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 had all limits over small and large diagrams, then for a functor that preserves all these limits we might define its would-be left adjoint by taking to be the limit
over the comma category (whose objects are pairs and whose morphisms are arrows in making the obvious triangle commute in ) of the projection functor :
Because with this definition there would be for every an obvious morphism
and hence an obvious morphism of cone tips
The problem with this would-be argument is that in general the comma category 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 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 an adjoint functor theorem for preorders:
If is any functor between (small) preorders such that has, and preserves, all small meets, then 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 and/or the functor .
Sufficient conditions for a limit-preserving functor to be a right adjoint include:
This is Freyd’s original version, sometimes called the “General Adjoint Functor Theorem”.
This is sometimes called the “Special Adjoint Functor Theorem”, and abbreviated to SAFT.
is locally small and cototal, and is locally small.
In the first two cases, which work by replacing large limits by small ones, it suffices to assume that preserves small limits (that it preserves all limits will follow). The third case works by assuming that has, while not all large limits, enough so that the theorem goes through; thus is this case must be already known to preserve large limits as well.
Here is a proof of the General Adjoint Functor Theorem: that a functor out of a locally small category 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 of an initial object in the comma category : an object such that for each there is a unique such that
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.
Let be a category. Call a small family of objects weakly initial if for every object of there exists and a morphism .
Suppose has small products. If is a weakly initial family, then is a weakly initial object.
Claim: Suppose is locally small and small complete. If is a weakly initial object, then the domain of the joint equalizer of all arrows is an initial object. Proof: clearly is weakly initial. Suppose given an object and arrows ; we must show . Let be the equalizer of and . There exists an arrow . The arrow equalizes and , so . Since is monic, . Thus is an epi, and follows.
If is locally small and small-complete and preserves limits, then is locally small and small-complete for every object of .
If in addition each has a weakly initial family (solution set condition), then by 2. and 3. each has an initial object. This restates the condition that has a left adjoint.
As before, the proof proceeds by constructing initial objects of comma categories. We assume that is locally small, small-complete, well-powered, has a cogenerating set , and that is a small-continuous functor into a locally small category .
As before, for each object of , the comma category 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 is a cogenerating set for .
It then remains to prove that any locally small, small-complete, well-powered category with a cogenerating set has an initial object. The initial object is constructed as the intersection = pullback of all subobjects of , i.e., the minimal subobject. Then, given , the equalizer is isomorphic to because is minimal, and so : there is at most one arrow for each .
On the other hand, for each the canonical map
is monic since the cogenerate. The following pullback of ,
gives a subobject of that maps to , and into which embeds. Thus there exists a map , and we conclude is initial.
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.
Let be a functor between locally presentable categories. Then
The second statement, characterizing when has a left adjoint, is (AdamekRosicky, theorem 1.66). This is an application of the general adjoint functor theorem: any accessible functor satisfies the solution set condition. The accessibility condition is indeed necessary, as is discussed below in the section In locally presentable categories.
The first statement, characterizing when 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.
A right adjoint to any cocontinuous functor between locally presentable categories can also be constructed directly. If is locally -presentable and is the subcategory of -small objects, then is equivalent to the full subcategory of of presheaves that preserve -small limits (AdamekRosicky, theorem 1.46). The presheaves in the image of the functor defined by preserve -small limits because is cocontinuous. So this functor factors through the subcategory . The functor so-constructed is a right adjoint to .
The following is a counter-example, showing that the necessity of the accessibility clause in theorem 3.
For every infinite cardinal number , let be a simple group of cardinality . Define the functor Group Set to be the product of all the representable functors . Since no group can admit a nontrivial homomorphism from proper-class-many of the , 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).
See the discussion at Grothendieck topos.
It follows that
It is instructive to spell out the construction of the right adjoint from a colimit preserving functor 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 and be small categories and
a colimit-preserving functor. Then its right adjoint is given by
as we shall check in a moment. But first notice that using the co-Yoneda lemma this may be rewritten as
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 thus obtained is indeed right adjoint to , by explicitly checking the hom-isomorphism of the pair of adjoint functors:
We compute . In the first step
we use the co-Yoneda lemma for . Then because preserves colimits this is
In total this establishes the hom-isomorphism
The classical adjoint functor theorems originate in the exercise section of ch.3 (pp.84ff) in
A more recent exposition is in
Careful discussions can be found in
Francis Borceux, Handbook of Categorical Algebra I, Cambridge UP, 1994. (sections 3.3, 6.6)
Saunders Mac Lane, Categories for the Working Mathematician, Springer Heidelberg, 1998². (sections V.6, V.8)
A brief introductory discussion is around theorem 5.4 of
A detailed expository survey is
The adjoint functor theorem in context with Yoneda embedding is discussed in
The case for locally presentable categories is discussed in
A relative version of Freyd’s classical results is in