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Boris Marinov
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@ -6,7 +6,9 @@ title: Sets
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Sets
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===
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Let's begin our inquiry by looking at the basic theory of sets. You will understand why shortly. For now, it suffices to say that sets are an example of a category.
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Let's begin our inquiry by looking at the basic theory of sets. Set theory and category theory share many similarities. We can view category theory is a *generalization* of set theory. That is it is is meant to describe the same thing as set theory (everything?), but to do it in a more absract manner, one that is more versatile and (hopefully) simpler.
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In other words sets are an *example of a category* (the *proto-example*, we might say), and it is useful to have examples.
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What is an Abstract Theory
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===
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@ -14,63 +16,69 @@ What is an Abstract Theory
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> Instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced.
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> Bertrand Russell, from Introduction to Mathematical Philosophy
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Most scientific and mathematical theories have a specific domain, to which they are related, and in which they are valid. They are created with this domain in mind and are not intended to be used outside of it. For example, Darwin's theory of evolution is created so it explains how different biological species came to be. And quantum mechanics is a description of what particles are at the low scale. Even the work of most mathematicians, although it is not bound to a specific domain, is strongly related to it, as differential equations are linked to the description of events that change over time.
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Most scientific and mathematical theories have a specific *domain*, to which they are tied to, and in which they are valid. They are created with this domain in mind and are not intended to be used outside of it. For example, Darwin's theory of evolution is created in order to explain how different *biological species* came to evolve using natural selection, quantum mechanics is a description of how particles behave at a specific scale etc.
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Set theory and category theory are different. They are not created to provide a rigorous explanation of how a particular phenomenon works, but to try to provide a more general framework for explaining all kinds of phenomena. Theories that are like that are called *abstract* theories. All theories *use abstraction*, else they would be pretty useless (without it Darwin would have to speak about specific animal species or even individual animals) but few are inherently abstract, so some of their core concepts are left unspecified. Or in other words, all theories are applicable outside of their domains, but set theory and category theory do not have a domain, to begin with.
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Even most mathematical theories, although not inherently *bound* to a specific domain, like the scientific ones, are often strongly related to one, as differential equations are linked to how events change over time.
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Set theory and category theory are different. They are not created to provide a rigorous explanation of how a particular phenomenon works. Instead they provide a more general framework for explaining all kinds of phenomena. They work less like tools and more like languages for defining tools. Theories that are like that are called *abstract* theories.
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The borders of the two are sometimes blurry, because all theories *use abstraction*, otherwise they would be pretty useless: without abstraction Darwin would have to speak about specific animal species or even individual animals. But theories have core concept that don't refer to anything in particular, but instead are left for people to generalize on. All theories are applicable outside of their domains, but set theory and category theory do not have a domain to begin with.
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Concrete theories, like the theory of evolution, are composed of concrete concepts. For example the concept of a *population*, also called a *gene-pool*, refers to a group of individuals that can interbreed. Abstract theories, like set theory, are composed of abstract concepts, like the concept of a set. The concept for a set by itself does not refer to anything. However, we cannot say that it is empty concept, as there are countless things that can be represented by sets, like for example a gene pools can be (very aptly) represented by sets of individual animals. And species of animals can too be represented by set - a set of all populations that can theoretically interbreed.
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You already see how abstract theories may be useful. Because they are so simple, they can be used as building blocks of many concrete theories. Because they are common, they can be used to unify and compare different concrete theories, by putting these theories in common grounds (this is very characteristic of category theory, as we will see later.) Moreover, good (abstract) theories can serve as *mental models* for developing our thoughts.
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<!-- comic - brain on category theory -->
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<!--
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People have tried to be precise and at the same time down to Earth for centuries, and only recently discovered that "precise and down to Earth" is an oxymoron. Let's take Euclidian geometry as an example. Yes, Euclidian geometry is precise, because it is valid for all sets of objects, called ("point" "line" "angle" and "circle" etc.), which have relationships, as defined by the five famous axioms. Yes, geometry does, in many instances, describe the natural world, because there are many sets of objects which have these relations. However, its "precise" part and it's "down to Earth" part have nothing to do with each other. We can, for example, define a point as any stain on the floor of your room and that a line as a piece of duct tape, put on the same floor - that will be a completely valid application of the Euclidian laws, albeit not very useful one. Or we can try to use geometry to reason about points on the surface of the Earth, which is a very useful application, of geometry, however not of Euclidian geometry, because Euclidian geometry only describes points on a flat plane, and the Earth is not flat. You can argue that these are actually two separate theories there, which just happen to be perceived as one. You have the axioms, or the postulates on one hand, which are not useful for anything on their own, and you have applications in science and engineering which are somewhat based on them, but not quite.
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-->
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Sets
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===
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Everything in set theory is defined in terms of sets. A set is a collection of things where the "things" can be anything you want. Consider, for example, these balls:
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Perhaps unsurprisingly, everything in set theory is defined in terms of sets. A set is an collection of things where the "things" can be anything you want (like individuals, populations, genes etc.) Consider, for example, these balls.
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For example, let's construct a set, call it **G** (as gray), consisting of *all* of them. This is how we can mark it:
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Let's construct a set, call it \\(G\\) (as gray) that contains *all* of them as elements. There can only be one such set: Because a set has no structure (there is not order, no ball goes before or after another, there are no members which are "special" with respect to their membership of the set) two sets that contain the same elements are just two pictures of the same set.
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The example may look childish but in fact, this set is just as valid as any other set.
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The things that are contained in the set are called its *elements*. A set is like a summary of its elements. It has no other structure, for example, there is not order, no ball goes before or after another, there are no members which are "special" with respect to their membership in the set. Two sets that contain the same elements are also the same.
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The key insight about what a set it is that it enables you to reason about several things as if they were one.
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This example may look overly-simple but in fact it is just as valid as any other one.
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The key insight that makes the concept useful is the fact that it enables you to reason about several things as if they were one.
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Subsets
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---
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Let's construct one more set. The set of *all balls that are warm color*. I will call it **Y**, because in the diagram is colored in **y**ellow.
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Let's construct one more set. The set of *all balls that are warm color*. Let's call it \\(Y\\) (because in the diagram is colored in **y**ellow.)
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Notice that **Y** contains just elements that are also present in **G**. That is, every element of the set of **Y** is also an element in the set **G**. When two sets have this relation, we may say that **Y** is a *subset* of **G** (or **Y ⊆ G**).
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Notice that \\(Y\\) contains only elements that are also present in \\(G\\). That is, every element of the set of \\(Y\\) is also an element in the set \\(G\\). When two sets have this relation, we may say that \\(Y\\) is a *subset* of \\(G\\) (or \\(Y \subseteq G \\) ). A subset resides completely inside its superset When the two are drawn together.
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Singleton Sets
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---
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The set of all *red balls* contains just one ball.
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The set of all *red balls* contains just one ball. We said above that sets summarize *several* elements into one. Still, sets that contain just one element are perfectly valid - simply put, there are things that are *one of a kind*. The set of queens of England is a singleton set. The set of books written by the American writer Harper Lee and published during her lifetime is a singleton set.
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Like we said above, sets are all about for summarizing *several* elements into one. Still, sets that contain just one element are perfectly valid. Simply, there are things which are one of a kind. Furthermore, if I have a function which expects a set of given items, here shouldn't be any issue if the "items" are just one item. Or to take a real-life example, the set of queens of England is a singleton set. The set of books written by the American writer Harper Lee and published during her lifetime is a singleton set - she has published just one novel.
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What's the point of the singleton set? Well, it is part of the language of set theory e.g. if we have a function which expects a set of given items, but if there is only one item that meets the criteria, we can just create a singleton set with that item.
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The Empty set
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---
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Of course if one is a valid answer, so can be zero. If we want a set of all *black balls* **B** or all the *white balls*, **W**, the answer to all these questions is the same - the empty set. Or in other words.
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Of course if one is a valid answer, so can be zero. If we want a set of all *black balls* \\(B\\) or all the *white balls*, \\(W\\), the answer to all these questions is the same - the empty set. Or in other words.
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Note that a set is defined only by the items it contains, which means that there is no difference between the set that contains zero *balls* and the set that contains zero *numbers*, for instance. In other words, the empty set is *unique* set, which makes it a very special one. Formally, the empty set is marked with the symbol \\(\varnothing\\) (so \\(B = W = \varnothing\\)).
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Note that a set is defined only by the items it contains, which means that there is no difference between the set that contains zero *balls* and the set that contains zero *numbers*, for instance. In other words, the empty set is *unique* set, which makes it a very special one. Formally, the empty set is marked with the symbol **∅** (so **B = W = ∅**).
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The empty set is a special one, for example, it is a subset of every other set (mathematically speaking, **∀ A \| ∅ ⊆ A**)
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We will encounter the empty set again.
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The empty set is a special one, for example, it is a subset of every other set or mathematically speaking, \\(\forall A \to \varnothing \subseteq A\\)) (\\(\forall\\) means "for all")
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Functions
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===
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@ -80,9 +88,9 @@ Functions
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A function is a relationship between two sets that matches each element of one set, called the *source set* of the function, with exactly one element from another set, called the **target set** of the function.
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These two sets are also called the *domain* and *codomain* of the function, or its *input* and *output*. In programming, they go by the name of *argument type* and *return type*. In logic, they correspond to the *premise* and *conclusion* (we will get there.) We might also say, depending on the situation, that a given function *goes* from this set to that other one, *connects* this set to the other. Or that it *converts* a value from this set to one of the the other one. These different terms show the many different faces of functions.
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These two sets are also called the *domain* and *codomain* of the function, or its *input* and *output*. In programming, they go by the name of *argument type* and *return type*. In logic, they correspond to the *premise* and *conclusion* (we will get there.) We might also say, depending on the situation, that a given function *goes* from this set to that other one, *connects* this set to the other, or that it *converts* a value from of this set to one of the the other one. These different terms demonstrate the multifaceted nature of the concept of function.
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Here is a function, **f** which converts each ball from the set **R** to the ball with the opposite colour in another set **G** (in mathematics a function's name is often accompanied by the names of its source and target sets, like this: **f: R → G**)
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Here is a function, \\(f\\) which converts each ball from the set \\(R\\) to the ball with the opposite colour in another set \\(G\\) (in mathematics a function's name is often accompanied by the names of its source and target sets, like this: \\(f: R → G\\))
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@ -220,6 +228,8 @@ The concepts of types and sets are related. The concept of sets is simpler - in
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- A type cannot contain other types, just values.
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- value can be a member of only one type (there exist the concept of are subtypes, just as there are subsets, but again things are more strict).
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{% if site.distribution == 'print'%}
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Russell's Paradox
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---
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@ -243,6 +253,8 @@ If we look at the definition, we recognize that the set that we just defined - *
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Hm, something is not quite right with this diagram as well - Because of the new adjustments that we made *R* **contains itself**. And removing it from the set would just bring us back to the previous situation. This is Russell's paradox. There are, of course, multiple theories that it does not apply to.
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{%endif%}
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Functional Composition
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===
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@ -98,12 +98,6 @@ In order to form the correct intuition about monoids, it is sometimes useful to
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In general, we use monoids and related structures as a way to model how a set of (associative) actions that are performed on a given object (or objects) alter it's state. We can do that, provided that the object's state is determined solely by the actions that are performed on it, this allows us to leave the object out of the equation and concentrate on how the actions are combined. And as per usual, the actions (and elements) can be anything, from mixing colors in paint, or adding a quantities to a given set of things etc.
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<!--
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TODO
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Free Monoids
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===
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-->
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Other monoid-like objects
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===
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@ -180,11 +174,11 @@ Enumerating all the rotations of a more complex geometrical figure looks quite m
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But it's much simpler to grasp if we notice the following: although our group has many actions, and there are more still for figures with more sides (if I am not mistaken, the number of actions is equal to the number of the sides), all of those actions can be reduced to the repetitive application of just the simplest action, (the 120-degree rotation for triangles and the 45-degree rotation for octagons). Let's make up a symbol for this rotation.
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But it's much simpler to grasp if we notice the following: although our group has many actions, and there are more still for figures with more sides (if I am not mistaken, the number of actions is equal to the number of the sides), *all those actions can be reduced to the repetitive application of just the simplest action*, (the 120-degree rotation for triangles and the 45-degree rotation for octagons). Let's make up a symbol for this action.
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Groups and monoids that have this "main" action (called a *generator*) that, when applied enough times, can get you to any state are called *cyclic groups*. All rotation groups are cyclic groups. Another example of a cyclic groups is, yes, the integers under addition. Here we can use **+1** or **-1** as generators.
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Groups and monoids that have this "main" action (called a *generator*) that, when applied enough times, can get you to any state of the group, are called *cyclic groups*. All rotation groups are cyclic groups. Another example of a cyclic groups is, yes, the integers under addition. Here we can use **+1** or **-1** as generators.
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@ -267,6 +261,8 @@ We already saw one way to create non-cyclic abelian groups - by creating a produ
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We can use this law to gain intuitive understanding of the what abelian groups are, but also to test whether a given group can be broken down to a product of more elementary groups.
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{% if site.distribution == 'print' %}
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Color-mixing monoid as a product
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---
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@ -288,6 +284,8 @@ Or alternatively, you can view it as having multiple states, representing the di
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In both cases the monoid would be cyclic.
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{%endif%}
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Groups/monoid of rotations and reflections
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---
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@ -374,7 +372,9 @@ Categories have an identity morphism for each object, so for categories with jus
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Categories provide a way to compose two morphisms with an appropriate type signature, and for categories with one object this means that all morphisms should be composable with one another. And the monoid operation does exactly that - given any two objects (or two morphisms, if we use the categorical terminology), it creates a third.
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Group presentations
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{% if site.distribution == 'print' %}
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Group/monoid presentations
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---
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When we view cyclic groups/monoids as categories, we would see that they correspond to categories that (besides having just one object) also have *just one morphism* (which, as we said is called *generator*), along with the morphisms that are created when this morphism is composed with itself. In fact the infinite cyclic monoid (which is isomorphic to the integers), can be completely described by this simple definition.
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@ -385,30 +385,47 @@ This is so, because applying the generator again and again yields all elements o
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Finite cyclic groups/monoids are the same, except that their definition contains an additional law, stating that that once you compose the generator with itself **n** number of times you get identity morphism. For the cyclic group **Z3** which can be visualized as the group of triangle rotations, this law states that composing the generator with itself **3** times yields the identity morphism.
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Finite cyclic groups/monoids are the same, except that their definition contains an additional law, stating that that once you compose the generator with itself **n** number of times you get identity morphism. For the cyclic group **Z3** (which can be visualized as the group of triangle rotations) this law states that composing the generator with itself **3** times yields the identity morphism.
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Composing the group generator with itself, according to the laws, yields the three morphisms of Z3.
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Composing the group generator with itself, and then applying the law, yields the three morphisms of **Z3**.
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Representing product groups in this way is just a little more complicated than composing the generators and their laws. Let's take Klein four as an example, Klein four has two generators that it inherits from the groups that form it (we can denote them **f** and **g**) and two laws. **f • f = id** and **g • g = id**.
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Representing product groups in this way is just a little more complex than composing the generators and their laws. But just a little. Let's take Klein four as an example, Klein four has two generators that it inherits from the groups that form it (which we viewed like vertical and horizontal rotation of a non-square rectangle) and two laws that it inherits from them.
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To make the representation complete, we add the law that we can combine the two generators: **f • g • f • g = id**
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To make the representation complete, we add the law that we can combine the two generators.
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And then, if we start applying the two generators and follow the laws, we get the four elements.
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**id**
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**f**
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**g**
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**f • g**
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The set of generators and laws that defines a given group is called the **presentation of a group**.
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The set of generators and laws that defines a given group is called the **presentation of a group**. Every group has a presentation (and finite groups always have a finite presentation.)
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Free groups/monoids
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Free monoids
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---
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We saw how different selections of laws give rise to different types of monoids. But what type of monoid would we get if we have *no laws*.
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We saw how picking a different selection of laws gives rise to different types of monoid. But what monoid would we get if we pick no laws at all? This monoid (or rather, monoids, as we get a different one depending on the set we picked) is called a *free monoid*. Here "free" is used in the sense that once you have the set, you can upgrade it to a monoid for free (i.e. without having to define anything else.)
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If you revisit the previous section you will notice that we already saw one such monoid. The free monoid with just one generator is the monoid of integers.
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We can make a free monoid from the set of colorful balls - the monoid's elements would be all possible combinations of the balls.
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The free monoid is a special one - each element of the free monoid over a given set, can be converted to a corresponding element in any other other monoid that uses the same set by just applying the monoid's laws. For example, here is how the elements above would look like if we apply the laws of the color-mixing monoid.
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**Task:** Write up the laws of the color-mixing monoid.
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If we put out programmers' hat, we will see that the type of the free monoid under the set T (which we can denote as `FreeMonoid<T>`) is isomorphic to the type `List<T>` (or `Array<T>`, if you prefer) and that the intuition behind the special property that we described above is actually very simple: it is the fact that keeping objects in a list allows you to convert them to any other structure i.e. when we want to perform some manipulation on a bunch of objects, but we don't know exactly what this manipulation is, we just keep a list of those objects until it's time to do it.
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{% endif %}
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390
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390
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Normal file
@ -0,0 +1,390 @@
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@ -345,6 +345,8 @@ And as a result of that, all "circle" relationships (e.g. where you have a weake
|
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|
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All of that structure arises naturally from the simple law of transitivity.
|
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|
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{% if site.distribution == 'print' %}
|
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|
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Maps as preorders
|
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---
|
||||
|
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@ -372,6 +374,8 @@ But as we saw, a finite state machine is similar to a preorder with a greatest a
|
||||
|
||||
Finite state machines are used in organization planning e.g. imagine a process where an item gets manifactured, gets checked by a quality control person, who, if they find some defficiencies, pass it to the necessary repairing departments and then they check it again and send it for shipping - this process can be modelled by the above diagram.
|
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{%endif%}
|
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|
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<!--
|
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|
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Or alternatively, a finite state machine is just a preorder with a greatest and least object, in which the relations between the objects are represented by functions.
|
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|
||||
@ -341,11 +341,11 @@ And now let's talk about one concept that is completely unrelated to functors, n
|
||||
|
||||
For example, the function that maps the current time to the distance travelled by some object is monotonoc because the distance travelled increases (or stays the same) as time increases.
|
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|
||||
|
||||
|
||||
|
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If we plot this or any other monotonic function on a line graph, we see that it goes just one direction.
|
||||
|
||||
|
||||
|
||||
|
||||
Now we are about to prove that monotonic functions are functors too, ready?
|
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|
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@ -372,10 +372,11 @@ If we compose the two morphisms in the source order, we get a morphism **g•f :
|
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|
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But both morphisms **F(g•f)** and **F(g)•F(f)** have the signature **F(a) ➞ F(c)** and so they must be equal to one another.
|
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|
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<!--
|
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|
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Forgetful Functors
|
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===
|
||||
|
||||
<!--
|
||||
Functors in logic
|
||||
===
|
||||
-->
|
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@ -383,7 +384,7 @@ Functors in logic
|
||||
The category of small categories
|
||||
===
|
||||
|
||||
Ha, I got you this time (or at least I hope I did). You probably thought that I won't introduce another category in this chapter, but this is exactly what I am going to do. And another surprise is that the new category won't be the category of functors (don't worry, we will introduce that in the next chapter.) Instead, we will examine the category of (small) categories.
|
||||
Ha, I got you this time (or at least I hope I did) - you probably thought that I won't introduce another category in this chapter, but this is exactly what I am going to do. And another surprise is that the new category won't be the category of functors (don't worry, we will introduce that in the next chapter.) Instead, we will examine the category of (small) categories.
|
||||
|
||||
This category has all the categories that we saw so far as objects and functors are its morphisms.
|
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|
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@ -395,11 +396,15 @@ There isn't much to say about this category, it is just a good exercise to try t
|
||||
|
||||
[The category of categories](category_of_categories)
|
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|
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{% if site.distribution == 'print' %}
|
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|
||||
The ghost of Russell's paradox
|
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---
|
||||
|
||||
We say *small* because otherwise the category will include itself and it's going to be Russell's paradox all over again.
|
||||
|
||||
{% endif %}
|
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|
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Categories all the way down
|
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---
|
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|
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|
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486
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486
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After Width: | Height: | Size: 27 KiB |
@ -8,6 +8,8 @@ name: Category Theory Illustrated
|
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# Short bio or description (displayed in the header)
|
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description: A book about category theory
|
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|
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distribution: print
|
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#
|
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# Flags below are optional
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#
|
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|
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@ -3,7 +3,14 @@
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<head>
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<title> {{site.name}} - {{ page.title }}</title>
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<link rel="stylesheet" type="text/css" href="{{site.baseurl}}/style.css">
|
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<link rel="stylesheet" type="text/css" href="{{site.baseurl}}/custom-style.css">
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<script type="text/javascript" id="MathJax-script" async
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src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
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</script>
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</head>
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<body>
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Loading…
Reference in New Issue
Block a user