Why Adjunctions Matter—A Functional Programmer Perspective

Abstract

For the average programmer, adjunctions are (if at all known) more respected than loved. At best, they are regarded as an algebraic device of theoretical interest only, not useful in common practice.

This paper is aimed at showing the opposite: that adjunctions underlie most of the work we do as programmers, in particular those using the functional paradigm. However, functions alone are not sufficient to express the whole spectrum of programming, with its dichotomy between specifications—what is (often vaguely) required—and implementations—how what is required is (hopefully well) implemented. For this, one needs to extend functions to relations.

Inspired by the pioneering work of Ralf Hinze on “adjoint (un)folds”, the core of the so-called (relational) Algebra of Programming is shown in this paper to arise from adjunctions. Moreover, the paper also shows how to calculate recursive programs from specifications expressed by Galois connections—a special kind of adjunction.

Because Galois connections are easier to understand than adjunctions in general, the paper adopts a tutorial style, starting from the former and leading to the latter (a path usually not followed in the literature). The main aim is to reconcile the functional programming community with a concept that is central to software design as a whole, but rarely accepted as such.

“(...) and Jim Thatcher proposed the name ADJ as a (terrible) pun on the title of the book that we had planned to write (...) [recalling] that adjointness is a very important concept in category theory (...)”

— Joseph A. Goguen, Memories of ADJ, EATCS nr. 36, 1989

A Properties of Adjunctions and Monads

Corollaries of $k \mathrel {=}\lceil f \rceil ~\Leftrightarrow ~\epsilon \mathbin {\cdot } \textsf{L} \; k \mathrel {=} f$ (17)  

reflection:

$$\begin{aligned} \lceil \epsilon \rceil \mathrel {=}{id} \end{aligned}$$

(64)

that is,

$$\begin{aligned} \epsilon \mathrel {=}\lfloor {id}\rfloor \end{aligned}$$

(65)

cancellation:

$$\begin{aligned} \epsilon \mathbin {\cdot } \textsf{L} \;\lceil f \rceil \mathrel {=} f \end{aligned}$$

(66)

fusion:

$$\begin{aligned} \lceil h \rceil \mathbin {\cdot } g \mathrel {=}\lceil h \mathbin {\cdot } \textsf{L} \; g \rceil \end{aligned}$$

(67)

absorption:

$$\begin{aligned} (\textsf{R} \; g ) \mathbin {\cdot } \lceil h \rceil \mathrel {=}\lceil g \mathbin {\cdot } h \rceil \end{aligned}$$

(68)

naturality:

$$\begin{aligned} h \mathbin {\cdot } \epsilon \mathrel {=}\epsilon \mathbin {\cdot } \textsf{L} \;(\textsf{R} \; h ) \end{aligned}$$

(69)

closed definition:

$$\begin{aligned} \lfloor k \rfloor \mathrel {=}\epsilon \mathbin {\cdot } (\textsf{L} \; k ) \end{aligned}$$

(70)

functor:

$$\begin{aligned} \textsf{R} \; h \mathrel {=}\lceil h \mathbin {\cdot } \epsilon \rceil \end{aligned}$$

(71)

isomorphism:

$$\begin{aligned} \lceil f \rceil \mathrel {=}\lceil g \rceil ~\Leftrightarrow ~ f \mathrel {=} g \end{aligned}$$

(72)

Dual corollaries of $k \mathrel {=}\lfloor f \rfloor ~\Leftrightarrow ~\textsf{R} \; k \mathbin {\cdot } \eta \mathrel {=} f$ (19)  

reflection:

$$\begin{aligned} \lfloor \eta \rfloor \mathrel {=}{id} \end{aligned}$$

(73)

that is,

$$\begin{aligned} \eta \mathrel {=}\lceil {id}\rceil \end{aligned}$$

(74)

cancellation:

$$\begin{aligned} \textsf{R} \;\lfloor f \rfloor \mathbin {\cdot } \eta \mathrel {=} f \end{aligned}$$

(75)

fusion:

$$\begin{aligned} g \mathbin {\cdot } \lfloor h \rfloor \mathrel {=}\lfloor \textsf{R} \; g \mathbin {\cdot } h \rfloor \end{aligned}$$

(76)

absorption:

$$\begin{aligned} \lfloor h \rfloor \mathbin {\cdot } \textsf{L} \; g \mathrel {=}\lfloor h \mathbin {\cdot } g \rfloor \end{aligned}$$

(77)

naturality:

$$\begin{aligned} h \mathbin {\cdot } \epsilon \mathrel {=}\epsilon \mathbin {\cdot } \textsf{L} \;(\textsf{R} \; h ) \end{aligned}$$

(78)

closed definition:

$$\begin{aligned} \lceil g \rceil \mathrel {=}(\textsf{R} \; g ) \mathbin {\cdot } \eta \end{aligned}$$

(79)

functor

$$\begin{aligned} \textsf{L} \; g \mathrel {=}\lfloor \eta \mathbin {\cdot } g \rfloor \end{aligned}$$

(80)

cancellation (corollary):

$$\begin{aligned} \epsilon \mathbin {\cdot } \textsf{L} \;\eta \mathrel {=}{id} \end{aligned}$$

(81)

Monads. Proof of (32):

Proof of (33):

Proof of (37):