adjoint functor

In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

By definition, an adjunction between categories $\mathcal$ and $\mathcal$ is a pair of functors (assumed to be covariant)

:$F: \mathcal \rightarrow \mathcal$   and   $G: \mathcal \rightarrow \mathcal$

and, for all objects $X$ in $\mathcal$ and $Y$ in $\mathcal$ a bijection between the respective morphism sets

:$\mathrm_(FY,X) \cong \mathrm_(Y,GX)$

such that this family of bijections is natural in $X$ and $Y$. Naturality here means that there are natural isomorphisms between the pair of functors $\mathcal(F-,X) : \mathcal \to \mathrm$ and $\mathcal(-,GX) : \mathcal \to \mathrm$ for a fixed $X$ in $\mathcal$, and also the pair of functors $\mathcal(FY,-) : \mathcal \to \mathrm$ and $\mathcal(Y,G-) : \mathcal \to \mathrm$ for a fixed $Y$ in $\mathcal$.

The functor $F$ is called a left adjoint functor or left adjoint to $G$, while $G$ is called a right adjoint functor or right adjoint to $F$.

An adjunction between categories $\mathcal$ and $\mathcal$ is somewhat akin to a "weak form" of an equivalence between $\mathcal$ and $\mathcal$, and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.

Terminology and notation

The terms '' adjoint'' and ''adjunct'' are both used, and are cognates: one is taken directly from Latin, the other from Latin via French. In the classic text ''Categories for the working mathematician'', Mac Lane makes a distinction between the two. Given a family

: $\varphi_: \mathrm_(FY,X) \cong \mathrm_(Y,GX)$

of hom-set bijections, we call $\varphi$ an adjunction or an adjunction between $F$ and $G$. If $f$ is an arrow in $\mathrm_(FY,X)$, $\varphi f$ is the right adjunct of $f$ (p. 81). The functor $F$ is left adjoint to $G$, and $G$ is right adjoint to $F$. (Note that $G$ may have itself a right adjoint that is quite different from $F$; see below for an example.)

In general, the phrases "$F$ is a left adjoint" and "$F$ has a right adjoint" are equivalent. We call $F$ a left adjoint because it is applied to the left argument of $\mathrm_$, and $G$ a right adjoint because it is applied to the right argument of $\mathrm_$.

If ''F'' is left adjoint to ''G'', we also write
:$F\dashv G.$

The terminology comes from the Hilbert space idea of adjoint operators $T$, $U$ with $\langle Ty,x\rangle = \langle y,Ux\rangle$, which is formally similar to the above relation between hom-sets. The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts.

Introduction and Motivation

Common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preserve colimits/limits (which are also found in every area of mathematics), and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.

Solutions to optimization problems

In a sense, an adjoint functor is a way of giving the ''most efficient'' solution to some problem via a method which is ''formulaic''. For example, an elementary problem in ring theory is how to turn a rng (which is like a ring that might not have a multiplicative identity) into a ring. The ''most efficient'' way is to adjoin an element '1' to the rng, adjoin all (and only) the elements which are necessary for satisfying the ring axioms (e.g. ''r''+1 for each ''r'' in the ring), and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is ''formulaic'' in the sense that it works in essentially the same way for any rng.

This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction is ''most efficient'' if it satisfies a universal property, and is ''formulaic'' if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual notions, it is only necessary to discuss one of them.

The idea of using an initial property is to set up the problem in terms of some auxiliary category ''E'', so that the problem at hand corresponds to finding an initial object of ''E''. This has an advantage that the ''optimization''—the sense that the process finds the ''most efficient'' solution—means something rigorous and recognisable, rather like the attainment of a supremum. The category ''E'' is also formulaic in this construction, since it is always the category of elements of the functor to which one is constructing an adjoint.

Back to our example: take the given rng ''R'', and make a category ''E'' whose ''objects'' are rng homomorphisms ''R'' → ''S'', with ''S'' a ring having a multiplicative identity. The ''morphisms'' in ''E'' between ''R'' → ''S''₁ and ''R'' → ''S''₂ are commutative triangles of the form (''R'' → ''S''₁, ''R'' → ''S''₂, ''S''₁ → ''S''₂) where S₁ → S₂ is a ring map (which preserves the identity). (Note that this is precisely the definition of the comma category of ''R'' over the inclusion of unitary rings into rng.) The existence of a morphism between ''R'' → ''S''₁ and ''R'' → ''S''₂ implies that ''S''₁ is at least as efficient a solution as ''S''₂ to our problem: ''S''₂ can have more adjoined elements and/or more relations not imposed by axioms than ''S''₁.
Therefore, the assertion that an object ''R'' → ''R*'' is initial in ''E'', that is, that there is a morphism from it to any other element of ''E'', means that the ring ''R''* is a ''most efficient'' solution to our problem.

The two facts that this method of turning rngs into rings is ''most efficient'' and ''formulaic'' can be expressed simultaneously by saying that it defines an ''adjoint functor''. More explicitly: Let ''F'' denote the above process of adjoining an identity to a rng, so ''F''(''R'')=''R*''. Let ''G'' denote the process of “forgetting″ whether a ring ''S'' has an identity and considering it simply as a rng, so essentially ''G''(''S'')=''S''. Then ''F'' is the ''left adjoint functor'' of ''G''.

Note however that we haven't actually constructed ''R*'' yet; it is an important and not altogether trivial algebraic fact that such a left adjoint functor ''R'' → ''R*'' actually exists.

Symmetry of optimization problems

It is also possible to ''start'' with the functor ''F'', and pose the following (vague) question: is there a problem to which ''F'' is the most efficient solution?

The notion that ''F'' is the ''most efficient solution'' to the problem posed by ''G'' is, in a certain rigorous sense, equivalent to the notion that ''G'' poses the ''most difficult problem'' that ''F'' solves.

This gives the intuition behind the fact that adjoint functors occur in pairs: if ''F'' is left adjoint to ''G'', then ''G'' is right adjoint to ''F''.

Formal definitions

There are various equivalent definitions for adjoint functors:

* The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving optimizations.
* The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word ''adjoint''.
* The definition via counit–unit adjunction is convenient for proofs about functors which are known to be adjoint, because they provide formulas that can be directly manipulated.

The equivalency of these definitions is quite useful. Adjoint functors arise everywhere, in all areas of mathematics. Since the structure in any of these definitions gives rise to the structures in the others, switching between them makes implicit use of many details that would otherwise have to be repeated separately in every subject area.

Conventions

The theory of adjoints has the terms ''left'' and ''right'' at its foundation, and there are many components which live in one of two categories ''C'' and ''D'' which are under consideration. Therefore it can be helpful to choose letters in alphabetical order according to whether they live in the "lefthand" category ''C'' or the "righthand" category ''D'', and also to write them down in this order whenever possible.

In this article for example, the letters ''X'', ''F'', ''f'', ε will consistently denote things which live in the category ''C'', the letters ''Y'', ''G'', ''g'', η will consistently denote things which live in the category ''D'', and whenever possible such things will be referred to in order from left to right (a functor ''F'' : ''D'' → ''C'' can be thought of as "living" where its outputs are, in ''C'').

Definition via universal morphisms

By definition, a functor
$F: D \to C$ is a left adjoint functor if for each object $X$ in $C$ there exists a universal morphism
from $F$ to $X$. Spelled out, this means that for each object $X$ in $C$ there exists an object
$G(X)$ in $D$ and a morphism $\epsilon_X: F(G(X)) \to X$ such that for every object
$Y$ in $D$ and every morphism $f: F(Y) \to X$ there exists a unique morphism
$g: Y \to G(X)$ with $\epsilon_X \circ F(g) = f$.

The latter equation is expressed by the following commutative diagram:

In this situation, one can show that $G$ can be t