Chapter 3.8 - F-Algebras

src/content/3.8/F-Algebras.tex

@@ -1,12 +1,4 @@
-\begin{quote}
+\lettrine[lhang=0.17]{W}{e've seen several} formulations of a monoid: as a set, as a
-This is part 24 of Categories for Programmers. Previously:
-\href{https://bartoszmilewski.com/2017/01/02/comonads/}{Comonads}. See
-the
-\href{https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/}{Table
-of Contents}.
-\end{quote}
-
-We've seen several formulations of a monoid: as a set, as a
 single-object category, as an object in a monoidal category. How much
 more juice can we squeeze out of this simple concept?
 
@@ -14,9 +6,9 @@ Let's try. Take this definition of a monoid as a set \code{m} with a
 pair of functions:
 
 \begin{verbatim}
-μ :: m × m -> m η :: 1 -> m
+μ :: m × m -> m
+η :: 1 -> m
 \end{verbatim}
-
 Here, 1 is the terminal object in \textbf{Set} --- the singleton set.
 The first function defines multiplication (it takes a pair of elements
 and returns their product), the second selects the unit element from
@@ -27,23 +19,21 @@ and just consider ``potential monoids.'' A pair of functions is an
 element of a cartesian product of two sets of functions. We know that
 these sets may be represented as exponential objects:
 
-\begin{verbatim}
+\begin{Verbatim}[commandchars=\\\{\}]
-μ ∈ m m×m η ∈ m1
+μ \ensuremath{\in} m \textsuperscript{m×m}
-\end{verbatim}
+η \ensuremath{\in} m\textsuperscript{1}
-
+\end{Verbatim}
 The cartesian product of these two sets is:
 
-\begin{verbatim}
+\begin{Verbatim}[commandchars=\\\{\}]
-m m×m × m1
+m m×m × m\textsuperscript{1}
-\end{verbatim}
+\end{Verbatim}
-
 Using some high-school algebra (which works in every cartesian closed
 category), we can rewrite it as:
 
-\begin{verbatim}
+\begin{Verbatim}[commandchars=\\\{\}]
-m m×m + 1
+m \textsuperscript{m×m + 1}
-\end{verbatim}
+\end{Verbatim}
-
 The plus sign stands for the coproduct in \textbf{Set}. We have just
 replaced a pair of functions with a single function --- an element of
 the set:
@@ -51,7 +41,6 @@ the set:
 \begin{verbatim}
 m × m + 1 -> m
 \end{verbatim}
-
 Any element of this set of functions is a potential monoid.
 
 The beauty of this formulation is that it leads to interesting
@@ -63,15 +52,15 @@ addition as a binary operation, zero as the unit, and negation as the
 inverse. To define a group we would start with a triple of functions:
 
 \begin{verbatim}
-m × m -> m m -> m 1 -> m
+m × m -> m
+m -> m
+1 -> m
 \end{verbatim}
-
 As before, we can combine all these triples into one set of functions:
 
 \begin{verbatim}
 m × m + m + 1 -> m
 \end{verbatim}
-
 We started with one binary operator (addition), one unary operator
 (negation), and one nullary operator (identity --- here zero). We
 combined them into one function. All functions with this signature
@@ -105,7 +94,6 @@ object \code{a}, and a morphism
 \begin{verbatim}
 F a -> a
 \end{verbatim}
-
 The object is often called the carrier, an underlying object or, in the
 context of programming, the carrier \newterm{type}. The morphism is often
 called the evaluation function or the structure map. Think of the
@@ -117,7 +105,6 @@ Here's the Haskell definition of an F-algebra:
 \begin{verbatim}
 type Algebra f a = f a -> a
 \end{verbatim}
-
 It identifies the algebra with its evaluation function.
 
 In the monoid example, the functor in question is:
@@ -125,27 +112,33 @@ In the monoid example, the functor in question is:
 \begin{verbatim}
 data MonF a = MEmpty | MAppend a a
 \end{verbatim}
-
+This is Haskell for \code{1 + a × a} (remember
-This is Haskell for \code{1\ +\ a\ ×\ a} (remember
+\hyperref[simple-algebraic-data-types]{algebraic
-\href{https://bartoszmilewski.com/2015/01/13/simple-algebraic-data-types/}{algebraic
 data structures}).
 
 A ring would be defined using the following functor:
 
 \begin{verbatim}
-data RingF a = RZero | ROne | RAdd a a | RMul a a | RNeg a
+data RingF a = RZero
+             | ROne
+             | RAdd a a
+             | RMul a a
+             | RNeg a
 \end{verbatim}
-
+which is Haskell for \code{1 + 1 + a × a + a × a + a}.
-which is Haskell for \code{1\ +\ 1\ +\ a\ ×\ a\ +\ a\ ×\ a\ +\ a}.
 
 An example of a ring is the set of integers. We can choose
 \code{Integer} as the carrier type and define the evaluation function
 as:
 
-\begin{verbatim}
+\begin{Verbatim}[commandchars=\\\{\}]
-evalZ :: Algebra RingF Integer evalZ RZero = 0 evalZ ROne = 1 evalZ (RAdd m n) = m + n evalZ (RMul m n) = m * n evalZ (RNeg n) = -n
+evalZ :: \textbf{Algebra} RingF Integer
-\end{verbatim}
+evalZ RZero      = 0
-
+evalZ ROne       = 1
+evalZ (RAdd m n) = m + n
+evalZ (RMul m n) = m * n
+evalZ (RNeg n)   = -n
+\end{Verbatim}
 There are more F-algebras based on the same functor \code{RingF}. For
 instance, polynomials form a ring and so do square matrices.
 
@@ -162,15 +155,22 @@ instance, an arbitrary expression in a ring is generated by this
 tree-like data structure:
 
 \begin{verbatim}
-data Expr = RZero | ROne | RAdd Expr Expr | RMul Expr Expr | RNeg Expr
+data Expr = RZero 
+          | ROne
+          | RAdd Expr Expr
+          | RMul Expr Expr
+          | RNeg Expr
 \end{verbatim}
-
 We can replace the original ring evaluator with its recursive version:
 
 \begin{verbatim}
-evalZ :: Expr -> Integer evalZ RZero = 0 evalZ ROne = 1 evalZ (RAdd e1 e2) = evalZ e1 + evalZ e2 evalZ (RMul e1 e2) = evalZ e1 * evalZ e2 evalZ (RNeg e) = -(evalZ e)
+evalZ :: Expr -> Integer
+evalZ RZero        = 0
+evalZ ROne         = 1
+evalZ (RAdd e1 e2) = evalZ e1 + evalZ e2
+evalZ (RMul e1 e2) = evalZ e1 * evalZ e2
+evalZ (RNeg e)     = -(evalZ e)
 \end{verbatim}
-
 This is still not very practical, since we are forced to represent all
 integers as sums of ones, but it will do in a pinch.
 
@@ -183,7 +183,6 @@ define a depth-one tree as:
 \begin{verbatim}
 type RingF1 a = RingF (RingF a)
 \end{verbatim}
-
 We are filling the holes in the definition of \code{RingF} with
 depth-zero trees generated by \code{RingF\ a}. Depth-2 trees are
 similarly obtained as:
@@ -191,19 +190,16 @@ similarly obtained as:
 \begin{verbatim}
 type RingF2 a = RingF (RingF (RingF a))
 \end{verbatim}
-
 which we can also write as:
 
 \begin{verbatim}
 type RingF2 a = RingF (RingF1 a)
 \end{verbatim}
-
 Continuing this process, we can write a symbolic equation:
 
-\begin{verbatim}
+\begin{Verbatim}[commandchars=\\\{\}]
-type RingFn+1 a = RingF (RingFn a)
+type RingF\textsubscript{n+1} a = RingF (RingF\textsubscript{n} a)
-\end{verbatim}
+\end{Verbatim}
-
 Conceptually, after repeating this process infinitely many times, we end
 up with our \code{Expr}. Notice that \code{Expr} does not depend on
 \code{a}. The starting point of our journey doesn't matter, we always
@@ -220,37 +216,33 @@ point}, an object defined as:
 \begin{verbatim}
 Fix f = f (Fix f)
 \end{verbatim}
-
 The intuition behind this definition is that, since we applied
-\code{f} infinitely many times to get \code{Fix\ f}, applying it one
+\code{f} infinitely many times to get \code{Fix f}, applying it one
 more time doesn't change anything. In Haskell, the definition of a fixed
 point is:
 
 \begin{verbatim}
 newtype Fix f = Fix (f (Fix f))
 \end{verbatim}
-
 Arguably, this would be more readable if the constructor's name were
 different than the name of the type being defined, as in:
 
 \begin{verbatim}
 newtype Fix f = In (f (Fix f))
 \end{verbatim}
-
 but I'll stick with the accepted notation. The constructor \code{Fix}
 (or \code{In}, if you prefer) can be seen as a function:
 
 \begin{verbatim}
 Fix :: f (Fix f) -> Fix f
 \end{verbatim}
-
 There is also a function that peels off one level of functor
 application:
 
 \begin{verbatim}
-unFix :: Fix f -> f (Fix f) unFix (Fix x) = x
+unFix :: Fix f -> f (Fix f)
+unFix (Fix x) = x
 \end{verbatim}
-
 The two functions are the inverse of each other. We'll use these
 functions later.
 
@@ -261,12 +253,12 @@ constructing some new objects, see if they form a category. Not
 surprisingly, algebras over a given endofunctor \code{F} form a
 category. Objects in that category are algebras --- pairs consisting of
 a carrier object \code{a} and a morphism
-\code{F\ a\ ->\ a}, both from the original category
+\code{F a -> a}, both from the original category
 \emph{C}.
 
 To complete the picture, we have to define morphisms in the category of
-F-algebras. A morphism must map one algebra \code{(a,\ f)} to another
+F-algebras. A morphism must map one algebra \code{(a, f)} to another
-algebra \code{(b,\ g)}. We'll define it as a morphism \code{m} that
+algebra \code{(b, g)}. We'll define it as a morphism \code{m} that
 maps the carriers --- it goes from \code{a} to \code{b} in the
 original category. Not any morphism will do: we want it to be compatible
 with the two evaluators. (We call such a structure-preserving morphism a
@@ -276,18 +268,21 @@ F-algebras. First, notice that we can lift \code{m} to the mapping:
 \begin{verbatim}
 F m :: F a -> F b
 \end{verbatim}
-
 we can then follow it with \code{g} to get to \code{b}.
-Equivalently, we can use \code{f} to go from \code{F\ a} to
+Equivalently, we can use \code{f} to go from \code{F a} to
 \code{a} and then follow it with \code{m}. We want the two paths to
 be equal:
 
 \begin{verbatim}
-g ∘ F m = m ∘ f
+g ◦ F m = m ◦ f
 \end{verbatim}
 
-\includegraphics[width=2.09375in]{images/alg.png}
+\begin{figure}[H]
+\centering
+\includegraphics[width=50mm]{images/alg.png}
+\end{figure}
 
+\noindent
 It's easy to convince yourself that this is indeed a category (hint:
 identity morphisms from \emph{C} work just fine, and a composition of
 homomorphisms is a homomorphism).
@@ -295,108 +290,124 @@ homomorphisms is a homomorphism).
 An initial object in the category of F-algebras, if it exists, is called
 the \newterm{initial algebra}. Let's call the carrier of this initial
 algebra \code{i} and its evaluator
-\code{j\ ::\ F\ i\ ->\ i}. It turns out that \code{j},
+\code{j :: F i -> i}. It turns out that \code{j},
 the evaluator of the initial algebra, is an isomorphism. This result is
 known as Lambek's theorem. The proof relies on the definition of the
 initial object, which requires that there be a unique homomorphism
 \code{m} from it to any other F-algebra. Since \code{m} is a
 homomorphism, the following diagram must commute:
 
-\includegraphics{images/alg2.png}
+\begin{figure}[H]
+\centering
+\includegraphics[width=50mm]{images/alg2.png}
+\end{figure}
 
-Now let's construct an algebra whose carrier is \code{F\ i}. The
+\noindent
-evaluator of such an algebra must be a morphism from \code{F\ (F\ i)}
+Now let's construct an algebra whose carrier is \code{F i}. The
-to \code{F\ i}. We can easily construct such an evaluator simply by
+evaluator of such an algebra must be a morphism from \code{F (F i)}
+to \code{F i}. We can easily construct such an evaluator simply by
 lifting \code{j}:
 
 \begin{verbatim}
 F j :: F (F i) -> F i
 \end{verbatim}
-
+Because \code{(i, j)} is the initial algebra, there must be a unique
-Because \code{(i,\ j)} is the initial algebra, there must be a unique
+homomorphism \code{m} from it to \code{(F i, F j)}. The following
-homomorphism \code{m} from it to \code{(F\ i,\ F\ j)}. The following
 diagram must commute:
 
-\includegraphics{images/alg3a.png}
+\begin{figure}[H]
+\centering
+\includegraphics[width=50mm]{images/alg3a.png}
+\end{figure}
 
+\noindent
 But we also have this trivially commuting diagram (both paths are the
 same!):
 
-\includegraphics{images/alg3.png}
+\begin{figure}[H]
+\centering
+\includegraphics[width=50mm]{images/alg3.png}
+\end{figure}
 
+\noindent
 which can be interpreted as showing that \code{j} is a homomorphism of
-algebras, mapping \code{(F\ i,\ F\ j)} to \code{(i,\ j)}. We can
+algebras, mapping \code{(F i, F j)} to \code{(i, j)}. We can
 glue these two diagrams together to get:
 
-\includegraphics[width=3.12500in]{images/alg4.png}
+\begin{figure}[H]
+\centering
+\includegraphics[width=80mm]{images/alg4.png}
+\end{figure}
 
+\noindent
 This diagram may, in turn, be interpreted as showing that
-\code{j\ ∘\ m} is a homomorphism of algebras. Only in this case the
+\code{j ◦ m} is a homomorphism of algebras. Only in this case the
-two algebras are the same. Moreover, because \code{(i,\ j)} is
+two algebras are the same. Moreover, because \code{(i, j)} is
 initial, there can only be one homomorphism from it to itself, and
-that's the identity morphism \code{idi} --- which we know is a
+that's the identity morphism \code{id\textsubscript{i}} --- which we know is a
-homomorphism of algebras. Therefore \code{j\ ∘\ m\ =\ idi}. Using this
+homomorphism of algebras. Therefore \code{j ◦ m = id\textsubscript{i}}. Using this
 fact and the commuting property of the left diagram we can prove that
-\code{m\ ∘\ j\ =\ idFi}. This shows that \code{m} is the inverse of
+\code{m ◦ j = id\textsubscript{Fi}}. This shows that \code{m} is the inverse of
 \code{j} and therefore \code{j} is an isomorphism between
-\code{F\ i} and \code{i}:
+\code{F i} and \code{i}:
-
-\begin{verbatim}
-F i ≅ i
-\end{verbatim}
 
+\begin{Verbatim}[commandchars=\\\{\}]
+F i \ensuremath{\cong} i
+\end{Verbatim}
 But that is just saying that \code{i} is a fixed point of \code{F}.
 That's the formal proof behind the original hand-waving argument.
 
-Back to Haskell: We recognize \code{i} as our \code{Fix\ f},
+Back to Haskell: We recognize \code{i} as our \code{Fix f},
 \code{j} as our constructor \code{Fix}, and its inverse as
 \code{unFix}. The isomorphism in Lambek's theorem tells us that, in
 order to get the initial algebra, we take the functor \code{f} and
-replace its argument \code{a} with \code{Fix\ f}. We also see why
+replace its argument \code{a} with \code{Fix f}. We also see why
 the fixed point does not depend on \code{a}.
 
-\subsection{Natural Numbers}\label{natural-numbers}
+\section{Natural Numbers}\label{natural-numbers}
 
 Natural numbers can also be defined as an F-algebra. The starting point
 is the pair of morphisms:
 
 \begin{verbatim}
-zero :: 1 -> N succ :: N -> N
+zero :: 1 -> N
+succ :: N -> N
 \end{verbatim}
-
 The first one picks the zero, and the second one maps all numbers to
 their successors. As before, we can combine the two into one:
 
 \begin{verbatim}
 1 + N -> N
 \end{verbatim}
-
 The left hand side defines a functor which, in Haskell, can be written
 like this:
 
 \begin{verbatim}
 data NatF a = ZeroF | SuccF a
 \end{verbatim}
-
 The fixed point of this functor (the initial algebra that it generates)
 can be encoded in Haskell as:
 
 \begin{verbatim}
 data Nat = Zero | Succ Nat
 \end{verbatim}
-
 A natural number is either zero or a successor of another number. This
 is known as the Peano representation for natural numbers.
 
 \section{Catamorphisms}\label{catamorphisms}
 
 Let's rewrite the initiality condition using Haskell notation. We call
-the initial algebra \code{Fix\ f}. Its evaluator is the contructor
+the initial algebra \code{Fix f}. Its evaluator is the contructor
 \code{Fix}. There is a unique morphism \code{m} from the initial
 algebra to any other algebra over the same functor. Let's pick an
 algebra whose carrier is \code{a} and the evaluator is \code{alg}.
 
-\includegraphics{images/alg5.png}\\
+\begin{figure}[H]
+\centering
+\includegraphics[width=60mm]{images/alg5.png}
+\end{figure}
+
+\noindent
 By the way, notice what \code{m} is: It's an evaluator for the fixed
 point, an evaluator for the whole recursive expression tree. Let's find
 a general way of implementing it.
@@ -405,18 +416,21 @@ Lambek's theorem tells us that the constructor \code{Fix} is an
 isomorphism. We called its inverse \code{unFix}. We can therefore flip
 one arrow in this diagram to get:
 
-\includegraphics{images/alg6.png}
+\begin{figure}[H]
+\centering
+\includegraphics[width=60mm]{images/alg6.png}
+\end{figure}
 
+\noindent
 Let's write down the commutation condition for this diagram:
 
 \begin{verbatim}
 m = alg . fmap m . unFix
 \end{verbatim}
-
 We can interpret this equation as a recursive definition of \code{m}.
 The recursion is bound to terminate for any finite tree created using
 the functor \code{f}. We can see that by noticing that
-\code{fmap\ m} operates underneath the top layer of the functor
+\code{fmap m} operates underneath the top layer of the functor
 \code{f}. In other words, it works on the children of the original
 tree. The children are always one level shallower than the original
 tree.
@@ -435,25 +449,25 @@ gives us the function we called \code{m}. This higher order function
 is called a catamorphism:
 
 \begin{verbatim}
-cata :: Functor f => (f a -> a) -> Fix f -> a cata alg = alg . fmap (cata alg) . unFix
+cata :: Functor f => (f a -> a) -> Fix f -> a
+cata alg = alg . fmap (cata alg) . unFix
 \end{verbatim}
-
 Let's see an example of that. Take the functor that defines natural
 numbers:
 
 \begin{verbatim}
 data NatF a = ZeroF | SuccF a
 \end{verbatim}
-
+Let's pick \code{(Int, Int)} as the carrier type and define our
-Let's pick \code{(Int,\ Int)} as the carrier type and define our
 algebra as:
 
 \begin{verbatim}
-fib :: NatF (Int, Int) -> (Int, Int) fib ZeroF = (1, 1) fib (SuccF (m, n)) = (n, m + n)
+fib :: NatF (Int, Int) -> (Int, Int)
+fib ZeroF = (1, 1)
+fib (SuccF (m, n)) = (n, m + n)
 \end{verbatim}
-
 You can easily convince yourself that the catamorphism for this algebra,
-\code{cata\ fib}, calculates Fibonacci numbers.
+\code{cata fib}, calculates Fibonacci numbers.
 
 In general, an algebra for \code{NatF} defines a recurrence relation:
 the value of the current element in terms of the previous element. A
@@ -466,14 +480,12 @@ A list of \code{e} is the initial algebra of the following functor:
 \begin{verbatim}
 data ListF e a = NilF | ConsF e a
 \end{verbatim}
-
 Indeed, replacing the variable \code{a} with the result of recursion,
-which we'll call \code{List\ e}, we get:
+which we'll call \code{List e}, we get:
 
 \begin{verbatim}
 data List e = Nil | Cons e (List e)
 \end{verbatim}
-
 An algebra for a list functor picks a particular carrier type and
 defines a function that does pattern matching on the two constructors.
 Its value for \code{NilF} tells us how to evaluate an empty list, and
@@ -484,10 +496,11 @@ For instance, here's an algebra that can be used to calculate the length
 of a list (the carrier type is \code{Int}):
 
 \begin{verbatim}
-lenAlg :: ListF e Int -> Int lenAlg (ConsF e n) = n + 1 lenAlg NilF = 0
+lenAlg :: ListF e Int -> Int
+lenAlg (ConsF e n) = n + 1
+lenAlg NilF = 0
 \end{verbatim}
-
+Indeed, the resulting catamorphism \code{cata lenAlg} calculates the
-Indeed, the resulting catamorphism \code{cata\ lenAlg} calculates the
 length of a list. Notice that the evaluator is a combination of (1) a
 function that takes a list element and an accumulator and returns a new
 accumulator, and (2) a starting value, here zero. The type of the value
@@ -498,22 +511,21 @@ Compare this to the traditional Haskell definition:
 \begin{verbatim}
 length = foldr (\e n -> n + 1) 0
 \end{verbatim}
-
 The two arguments to \code{foldr} are exactly the two components of
 the algebra.
 
 Let's try another example:
 
 \begin{verbatim}
-sumAlg :: ListF Double Double -> Double sumAlg (ConsF e s) = e + s sumAlg NilF = 0.0
+sumAlg :: ListF Double Double -> Double
+sumAlg (ConsF e s) = e + s
+sumAlg NilF = 0.0
 \end{verbatim}
-
 Again, compare this with:
 
 \begin{verbatim}
 sum = foldr (\e s -> e + s) 0.0
 \end{verbatim}
-
 As you can see, \code{foldr} is just a convenient specialization of a
 catamorphism to lists.
 
@@ -525,23 +537,25 @@ direction of the morphism is reversed:
 \begin{verbatim}
 a -> F a
 \end{verbatim}
-
 Coalgebras for a given functor also form a category, with homomorphisms
 preserving the coalgebraic structure. The terminal object
-\code{(t,\ u)} in that category is called the terminal (or final)
+\code{(t, u)} in that category is called the terminal (or final)
-coalgebra. For every other algebra \code{(a,\ f)} there is a unique
+coalgebra. For every other algebra \code{(a, f)} there is a unique
 homomorphism \code{m} that makes the following diagram commute:
 
-\includegraphics{images/alg7.png}
+\begin{figure}[H]
+\centering
+\includegraphics[width=50mm]{images/alg7.png}
+\end{figure}
 
+\noindent
 A terminal colagebra is a fixed point of the functor, in the sense that
-the morphism \code{u\ ::\ t\ ->\ F\ t} is an isomorphism
+the morphism \code{u :: t -> F t} is an isomorphism
 (Lambek's theorem for coalgebras):
 
-\begin{verbatim}
+\begin{Verbatim}[commandchars=\\\{\}]
-F t ≅ t
+F t \ensuremath{\cong} t
-\end{verbatim}
+\end{Verbatim}
-
 A terminal coalgebra is usually interpreted in programming as a recipe
 for generating (possibly infinite) data structures or transition
 systems.
@@ -550,24 +564,23 @@ Just like a catamorphism can be used to evaluate an initial algebra, an
 anamorphism can be used to coevaluate a terminal coalgebra:
 
 \begin{verbatim}
-ana :: Functor f => (a -> f a) -> a -> Fix f ana coalg = Fix . fmap (ana coalg) . coalg
+ana :: Functor f => (a -> f a) -> a -> Fix f
+ana coalg = Fix . fmap (ana coalg) . coalg
 \end{verbatim}
-
 A canonical example of a coalgebra is based on a functor whose fixed
 point is an infinite stream of elements of type \code{e}. This is the
 functor:
 
 \begin{verbatim}
-data StreamF e a = StreamF e a deriving Functor
+data StreamF e a = StreamF e a
+  deriving Functor
 \end{verbatim}
-
 and this is its fixed point:
 
 \begin{verbatim}
 data Stream e = Stream e (Stream e)
 \end{verbatim}
-
+A coalgebra for \code{StreamF e} is a function that takes the seed of
-A coalgebra for \code{StreamF\ e} is a function that takes the seed of
 type \code{a} and produces a pair (\code{StreamF} is a fancy name
 for a pair) consisting of an element and the next seed.
 
@@ -584,24 +597,26 @@ the sieve of Eratosthenes. This coalgebra is implemented by the
 following function:
 
 \begin{verbatim}
-era :: [Int] -> StreamF Int [Int] era (p : ns) = StreamF p (filter (notdiv p) ns) where notdiv p n = n `mod` p /= 0
+era :: [Int] -> StreamF Int [Int]
+era (p : ns) = StreamF p (filter (notdiv p) ns)
+    where notdiv p n = n `mod` p /= 0
 \end{verbatim}
-
 The anamorphism for this coalgebra generates the list of primes:
 
 \begin{verbatim}
 primes = ana era [2..]
 \end{verbatim}
-
 A stream is an infinite list, so it should be possible to convert it to
 a Haskell list. To do that, we can use the same functor \code{StreamF}
 to form an algebra, and we can run a catamorphism over it. For instance,
 this is a catamorphism that converts a stream to a list:
 
 \begin{verbatim}
-toListC :: Fix (StreamF e) -> [e] toListC = cata al where al :: StreamF e [e] -> [e] al (StreamF e a) = e : a
+toListC :: Fix (StreamF e) -> [e]
+toListC = cata al
+    where al :: StreamF e [e] -> [e] 
+          al (StreamF e a) = e : a
 \end{verbatim}
-
 Here, the same fixed point is simultaneously an initial algebra and a
 terminal coalgebra for the same endofunctor. It's not always like this,
 in an arbitrary category. In general, an endofunctor may have many (or
@@ -616,16 +631,15 @@ functor is modified to produce a \code{Maybe} pair:
 \begin{verbatim}
 unfoldr :: (b -> Maybe (a, b)) -> b -> [a]
 \end{verbatim}
-
 The value of \code{Nothing} will terminate the generation of the list.
 
 An interesting case of a coalgebra is related to lenses. A lens can be
 represented as a pair of a getter and a setter:
 
 \begin{verbatim}
-set :: a -> s -> a get :: a -> s
+set :: a -> s -> a
+get :: a -> s
 \end{verbatim}
-
 Here, \code{a} is usually some product data type with a field of type
 \code{s}. The getter retrieves the value of that field and the setter
 replaces this field with a new value. These two functions can be
@@ -634,24 +648,21 @@ combined into one:
 \begin{verbatim}
 a -> (s, s -> a)
 \end{verbatim}
-
 We can rewrite this function further as:
 
 \begin{verbatim}
 a -> Store s a
 \end{verbatim}
-
 where we have defined a functor:
 
 \begin{verbatim}
 data Store s a = Store (s -> a) s
 \end{verbatim}
-
 Notice that this is not a simple algebraic functor constructed from sums
 of products. It involves an exponential \code{as}.
 
 A lens is a coalgebra for this functor with the carrier type \code{a}.
-We've seen before that \code{Store\ s} is also a comonad. It turns out
+We've seen before that \code{Store s} is also a comonad. It turns out
 that a well-behaved lens corresponds to a coalgebra that is compatible
 with the comonad structure. We'll talk about this in the next section.
 
@@ -662,12 +673,12 @@ with the comonad structure. We'll talk about this in the next section.
 \item
   Implement the evaluation function for a ring of polynomials of one
   variable. You can represent a polynomial as a list of coefficients in
-  front of powers of \code{x}. For instance, \code{4x2-1} would be
+  front of powers of \code{x}. For instance, \code{4x\textsuperscript{2}-1} would be
   represented as (starting with the zero'th power)
-  \code{{[}-1,\ 0,\ 4{]}}.
+  \code{{[}-1, 0, 4{]}}.
 \item
   Generalize the previous construction to polynomials of many
-  independent variables, like \code{x2y-3y3z}.
+  independent variables, like \code{x\textsuperscript{2}y-3y\textsuperscript{3}z}.
 \item
   Implement the algebra for the ring of 2×2 matrices.
 \item
@@ -677,7 +688,3 @@ with the comonad structure. We'll talk about this in the next section.
   Use \code{unfoldr} to generate a list of the first \code{n}
   primes.
 \end{enumerate}
-
-Next:
-\href{https://bartoszmilewski.com/2017/03/14/algebras-for-monads/}{Algebras
-for Monads}.

src/ctfp.texi

@@ -95,7 +95,7 @@ PDF compiled by @url{https://github.com/hmemcpy/milewski-ctfp-pdf, Igal Tabachni
 \chapter{Products and Coproducts}\label{products-and-coproducts}
 \subfile{content/1.5/Products and Coproducts}
 
-\chapter{Simple Algebraic Data Types}
+\chapter{Simple Algebraic Data Types}\label{simple-algebraic-data-types}
 \subfile{content/1.6/Simple Algebraic Data Types}
 
 \chapter{Functors}
@@ -135,6 +135,10 @@ PDF compiled by @url{https://github.com/hmemcpy/milewski-ctfp-pdf, Igal Tabachni
 \chapter{Comonads}\label{comonads}
 \subfile{content/3.7/Comonads}
 
+\chapter{F-Algebras}\label{f-algebras}
+\subfile{content/3.8/F-Algebras}
+
+
 \backmatter
 
 @unnumbered Acknowledgments