From 2d745c6c68a1c5aaaef7ad630837dec117037d4b Mon Sep 17 00:00:00 2001 From: Igal Tabachnik Date: Mon, 25 Sep 2017 12:22:52 +0300 Subject: [PATCH] Chapter 1.10 - Natural Transformations + tweaks Fixes #13 --- ctfp.pdf | Bin 3642230 -> 4140213 bytes src/content/1.10/Natural Transformations.tex | 366 +++++++++---------- src/content/1.5/Products and Coproducts.tex | 2 +- src/content/1.7/Functors.tex | 4 +- src/content/1.8/Functoriality.tex | 4 +- src/content/1.9/Function Types.tex | 4 +- src/sicp.texi | 3 + 7 files changed, 187 insertions(+), 196 deletions(-) diff --git a/ctfp.pdf b/ctfp.pdf index f4fbb1982c00b9c3e14fbc0e69355a38ba95c2d7..fe7d4b2cc8116d65c5014603fb643233b40bbcd5 100644 GIT binary patch delta 583619 zcmZsC1yodR*ES3=bcb|L(4ErV4FZa^APCY8Lzi?7-7O+L2!eo;(kYD~jUX*0|ESOR zyzlp|f7a}^=04}l9edw%u6?d+&qetPYV-UpYTb+vKu{3=)sPJg6XchY!g2R-v$b@> z@mcturT%hJl&JschV3WDr%gP&pC|x2oF)vv#fX=$Bz;6wwI!QY?{)On<)_gy=%`c{ z7+pBk$;7e?wvQ`%y;0>Nq=_gn#8|LRZtpdrgE${yn;)fR$Y8Li`8UUg0irR|(jh1y 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Previously: -\href{https://bartoszmilewski.com/2015/03/13/function-types/}{Function -Types}. See the -\href{https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/}{Table -of Contents}. -\end{quote} +\lettrine[lhang=0.17]{W}{e talked about} functors as mappings between categories that preserve +their structure. -We talked about functors as mappings between categories that preserve -their structure. A functor ``embeds'' one category in another. It may +\begin{wrapfigure}[13]{R}{0pt} +\raisebox{0pt}[\dimexpr\height-0.75\baselineskip\relax]{ +\fbox{\includegraphics[width=60mm]{images/1_functors.jpg}}}% +\end{wrapfigure} + +\noindent +A functor ``embeds'' one category in another. It may collapse multiple things into one, but it never breaks connections. One way of thinking about it is that with a functor we are modeling one category inside another. The source category serves as a model, a blueprint, for some structure that's part of the target category. -\includegraphics[width=5.31250in]{images/1_functors.jpg} - There may be many ways of embedding one category in another. Sometimes they are equivalent, sometimes very different. One may collapse the whole source category into one object, another may map every object to a @@ -25,78 +23,86 @@ functors --- special mappings that preserve their functorial nature. Consider two functors \code{F} and \code{G} between categories \emph{C} and \emph{D}. If you focus on just one object \code{a} in -\emph{C}, it is mapped to two objects: \code{F\ a} and \code{G\ a}. -A mapping of functors should therefore map \code{F\ a} to -\code{G\ a}. +\emph{C}, it is mapped to two objects: \code{F a} and \code{G a}. +A mapping of functors should therefore map \code{F a} to +\code{G a}. -\includegraphics[width=3.12500in]{images/2_natcomp.jpg} +\begin{figure}[H] +\centering +\fbox{\includegraphics[width=80mm]{images/2_natcomp.jpg}} +\end{figure} -Notice that \code{F\ a} and \code{G\ a} are objects in the same +\noindent +Notice that \code{F a} and \code{G a} are objects in the same category \emph{D}. Mappings between objects in the same category should not go against the grain of the category. We don't want to make artificial connections between objects. So it's \emph{natural} to use existing connections, namely morphisms. A natural transformation is a selection of morphisms: for every object \code{a}, it picks one -morphism from \code{F\ a} to \code{G\ a}. If we call the natural +morphism from \code{F a} to \code{G a}. If we call the natural transformation \code{α}, this morphism is called the \newterm{component} -of \code{α} at \code{a}, or \code{αa}. +of \code{α} at \code{a}, or \code{α\textsubscript{a}}. -\begin{verbatim} -αa :: F a -> G a -\end{verbatim} - -Keep in mind that \code{a} is an object in \emph{C} while \code{αa} +\begin{Verbatim}[commandchars=\\\{\}] +α\textsubscript{a} :: F a -> G a +\end{Verbatim} +Keep in mind that \code{a} is an object in \emph{C} while \code{α\textsubscript{a}} is a morphism in \emph{D}. -If, for some \code{a}, there is no morphism between \code{F\ a} and -\code{G\ a} in \emph{D}, there can be no natural transformation +If, for some \code{a}, there is no morphism between \code{F a} and +\code{G a} in \emph{D}, there can be no natural transformation between \code{F} and \code{G}. Of course that's only half of the story, because functors not only map objects, they map morphisms as well. So what does a natural transformation do with those mappings? It turns out that the mapping of morphisms is fixed --- under any natural transformation between F and G, -\code{F\ f} must be transformed into \code{G\ f}. What's more, the +\code{F f} must be transformed into \code{G f}. What's more, the mapping of morphisms by the two functors drastically restricts the choices we have in defining a natural transformation that's compatible with it. Consider a morphism \code{f} between two objects \code{a} -and \code{b} in \emph{C}. It's mapped to two morphisms, \code{F\ f} -and \code{G\ f} in \emph{D}: +and \code{b} in \emph{C}. It's mapped to two morphisms, \code{F f} +and \code{G f} in \emph{D}: \begin{verbatim} F f :: F a -> F b G f :: G a -> G b \end{verbatim} +\begin{wrapfigure}[5]{R}{0pt} +\raisebox{0pt}[\dimexpr\height-0.75\baselineskip\relax]{ +\fbox{\includegraphics[width=40mm]{images/3_naturality.jpg}}}% +\end{wrapfigure} +\noindent The natural transformation \code{α} provides two additional morphisms that complete the diagram in \emph{D}: -\begin{verbatim} -αa :: F a -> G a -αb :: F b -> G b -\end{verbatim} - -\includegraphics[width=3.12500in]{images/3_naturality.jpg} - -Now we have two ways of getting from \code{F\ a} to \code{G\ b}. To +\begin{Verbatim}[commandchars=\\\{\}] +α\textsubscript{a} :: F a -> G a +α\textsubscript{b} :: F b -> G b +\end{Verbatim} +Now we have two ways of getting from \code{F a} to \code{G b}. To make sure that they are equal, we must impose the \newterm{naturality condition} that holds for any \code{f}: -\begin{verbatim} -G f ∘ αa = αb ∘ F f -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +G f ◦ α\textsubscript{a} = α\textsubscript{b} ◦ F f +\end{Verbatim} The naturality condition is a pretty stringent requirement. For -instance, if the morphism \code{F\ f} is invertible, naturality -determines \code{αb} in terms of \code{αa}. It \newterm{transports} -\code{αa} along \code{f}: +instance, if the morphism \code{F f} is invertible, naturality +determines \code{α\textsubscript{b}} in terms of \code{α\textsubscript{a}}. It \newterm{transports} +\code{α\textsubscript{a}} along \code{f}: -\begin{verbatim} -αb = (G f) ∘ αa ∘ (F f)-1 -\end{verbatim} +\begin{Verbatim}[commandchars=\\\{\}] +α\textsubscript{b} = (G f) ◦ α\textsubscript{a} ◦ (F f)\textsuperscript{-1} +\end{Verbatim} -\includegraphics[width=3.12500in]{images/4_transport.jpg} +\begin{wrapfigure}[7]{R}{0pt} +\raisebox{0pt}[\dimexpr\height-0.75\baselineskip\relax]{ +\fbox{\includegraphics[width=40mm]{images/4_transport.jpg}}}% +\end{wrapfigure} +\noindent If there is more than one invertible morphism between two objects, all these transports have to agree. In general, though, morphisms are not invertible; but you can see that the existence of natural @@ -111,8 +117,12 @@ maps objects to morphisms. Because of the naturality condition, one may also say that it maps morphisms to commuting squares --- there is one commuting naturality square in \emph{D} for every morphism in \emph{C}. -\includegraphics[width=3.12500in]{images/naturality.jpg} +\begin{figure}[H] +\centering +\fbox{\includegraphics[width=60mm]{images/naturality.jpg}} +\end{figure} +\noindent This property of natural transformations comes in very handy in a lot of categorical constructions, which often include commuting diagrams. With a judicious choice of functors, a lot of these commutativity conditions @@ -135,29 +145,26 @@ mapping is implemented by a higher order function \code{fmap} (or To construct a natural transformation we start with an object, here a type, \code{a}. One functor, \code{F}, maps it to the type -\code{F\ a}. Another functor, \code{G}, maps it to \code{G\ a}. +\code{F a}. Another functor, \code{G}, maps it to \code{G a}. The component of a natural transformation \code{alpha} at \code{a} -is a function from \code{F\ a} to \code{G\ a}. In pseudo-Haskell: - -\begin{verbatim} -alphaa :: F a -> G a -\end{verbatim} +is a function from \code{F a} to \code{G a}. In pseudo-Haskell: +\begin{Verbatim}[commandchars=\\\{\}] +alpha\textsubscript{a} :: F a -> G a +\end{Verbatim} A natural transformation is a polymorphic function that is defined for all types \code{a}: \begin{verbatim} alpha :: forall a . F a -> G a \end{verbatim} - -The \code{forall\ a} is optional in Haskell (and in fact requires +The \code{forall a} is optional in Haskell (and in fact requires turning on the language extension \code{ExplicitForAll}). Normally, you would write it like this: \begin{verbatim} alpha :: F a -> G a \end{verbatim} - Keep in mind that it's really a family of functions parameterized by \code{a}. This is another example of the terseness of the Haskell syntax. A similar construct in C++ would be slightly more verbose: @@ -165,14 +172,13 @@ syntax. A similar construct in C++ would be slightly more verbose: \begin{verbatim} template G alpha(F); \end{verbatim} - There is a more profound difference between Haskell's polymorphic functions and C++ generic functions, and it's reflected in the way these functions are implemented and type-checked. In Haskell, a polymorphic function must be defined uniformly for all types. One formula must work across all types. This is called \newterm{parametric polymorphism}. -C++, on the other hand, supports by default~\newterm{ad hoc polymorphism}, +C++, on the other hand, supports by default \newterm{ad hoc polymorphism}, which means that a template doesn't have to be well-defined for all types. Whether a template will work for a given type is decided at instantiation time, where a concrete type is substituted for the type @@ -190,30 +196,26 @@ polymorphic function of the type: \begin{verbatim} alpha :: F a -> G a \end{verbatim} - where \code{F} and \code{G} are functors, automatically satisfies the naturality condition. Here it is in categorical notation (\code{f} is a function \code{f::a->b}): -\begin{verbatim} -G f ∘ αa = αb ∘ F f -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +G f ◦ α\textsubscript{a} = α\textsubscript{b} ◦ F f +\end{Verbatim} In Haskell, the action of a functor \code{G} on a morphism \code{f} is implemented using \code{fmap}. I'll first write it in pseudo-Haskell, with explicit type annotations: -\begin{verbatim} -fmapG f . alphaa = alphab . fmapF f -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +fmap\textsubscript{G} f . alpha\textsubscript{a} = alpha\textsubscript{b} . fmap\textsubscript{F} f +\end{Verbatim} Because of type inference, these annotations are not necessary, and the following equation holds: \begin{verbatim} fmap f . alpha = alpha . fmap f \end{verbatim} - This is still not real Haskell --- function equality is not expressible in code --- but it's an identity that can be used by the programmer in equational reasoning; or by the compiler, to implement optimizations. @@ -224,10 +226,10 @@ define natural transformations in Haskell, imposes very strong limitations on the implementation --- one formula for all types. These limitations translate into equational theorems about such functions. In the case of functions that transform functors, free theorems are the -naturality conditions. {[}You may read more about free theorems in my -blog -\href{https://bartoszmilewski.com/2014/09/22/parametricity-money-for-nothing-and-theorems-for-free/}{Parametricity: -Money for Nothing and Theorems for Free}.{]} +naturality conditions.\footnote{ +You may read more about free theorems in my +blog \href{https://bartoszmilewski.com/2014/09/22/parametricity-money-for-nothing-and-theorems-for-free/}{Parametricity: +Money for Nothing and Theorems for Free}.} One way of thinking about functors in Haskell that I mentioned earlier is to consider them generalized containers. We can continue this analogy @@ -253,7 +255,6 @@ safeHead :: [a] -> Maybe a safeHead [] = Nothing safeHead (x:xs) = Just x \end{verbatim} - It's a function polymorphic in \code{a}. It works for any type \code{a}, with no limitations, so it is an example of parametric polymorphism. Therefore it is a natural transformation between the two @@ -263,7 +264,6 @@ condition. \begin{verbatim} fmap f . safeHead = safeHead . fmap f \end{verbatim} - We have two cases to consider; an empty list: \begin{verbatim} @@ -273,45 +273,39 @@ fmap f (safeHead []) = fmap f Nothing = Nothing \begin{verbatim} safeHead (fmap f []) = safeHead [] = Nothing \end{verbatim} - and a non-empty list: \begin{verbatim} fmap f (safeHead (x:xs)) = fmap f (Just x) = Just (f x) \end{verbatim} - -\begin{verbatim} +\begin{minted}[breaklines]{text} safeHead (fmap f (x:xs)) = safeHead (f x : fmap f xs) = Just (f x) -\end{verbatim} - +\end{minted} I used the implementation of \code{fmap} for lists: \begin{verbatim} fmap f [] = [] fmap f (x:xs) = f x : fmap f xs \end{verbatim} - and for \code{Maybe}: \begin{verbatim} fmap f Nothing = Nothing fmap f (Just x) = Just (f x) \end{verbatim} - An interesting case is when one of the functors is the trivial \code{Const} functor. A natural transformation from or to a \code{Const} functor looks just like a function that's either polymorphic in its return type or in its argument type. For instance, \code{length} can be thought of as a natural -transformation from the list functor to the \code{Const\ Int} functor: +transformation from the list functor to the \code{Const Int} functor: \begin{verbatim} length :: [a] -> Const Int a length [] = Const 0 length (x:xs) = Const (1 + unConst (length xs)) \end{verbatim} - Here, \code{unConst} is used to peel off the \code{Const} constructor: @@ -319,13 +313,11 @@ constructor: unConst :: Const c a -> c unConst (Const x) = x \end{verbatim} - Of course, in practice \code{length} is defined as: \begin{verbatim} length :: [a] -> Int \end{verbatim} - which effectively hides the fact that it's a natural transformation. Finding a parametrically polymorphic function \emph{from} a @@ -336,7 +328,6 @@ creation of a value from nothing. The best we can do is: scam :: Const Int a -> Maybe a scam (Const x) = Nothing \end{verbatim} - Another common functor that we've seen already, and which will play an important role in the Yoneda lemma, is the \code{Reader} functor. I will rewrite its definition as a \code{newtype}: @@ -344,20 +335,18 @@ will rewrite its definition as a \code{newtype}: \begin{verbatim} newtype Reader e a = Reader (e -> a) \end{verbatim} - It is parameterized by two types, but is (covariantly) functorial only in the second one: \begin{verbatim} -instance Functor (Reader e) where fmap f (Reader g) = Reader (\x -> f (g x)) +instance Functor (Reader e) where + fmap f (Reader g) = Reader (\x -> f (g x)) \end{verbatim} - For every type \code{e}, you can define a family of natural transformations from \code{Reader\ e} to any other functor \code{f}. We'll see later that the members of this family are always in one to one -correspondence with the elements of \code{f\ e} (the -\href{https://bartoszmilewski.com/2015/09/01/the-yoneda-lemma/}{Yoneda -lemma}). +correspondence with the elements of \code{f e} (the +\hyperref[the-yoneda-lemma]{Yoneda lemma}). For instance, consider the somewhat trivial unit type \code{()} with one element \code{()}. The functor \code{Reader\ ()} takes any type @@ -370,25 +359,22 @@ to the \code{Maybe} functor: \begin{verbatim} alpha :: Reader () a -> Maybe a \end{verbatim} - There are only two of these, \code{dumb} and \code{obvious}: \begin{verbatim} dumb (Reader _) = Nothing \end{verbatim} - and \begin{verbatim} obvious (Reader g) = Just (g ()) \end{verbatim} - (The only thing you can do with \code{g} is to apply it to the unit value \code{()}.) And, indeed, as predicted by the Yoneda lemma, these correspond to the -two elements of the \code{Maybe\ ()} type, which are \code{Nothing} -and \code{Just\ ()}. We'll come back to the Yoneda lemma later --- +two elements of the \code{Maybe ()} type, which are \code{Nothing} +and \code{Just ()}. We'll come back to the Yoneda lemma later --- this was just a little teaser. \section{Beyond Naturality}\label{beyond-naturality} @@ -416,20 +402,18 @@ at before: \begin{verbatim} newtype Op r a = Op (a -> r) \end{verbatim} - This functor is contravariant in \code{a}: \begin{verbatim} -instance Contravariant (Op r) where contramap f (Op g) = Op (g . f) +instance Contravariant (Op r) where + contramap f (Op g) = Op (g . f) \end{verbatim} - -We can write a polymorphic function from, say, \code{Op\ Bool} to -\code{Op\ String}: +We can write a polymorphic function from, say, \code{Op Bool} to +\code{Op String}: \begin{verbatim} predToStr (Op f) = Op (\x -> if f x then "T" else "F") \end{verbatim} - But since the two functors are not covariant, this is not a natural transformation in \textbf{Hask}. However, because they are both contravariant, they satisfy the ``opposite'' naturality condition: @@ -437,7 +421,6 @@ contravariant, they satisfy the ``opposite'' naturality condition: \begin{verbatim} contramap f . predToStr = predToStr . contramap f \end{verbatim} - Notice that the function \code{f} must go in the opposite direction than what you'd use with \code{fmap}, because of the signature of \code{contramap}: @@ -445,14 +428,12 @@ than what you'd use with \code{fmap}, because of the signature of \begin{verbatim} contramap :: (b -> a) -> (Op Bool a -> Op Bool b) \end{verbatim} - Are there any type constructors that are not functors, whether covariant or contravariant? Here's one example: \begin{verbatim} a -> a \end{verbatim} - This is not a functor because the same type \code{a} is used both in the negative (contravariant) and positive (covariant) position. You can't implement \code{fmap} or \code{contramap} for this type. @@ -461,7 +442,6 @@ Therefore a function of the signature: \begin{verbatim} (a -> a) -> f a \end{verbatim} - where \code{f} is an arbitrary functor, cannot be a natural transformation. Interestingly, there is a generalization of natural transformations, called dinatural transformations, that deals with such @@ -483,42 +463,47 @@ we know how to compose morphisms. Indeed, let's take a natural transformation α from functor F to G. Its component at object \code{a} is some morphism: -\begin{verbatim} -αa :: F a -> G a -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +α\textsubscript{a} :: F a -> G a +\end{Verbatim} We'd like to compose α with β, which is a natural transformation from functor G to H. The component of β at \code{a} is a morphism: -\begin{verbatim} -βa :: G a -> H a -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +β\textsubscript{a} :: G a -> H a +\end{Verbatim} These morphisms are composable and their composition is another morphism: -\begin{verbatim} -βa ∘ αa :: F a -> H a -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +β\textsubscript{a} ◦ α\textsubscript{a} :: F a -> H a +\end{Verbatim} We will use this morphism as the component of the natural transformation β ⋅ α --- the composition of two natural transformations β after α: -\begin{verbatim} -(β ⋅ α)a = βa ∘ αa -\end{verbatim} +\begin{Verbatim}[commandchars=\\\{\}] +(β \ensuremath{⋅} α)a = β\textsubscript{a} ◦ α\textsubscript{a} +\end{Verbatim} -\includegraphics[width=3.12500in]{images/5_vertical.jpg} +\begin{figure}[H] +\centering +\fbox{\includegraphics[width=60mm]{images/5_vertical.jpg}} +\end{figure} +\noindent One (long) look at a diagram convinces us that the result of this composition is indeed a natural transformation from F to H: -\begin{verbatim} -H f ∘ (β ⋅ α)a = (β ⋅ α)b ∘ F f -\end{verbatim} +\begin{Verbatim}[commandchars=\\\{\}] +H f ◦ (β \ensuremath{⋅} α)\textsubscript{a} = (β \ensuremath{⋅} α)\textsubscript{b} ◦ F f +\end{Verbatim} -\includegraphics[width=3.12500in]{images/6_verticalnaturality.jpg} +\begin{figure}[H] +\centering +\fbox{\includegraphics[width=60mm]{images/6_verticalnaturality.jpg}} +\end{figure} +\noindent Composition of natural transformations is associative, because their components, which are regular morphisms, are associative with respect to their composition. @@ -526,10 +511,9 @@ their composition. Finally, for each functor F there is an identity natural transformation 1\textsubscript{F} whose components are the identity morphisms: -\begin{verbatim} -idF a :: F a -> F a -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +id\textsubscript{F a} :: F a -> F a +\end{Verbatim} So, indeed, functors form a category. A word about notation. Following Saunders Mac Lane I use the dot for the @@ -540,10 +524,13 @@ usually stacked up vertically in the diagrams that describe it. Vertical composition is important in defining the functor category. I'll explain horizontal composition shortly. -\includegraphics[width=2.29167in]{images/6a_vertical.jpg} +\begin{figure} +\centering +\fbox{\includegraphics[width=60mm]{images/6a_vertical.jpg}} +\end{figure} The functor category between categories C and D is written as -\code{Fun(C,\ D)}, or \code{{[}C,\ D{]}}, or sometimes as +\code{Fun(C, D)}, or \code{{[}C, D{]}}, or sometimes as \code{DC}. This last notation suggests that a functor category itself might be considered a function object (an exponential) in some other category. Is this indeed the case? @@ -556,7 +543,10 @@ in a higher-level category \textbf{Cat}. Morphisms in that category are functors. A Hom-set in \textbf{Cat} is a set of functors. For instance Cat(C, D) is a set of functors between two categories C and D. -\includegraphics[width=2.23958in]{images/7_cathomset.jpg} +\begin{figure} +\centering +\fbox{\includegraphics[width=60mm]{images/7_cathomset.jpg}} +\end{figure} A functor category {[}C, D{]} is also a set of functors between two categories (plus natural transformations as morphisms). Its objects are @@ -572,12 +562,12 @@ As you may remember, in order to construct an exponential, we need to first define a product. In \textbf{Cat}, this turns out to be relatively easy, because small categories are \emph{sets} of objects, and we know how to define cartesian products of sets. So an object in a product -category C × D is just a pair of objects, \code{(c,\ d)}, one from C +category C × D is just a pair of objects, \code{(c, d)}, one from C and one from D. Similarly, a morphism between two such pairs, -\code{(c,\ d)} and \code{(c\',\ d\')}, is a pair of -morphisms, \code{(f,\ g)}, where -\code{f\ ::\ c\ ->\ c\'} and -\code{g\ ::\ d\ ->\ d\'}. These pairs of morphisms +\code{(c, d)} and \code{(c', d')}, is a pair of +morphisms, \code{(f, g)}, where +\code{f :: c -> c'} and +\code{g :: d -> d'}. These pairs of morphisms compose component-wise, and there is always an identity pair that is just a pair of identity morphisms. To make the long story short, \textbf{Cat} is a full-blown cartesian closed category in which there is @@ -612,6 +602,12 @@ In the case of \textbf{Cat} seen as a 2-category we have: 2-morphisms: Natural transformations between functors. \end{itemize} +\begin{wrapfigure}[11]{R}{0pt} +\raisebox{0pt}[\dimexpr\height-0.75\baselineskip\relax]{ +\fbox{\includegraphics[width=2.25000in]{images/8_cat-2-cat.jpg}}}% +\end{wrapfigure} + +\noindent Instead of a Hom-set between two categories C and D, we have a Hom-category --- the functor category D\textsuperscript{C}. We have regular functor composition: a functor F from D\textsuperscript{C} @@ -620,8 +616,7 @@ E\textsuperscript{C}. But we also have composition inside each Hom-category --- vertical composition of natural transformations, or 2-morphisms, between functors. -\includegraphics[width=2.25000in]{images/8_cat-2-cat.jpg} - +\noindent With two kinds of composition in a 2-category, the question arises: How do they interact with each other? @@ -631,13 +626,11 @@ Let's pick two functors, or 1-morphisms, in \textbf{Cat}: F :: C -> D G :: D -> E \end{verbatim} - and their composition: \begin{verbatim} -G ∘ F :: C -> E +G ◦ F :: C -> E \end{verbatim} - Suppose we have two natural transformations, α and β, that act, respectively, on functors F and G: @@ -646,8 +639,12 @@ respectively, on functors F and G: β :: G -> G' \end{verbatim} -\includegraphics[width=3.12500in]{images/10_horizontal.jpg} +\begin{figure}[H] +\centering +\fbox{\includegraphics[width=60mm]{images/10_horizontal.jpg}} +\end{figure} +\noindent Notice that we cannot apply vertical composition to this pair, because the target of α is different from the source of β. In fact they are members of two different functor categories: D \textsuperscript{C} and E @@ -658,50 +655,49 @@ category D. What's the relation between the functors G'∘ F' and G ∘ F? Having α and β at our disposal, can we define a natural transformation from G ∘ F to G'∘ F'? Let me sketch the construction. -\includegraphics[width=3.84375in]{images/9_horizontal.jpg} +\begin{figure}[H] +\centering +\fbox{\includegraphics[width=90mm]{images/9_horizontal.jpg}} +\end{figure} +\noindent As usual, we start with an object \code{a} in C. Its image splits into -two objects in D: \code{F\ a} and \code{F\'a}. There is also a +two objects in D: \code{F a} and \code{F'a}. There is also a morphism, a component of α, connecting these two objects: -\begin{verbatim} -αa :: F a -> F'a -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +α\textsubscript{a} :: F a -> F'a +\end{Verbatim} When going from D to E, these two objects split further into four objects: \begin{verbatim} G (F a), G'(F a), G (F'a), G'(F'a) \end{verbatim} - We also have four morphisms forming a square. Two of these morphisms are the components of the natural transformation β: -\begin{verbatim} -βF a :: G (F a) -> G'(F a) -βF'a :: G (F'a) -> G'(F'a) -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +β\textsubscript{F a} :: G (F a) -> G'(F a) +β\textsubscript{F'a} :: G (F'a) -> G'(F'a) +\end{Verbatim} The other two are the images of α\textsubscript{a} under the two functors (functors map morphisms): -\begin{verbatim} -G αa :: G (F a) -> G (F'a) -G'αa :: G'(F a) -> G'(F'a) -\end{verbatim} - +\begin{Verbatim}[commandchars=\\\{\}] +G α\textsubscript{a} :: G (F a) -> G (F'a) +G'α\textsubscript{a} :: G'(F a) -> G'(F'a) +\end{Verbatim} That's a lot of morphisms. Our goal is to find a morphism that goes from -\code{G\ (F\ a)} to \code{G\'(F\'a)}, a candidate for the +\code{G (F a)} to \code{G'(F'a)}, a candidate for the component of a natural transformation connecting the two functors G ∘ F and G'∘ F'. In fact there's not one but two paths we can take from -\code{G\ (F\ a)} to \code{G\'(F\'a)}: - -\begin{verbatim} -G'αa ∘ βF a -βF'a ∘ G αa -\end{verbatim} +\code{G (F a)} to \code{G'(F'a)}: +\begin{Verbatim}[commandchars=\\\{\}] +G'α\textsubscript{a} ◦ β\textsubscript{F a} +β\textsubscript{F'a} ◦ G α\textsubscript{a} +\end{Verbatim} Luckily for us, they are equal, because the square we have formed turns out to be the naturality square for β. @@ -713,9 +709,8 @@ We call this natural transformation the \newterm{horizontal composition} of α and β: \begin{verbatim} -β ∘ α :: G ∘ F -> G'∘ F' +β ◦ α :: G ◦ F -> G' ◦ F' \end{verbatim} - Again, following Mac Lane I use the small circle for horizontal composition, although you may also encounter star in its place. @@ -730,8 +725,12 @@ between categories. A natural transformation sits between two categories, the ones that are connected by the functors it transforms. We can think of it as connecting these two categories. -\includegraphics[width=3.12500in]{images/sideways.jpg} +\begin{figure}[H] +\centering +\fbox{\includegraphics[width=60mm]{images/sideways.jpg}} +\end{figure} +\noindent Let's focus on two objects of \textbf{Cat} --- categories C and D. There is a set of natural transformations that go between functors that connect C to D. These natural transformations are our new arrows from C @@ -762,9 +761,8 @@ transformation acting on this functor. So you'll often see this notation: \begin{verbatim} -F ∘ α +F ◦ α \end{verbatim} - where F is a functor from D to E, and α is a natural transformation between two functors going from C to D. Since you can't compose a functor with a natural transformation, this is interpreted as a @@ -774,9 +772,8 @@ horizontal composition of the identity natural transformation Similarly: \begin{verbatim} -α ∘ F +α ◦ F \end{verbatim} - is a horizontal composition of α after 1\textsubscript{F}. \section{Conclusion}\label{conclusion} @@ -808,15 +805,16 @@ natural transformation is a special type of polymorphic function. \section{Challenges}\label{challenges} \begin{enumerate} +\tightlist \item Define a natural transformation from the \code{Maybe} functor to the list functor. Prove the naturality condition for it. \item Define at least two different natural transformations between - \code{Reader\ ()} and the list functor. How many different lists of + \code{Reader ()} and the list functor. How many different lists of \code{()} are there? \item - Continue the previous exercise with \code{Reader\ Bool} and + Continue the previous exercise with \code{Reader Bool} and \code{Maybe}. \item Show that horizontal composition of natural transformation satisfies @@ -842,13 +840,3 @@ f :: String -> Int f x = read x \end{verbatim} \end{enumerate} - -Next: -\href{https://bartoszmilewski.com/2015/04/15/category-theory-and-declarative-programming/}{Declarative -Programming}. - -\section{Acknowledgments}\label{acknowledgments} - -I'd like to thank Gershom Bazerman for checking my math and logic, and -André van Meulebrouck, who has been volunteering his editing help.\\ -\href{https://twitter.com/BartoszMilewski}{Follow @BartoszMilewski} diff --git a/src/content/1.5/Products and Coproducts.tex b/src/content/1.5/Products and Coproducts.tex index f5bfef3..6393b1f 100644 --- a/src/content/1.5/Products and Coproducts.tex +++ b/src/content/1.5/Products and Coproducts.tex @@ -1,4 +1,4 @@ -The Ancient Greek playwright Euripides once said: ``Every man is like +\lettrine[lhang=0.17]{T}{he Ancient Greek} playwright Euripides once said: ``Every man is like the company he is wont to keep.'' We are defined by our relationships. Nowhere is this more true than in category theory. If we want to single out a particular object in a category, we can only do this by describing diff --git a/src/content/1.7/Functors.tex b/src/content/1.7/Functors.tex index ee842eb..e29a0de 100644 --- a/src/content/1.7/Functors.tex +++ b/src/content/1.7/Functors.tex @@ -795,9 +795,9 @@ object to itself, and every morphism to itself. So functors have all the same properties as morphisms in some category. But what category would that be? It would have to be a category in which objects are categories and morphisms are functors. It's a category of categories. But a -category of \newterm{all} categories would have to include itself, and we +category of \emph{all} categories would have to include itself, and we would get into the same kinds of paradoxes that made the set of all sets -impossible. There is, however, a category of all \newterm{small} categories +impossible. There is, however, a category of all \emph{small} categories called \textbf{Cat} (which is big, so it can't be a member of itself). A small category is one in which objects form a set, as opposed to something larger than a set. Mind you, in category theory, even an diff --git a/src/content/1.8/Functoriality.tex b/src/content/1.8/Functoriality.tex index fbcea95..2894ea8 100644 --- a/src/content/1.8/Functoriality.tex +++ b/src/content/1.8/Functoriality.tex @@ -1,4 +1,4 @@ -Now that you know what a functor is, and have seen a few examples, let's +\lettrine[lhang=0.17]{N}{ow that you} know what a functor is, and have seen a few examples, let's see how we can build larger functors from smaller ones. In particular it's interesting to see which type constructors (which correspond to mappings between objects in a category) can be extended to functors @@ -639,7 +639,7 @@ the defaults for \code{lmap} and \code{rmap}, or implementing both \code{lmap} and \code{rmap} and accepting the default for \code{dimap}. -\begin{figure} +\begin{figure}[H] \centering \fbox{\includegraphics[width=60mm]{images/dimap.jpg}} \caption{dimap} diff --git a/src/content/1.9/Function Types.tex b/src/content/1.9/Function Types.tex index bb8afce..abfa1fc 100644 --- a/src/content/1.9/Function Types.tex +++ b/src/content/1.9/Function Types.tex @@ -1,4 +1,4 @@ -So far I've been glossing over the meaning of function types. A function +\lettrine[lhang=0.17]{S}{o far I've} been glossing over the meaning of function types. A function type is different from other types. \begin{wrapfigure}[13]{R}{0pt} @@ -111,7 +111,7 @@ That's when we apply our secret weapon: ranking. This is usually done by requiring that there be a unique mapping between candidate objects --- a mapping that somehow factorizes our construction. In our case, we'll decree that \code{z} together with the morphism \code{g} from -\code{z×a} to \code{b} is \newterm{better} than some other +\code{z×a} to \code{b} is \emph{better} than some other \code{z'} with its own application \code{g'}, if and only if there is a unique mapping \code{h} from \code{z'} to \code{z} such that the application of \code{g'} factors diff --git a/src/sicp.texi b/src/sicp.texi index 3c7ad4f..301320b 100644 --- a/src/sicp.texi +++ b/src/sicp.texi @@ -114,6 +114,9 @@ PDF compiled by @url{https://github.com/hmemcpy/milewski-ctfp-pdf, Igal Tabachni \chapter{Function Types} \subfile{content/1.9/Function Types} +\chapter{Natural Transformations} +\subfile{content/1.10/Natural Transformations} + \backmatter @unnumbered Acknowledgments