A) \right) \;=\; \widehat{C}(X, R A) \,,$$

where on the right we identified (1).

The composition of this sequence of natural isomorphisms is hence the desired hom-isomorphism:

$$\widehat{D}(L(X), A) \simeq \widehat{C}(X, R(A)) \,.$$

Related concepts

• (co)continuous functor

• indexed adjoint functor theorem

• adjoint (∞,1)-functor theorem

• adjoint triangle theorem

• adjoint lifting theorem

References

The classical adjoint functor theorems originate in the exercise section of ch.3 (pp.84ff) in

• Peter Freyd, Abelian Categories, Harper & Row New York 1964. (Reprinted with author’s comment as TAC reprints no. 3 (2003))

A more recent exposition is in

• Peter Freyd, Andre Scedrov, Categories, Allegories, North-Holland, 1990. (Also Dover reprint New York 2014, pp.144-148)

where the solution set condition is called “pre-adjointness”.

Careful discussions can be found in

• Francis Borceux, Handbook of Categorical Algebra I, Cambridge UP, 1994. (sections 3.3, 6.6)

• Saunders MacLane, §V.6, V.8 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

A brief introductory discussion is around theorem 5.4 of

• Jaap van Oosten, Basic category theory, ms. (pdf)

A detailed expository survey is

• Oliver Kullmann, The adjoint functor theorem, ms. (pdf slides)

The adjoint functor theorem in context with Yoneda embedding is discussed in

• Friedrich Ulmer, The adjoint functor theorem and the Yoneda embedding, Illinois J. Math. 15 no.3 (1971), pp. 355-361. (euclid)

The connection between the solution set condition and the Čech homology construction is discussed in

• Renato Betti, Čech methods and the adjoint functor theorem , Cah. Top. Géom. Diff. Cat. XXVI no.3 (1985) pp.245-257. (numdam)

An enriched adjoint functor theorem is given in:

• Francis Borceux. Limites enrichies et existence de $V$-foncteur adjoint. Cahiers de topologie et géométrie différentielle catégoriques 16.4 (1975): 395-408. (pdf)

The parallels between the adjoint functor theorem for categories and the computation of colimits from limits in a lattice as well as similar parallels between co/completeness for Boolean algebras and Paré’s theorem for finite completeness of toposes is studied from a type-theoretical perspective in

• Duško Pavlović, On completeness and cocompleteness in and around small categories , APAL 74 (1995) pp.121-152.

The case for locally presentable categories is discussed in

• Jiří Adámek, Jiri Rosicky, Locally presentable and accessible categories, Cambridge UP, 1994.

• Jiří Adámek, V. Koubek and V. Trnková, How large are left exact functors?, Theory and Applications of Categories 8 (2001), pp. 377-390. (abstract)

A specialization to the locally $\alpha$-presentable case is given in Theorem 2.11 of

• Giacomo Tendas, On continuity of accessible functors, Applied Categorical Structures, 2022, doi.

A relative version of Freyd’s classical results is in

• Brian Day, An adjoint-functor theorem over topoi, Bull. Austral. Math. Soc. 15 (1976), pp. 381-394.

Adjoint functor theorems for indexed categories are discussed in

• Robert Paré, Dietmar Schumacher, Abstract Families and the Adjoint Functor Theorems , pp.1-125 in LNM 661 Springer Heidelberg 1978.

A history of the general adjoint functor theorem is discussed in:

• Hans-E. Porst, The history of the General Adjoint Functor Theorem [arxiv:2310.19528]

A stronger version for finitary functors between locally finitely presentable categories whose domain is ranked, requiring only the preservation of countable limits for the existence of a left adjoint, is discussed in

• Jiří Adámek, Lurdes Sousa, A Finitary Adjoint Functor Theorem, arXiv.