Glossary

This is a catalogue of the major functions, type classes, and data types in Cats. It serves as a bird's-eye view of each class capabilities. It is also intended as a go-to reference for Cats users, who may not recall the answer to questions like these:

• What is the difference between unit and void?
• To discard the first value and keep only the first effect, is it <* or *>?
• How do I make a computation F[A] fail by checking a condition on the value?

The signatures and type-classes have been simplified, are described below. If you want a printable version, you can also check out this cats-cheatsheet.

WARNING: this page is written manually, and not automatically generated, so many things may be missing. If you find a mistake, or addition, please submit a PR following the guidelines below.

Type-Classes over an F[_]

Functor

Type	Method Name	Notes
F[A] => F[Unit]	void
F[A] => B => F[B]	as
F[A] => (A => B) => F[B]	map
F[A] => (A => A1) => F[A1])	mapOrKeep	A1 >: A, the (A => A1) is a PartialFunction
F[A] => (A => B) => F[(A,B)]	fproduct
F[A] => (A => B) => F[(B,A)]	fproductLeft
F[A] => B => F[(B, A)]	tupleLeft
F[A] => B => F[(A, B)]	tupleRight
(A => B) => (F[A] => F[B])	lift

Apply

Type	Method Name	Symbol
F[A] => F[B] => F[A]	productL	<*
F[A] => F[B] => F[B]	productR	*>
F[A] => F[B] => F[(A,B)]	product
F[A => B] => F[A] => F[B]	ap	<*>
F[A => B => C] => F[A] => F[B] => F[C]	ap2
F[A] => F[B] => (A => B => C) => F[C]	map2

Applicative

Type	Method Name	Notes
A => F[A]	pure
=> F[Unit]	unit
Boolean => F[Unit] => F[Unit]	whenA	Performs effect iff condition is true
	unlessA	Adds effect iff condition is false

FlatMap

Type	Method Name
F[F[A]] => F[A]	flatten
F[A] => (A => F[B]) => F[B]	flatMap
F[A] => (A => F[B]) => F[(A,B)]	mproduct
F[Boolean] => F[A] => F[A] => F[A]	ifM
F[A] => (A => F[B]) => F[A]	flatTap

FunctorFilter

Type	Method Name	Notes
F[A] => (A => Boolean) => F[A]	filter
F[A] => (A => Option[B]) => F[B]	mapFilter
F[A] => (A => B) => F[B]	collect	The A => B is a PartialFunction
F[Option[A]] => F[A]	flattenOption

ApplicativeError

The source code of Cats uses the E type variable for the error type.

Type	Method Name	Notes
E => F[A]	raiseError
F[A] => F[Either[E,A]]	attempt
F[A] => (E => A) => F[A]	handleError
F[A] => (E => F[A]) => F[A]	handleErrorWith
F[A] => (E => A) => F[A]	recover	The E => A is a PartialFunction.
F[A] => (E => F[A]) => F[A]	recoverWith	The E => F[A] is a PartialFunction.
F[A] => (E => F[Unit]) => F[A]	onError	The E => F[Unit] is a PartialFunction.
Either[E,A] => F[A]	fromEither
Option[A] => E => F[A]	liftFromOption

MonadError

Like the previous section, we use the E for the error parameter type.

Type	Method Name	Notes
F[A] => E => (A => Boolean) => F[A]	ensure
F[A] => (A => E) => (A => Boolean) => F[A]	ensureOr
F[A] => (E => E) => F[A]	adaptError	The E => E is a PartialFunction.
F[Either[E,A]] => F[A]	rethrow

UnorderedFoldable

Type	Method Name	Constraints
F[A] => Boolean	isEmpty
F[A] => Boolean	nonEmpty
F[A] => Long	size
F[A] => (A => Boolean) => Boolean	forall
F[A] => (A => Boolean) => Boolean	exists
F[A] => A	unorderedFold	A: CommutativeMonoid
F[A] => (A => B) => B	unorderedFoldMap	B: CommutativeMonoid

Foldable

Type	Method Name	Constraints
F[A] => A	fold	A: Monoid
F[A] => B => ((B,A) => B) => B	foldLeft
F[A] => (A => B) => B	foldMap	B: Monoid
F[A] => (A => G[B]) => G[B]	foldMapM	G: Monad and B: Monoid
F[A] => (A => B) => Option[B]	collectFirst	The A => B is a PartialFunction
F[A] => (A => Option[B]) => Option[B]	collectFirstSome
F[A] => (A => G[B]) => G[Unit]	traverse_	G: Applicative
F[G[A]] => G[Unit]	sequence_	G: Applicative
F[A] => (A => Either[B, C]) => (F[B], F[C])	partitionEither	G: Applicative

Reducible

Type	Method Name	Constraints
F[A] => ((A,A) => A) => A	reduceLeft
F[A] => A	reduce	A: Semigroup

Traverse

Type	Method Name	Constraints
F[G[A]] => G[F[A]]	sequence	G: Applicative
F[A] => (A => G[B]) => G[F[B]]	traverse	G: Applicative
F[A] => (A => G[F[B]]) => G[F[B]]	flatTraverse	F: FlatMap and G: Applicative
F[G[F[A]]] => G[F[A]]	flatSequence	G: Applicative and F: FlatMap
F[A] => F[(A,Int)]	zipWithIndex
F[A] => ((A,Int) => B) => F[B]	mapWithIndex
F[A] => ((A,Int) => G[B]) => G[F[B]]	traverseWithIndexM	F: Monad

SemigroupK

Type	Method Name	Constraints
F[A] => F[A] => F[A]	combineK
F[A] => Int => F[A]	combineNK
F[A] => F[B] => F[Either[A, B]]	sum	F: Functor
IterableOnce[F[A]] => Option[F[A]]	combineAllOptionK

MonoidK

Type	Method Name	Constraints
F[A]	empty
F[A] => Boolean	isEmpty
IterableOnce[F[A]] => F[A]	combineAllK

Alternative

Type	Method Name	Constraints
F[G[A]] => F[A]	unite	F: FlatMap and G: Foldable
F[G[A, B]] => (F[A], F[B])	separate	F: FlatMap and G: Bifoldable
F[G[A, B]] => (F[A], F[B])	separateFoldable	F: Foldable and G: Bifoldable
Boolean => F[Unit]	guard
IterableOnce[A] => F[A]	fromIterableOnce
G[A] => F[A]	fromFoldable	G: Foldable

NonEmptyAlternative

Type	Method Name	Constraints
A => F[A] => F[A]	prependK
F[A] => A => F[A]	appendK

Transformers

Constructors and wrappers

Data Type	is an alias or wrapper of
OptionT[F[_], A]	F[Option[A]]
EitherT[F[_], A, B]	F[Either[A,B]
Kleisli[F[_], A, B]	A => F[B]
Reader[A, B]	A => B
ReaderT[F[_], A, B]	Kleisli[F, A, B]
Writer[A, B]	(A,B)
WriterT[F[_], A, B]	F[(A,B)]
Tuple2K[F[_], G[_], A]	(F[A], G[A])
EitherK[F[_], G[_], A]	Either[F[A], G[A]]
FunctionK[F[_], G[_]]	F[X] => G[X] for every X
F ~> G	Alias of FunctionK[F, G]

OptionT

For convenience, in these types we use the symbol OT to abbreviate OptionT.

Type	Method Name	Constraints
=> OT[F, A]	none	F: Applicative
A => OT[F, A]	some or pure	F: Applicative
F[A] => OT[F, A]	liftF	F: Functor
Boolean => F[A] => OT[F, A]	whenF	F: Applicative
F[Boolean] => F[A] => OT[F, A]	whenM	F: Monad
OT[F, A] => F[Option[A]]	value
OT[F, A] => A => Boolean => OT[F, A]	filter	F: Functor
OT[F, A] => A => F[Boolean] => OT[F, A]	filterF	F: Monad
OT[F, A] => (A => B) => OT[F, B]	map	F: Functor
OT[F, A] => (F ~> G) => OT[G, B]	mapK
OT[F, A] => (A => Option[B]) => OT[F, B]	mapFilter	F: Functor
OT[F, A] => B => (A => B) => F[B]	fold or cata
OT[F, A] => (A => OT[F, B]) => OT[F,B]	flatMap
OT[F, A] => (A => F[Option[B]]) => OT[F,B]	flatMapF	F: Monad
OT[F, A] => A => F[A]	getOrElse	F: Functor
OT[F, A] => F[A] => F[A]	getOrElseF	F: Monad
OT[F, A] => OT[F, A] => OT[F, A]

EitherT

Here, we use ET to abbreviate EitherT; and we use A and B as type variables for the left and right sides of the Either.

Type	Method Name	Constraints
A => ET[F, A, B]	leftT	F: Applicative
B => ET[F, A, B]	rightT	F: Applicative
	pure	F: Applicative
F[A] => ET[F, A, B]	left	F: Applicative
F[B] => ET[F, A, B]	right	F: Applicative
	liftF	F: Applicative
Either[A, B] => ET[F, A, B]	fromEither	F: Applicative
Option[B] => A => ET[F, A, B]	fromOption	F: Applicative
F[Option[B]] => A => ET[F, A, B]	fromOptionF	F: Functor
F[Option[B]] => F[A] => ET[F, A, B]	fromOptionM	F: Monad
Boolean => B => A => ET[F, A, B]	cond	F: Applicative
ET[F, A, B] => (A => C) => (B => C) => F[C]	fold	F: Functor
ET[F, A, B] => ET[F, B, A]	swap	F: Functor
ET[F, A, A] => F[A]	merge

Kleisli (or ReaderT)

Here, we use Ki as a short-hand for Kleisli.

Type	Method Name	Constraints
Ki[F, A, B] => (A => F[B])	run
Ki[F, A, B] => A => F[B]	apply
A => Ki[F, A, A]	ask	F: Applicative
B => Ki[F, A, B]	pure	F: Applicative
F[B] => Ki[F, A, B]	liftF
Ki[F, A, B] => (C => A) => Ki[F, C, B]	local
Ki[F, A, B] => Ki[F, A, A]	tap
Ki[F, A, B] => (B => C) => Ki[F, A, C]	map
Ki[F, A, B] => (F ~> G) => Ki[G, A, B]	mapK
Ki[F, A, B] => (F[B] => G[C]) => Ki[F, A, C]	mapF
Ki[F, A, B] => Ki[F, A, F[B]]	lower

Type Classes for types F[_, _]

Bifunctor

Type	Method Name
F[A,B] => (A => C) => F[C,B]	leftMap
F[A,B] => (B => D) => F[A,D]	.rightFunctor and .map
F[A,B] => (A => C) => (B => D) => F[C,D]	bimap

Profunctor

Type	Method Name
F[A, B] => (B => C) => F[A, C]	rmap
F[A, B] => (C => A) => F[C, B]	lmap
F[A, B] => (C => A) => (B => D) => F[C,D]	dimap

Strong Profunctor

Type	Method Name
F[A, B] => F[(A,C), (B,C)]	first
F[A, B] => F[(C,A), (C,B)]	second

Compose, Category, Choice

Type	Method Name	Symbol
F[A, B] => F[C, A] => F[C, B]	compose	<<<
F[A, B] => F[B, C] => F[A, C]	andThen	>>>
=> F[A,A]	id
F[A, B] => F[C, B] => F[Either[A, C], B]	choice	`
=> F[ Either[A, A], A]	codiagonal

Arrow

Type	Method Name	Symbol
(A => B) => F[A, B]	lift
F[A,B] => F[C,D] => F[(A,C), (B,D)]	split	***
F[A,B] => F[A,C] => F[A, (B,C)]	merge	&&&

ArrowChoice

Type	Method Name	Symbol
F[A,B] => F[C,D] => F[Either[A, C], Either[B, D]]	choose	+++
F[A,B] => F[Either[A, C], Either[B, C]]	left
F[A,B] => F[Either[C, A], Either[C, B]]	right

Simplifications

Because Сats is a Scala library and Scala has many knobs and switches, the actual definitions and the implementations of the functions and type-classes in Сats can be a bit obfuscated at first. To alleviate this, in this glossary we focus on the plain type signatures of the method, and ignore many of the details from Scala. In particular, in our type signatures:

• We use A,B,C for type variables of kind *, and F, G, H for type variables of a higher kind.
• We write type signatures in currified form: parameters are taken one at a time, and they are separated with the arrow => operation. In Scala, a method's parameters may be split in several comma-separated lists.
• We do not differentiate between methods from the type-class trait (e.g. trait Functor), or the companion object, or the syntax companion (implicit class).
• For functions defined as method of the typeclass trait, we ignore the receiver object.
• We ignore implicit parameters that represent type-class constraints; and write them on a side column instead.
• We use A => B for both Function1[A, B] and PartialFunction[A, B] parameters, without distinction. We add a side note when one is a PartialFunction.
• Some functions are defined through the Partially Applied Type Params pattern. We ignore this.
• We ignore the distinction between by-name and by-value input parameters. We use the notation => A, without parameters, to indicate constant functions.
• We ignore Scala variance annotations. We also ignore extra type parameters, which in some methods are added with a subtype-constraint, (e.g. B >: A). These are usually meant for flexibility, but we replace each one by its bound.