Functortown: A Map of the Territory

This course as a whole will be a full tour of the functors. We will start by examining the motivation for functors and the Functor typeclass. From there, we’ll explore monoidal functors, bifunctors, contravariant functors, and profunctors in depth, discussing the laws and the mathematical basis for these structures along the way.

In order to simplify navigation and provide more context, we’ve divided this course into four major sub-courses. The first covers Functor and Bifunctor and provides a complete survey of those two typeclasses, how they relate, and why they exist. This first part assumes little in the way of prior Haskell knowledge, although some basic understanding of types and typeclasses will be helpful, along with some prior experience reading and understanding Haskell syntax.

While all the remaining components of this course rely on a solid understanding of fmap and will refer back to lessons from the Functor series, if you already understand that well, you may be looking to jump straight into Applicative. Or Monad, or one of the other, later topics, once those are written. The Applicative series covers that typeclass in great detail as well as two closely related typeclasses, Alternative and Traversable. Monads get a lot of attention, for historical reasons (to wit: they appeared in Haskell long before applicatives did), but the trio of typeclasses featured in this series is incredibly powerful and should not be underestimated.

The next series within the Functortown course covers the Monad class and related topics, including monad transformers. We will survey some of the most important, need-to-know monads and how they work and how they work in example programs. We’ll then take a brief pause to consider an X that is not a Y: a functor that is not a Functor, a Functor that is not an Applicative, and so on, and what this tells us about typeclasses.

Finally, the course will conclude with a series on some oddballs: Contravariant, Profunctor, and Arrow. Interestingly, the first two are becoming more important in contemporary Haskell than they formerly were, while the popularity of Arrow seems to be declining somewhat. We think, however, that all three deserve comprehensive treatment and discussion of how they fit into the larger body of functorial typeclasses within Haskell. The Functortown course will conclude with a summary of what functors are.

Each series within this course starts from a motivating example: we have a problem that we’re trying to solve, and we write a function to solve it. That function turns out to be a concrete instance of a more generic, reusable function – the function that is at the heart of the typeclass we’ll be discussing. Each series covers representative instances and practical uses of each typeclass. Additionally, we cover the laws of each typeclass and how to property test your instances using the hedgehog testing library; however, in each case, we do this after the functions and uses of the typeclass have been thoroughly discussed.

The complete course plan

Part I: Functor and Bifunctor

Part II: Applicative

Part III: Traversable and Alternative

  • A monoid for applicative functors
  • Traversable: what and why and what does this have to do with Applicative
  • Concurrency!

Part IV: Monad and monad transformers

  • Introduction

  • Monad: what and why

  • Instances

  • Relationship to Functor and Applicative

  • Examples of >>= usage

  • The Kleisli fish

  • ReaderReader everywhere

  • The Monad laws and property tests

  • Monads do not compose: monad transformers

  • Interlude: An X that is not a Y

Part V: Contravariant, Profunctor, Arrow

  • Introduction
  • Contravariant: what and why
  • What kinds of types are contravariant
  • Relationship to an earlier functor
  • Examples of contramap usage
  • The Contravariant laws and property tests
  • Profunctor: what and why
  • Breaking down dimap
  • What kinds of types are profunctors
  • Examples, from basic to complex
  • The Profunctor laws and property tests
  • Arrow: What is this madness?
  • Situating Arrow
  • Functors and computation models

Wrapping up: So, then, what is functoriality?

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