Video Infoblog - Category Theory: The Beginner’s Introduction (Lesson 1 Video 1)
VIDEO
15 Jun 2015
Lesson 1 is concerned with defining the category of Abstract Sets and Arbitrary Mappings. We also define our first Limit and Co-Limit: The Terminal Object, and the Initial Object.
Other topics discussed include Duality and the Opposite (or Mirror) Category.
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The Facebook Group is: http://is.gd/SJlp7I
Martin Codrington
welcome to category theory the beginners introduction I am Martin Cottington and this is lesson 1 video 1 of 6 welcome to the first video of the first lesson of what I hope will be a complete accessible and excitin aspiration of I am dr. Martin Codrington and I'll tell you a little bit about me at the end of this video let's first discuss the intent and the intended audience of this video series intend to provide a self-contained introduction to category theory no mathematics prerequisites it will focus on the ways in which it can be practically applied to solve in many different types of real-world problems for this reason our exploration be project-based it will be divided into sections that will consist of five to ten lessons at the end of each section we will apply what we have learned to design a software that implements it for the first few sections the area focus will be music theory there are two reasons for choosing music rather than say data analysis the area to which I've done most of my category theory work the first is because I wanted to reach the widest possible audience category theory can be applied to everything from pure and applied mathematics anthropology economics financial engineering philosophy and linguistics just to name I would like anyone who is interested to be able to follow along not just in the theoretical discussions but in the application phase as well also many of the sets we'll be dealing with in the beginning with music are very small on the order of seven twelve seventeen elements so we can easily visualize the internal diagrams with the maps we discuss and with music you can hear the results of our theoretical aspirations in a few weeks when we were able to write software that creates a melody and harmonizes it beautifully and takes that melody and translates it to another mode and modifies it reharmonization satellite what it did use a normal language that you can understand you will quickly see the power of your theory as this sounds like a lot are difficult to design in program stick around for the first seven lessons where we will be exploring the category of abstract sets and arbitrary mappings and introduced a category of sets of an endomorphism commonly interpreted as dynamical systems or tone of tones and it's subcategory the category of sets of a permutation with these categories alone we can do all these things rather easily this first lesson is intended to give you an overview of the category theory way of thinking we will go through the definition of a category and get used to constructing and reasoning with commutative diagrams in this video you will go over the plan structure of our exploration in category theory and I will discuss the one music prerequisite you need understanding of the 12 pitch in video 2 I will give the definition of a general category enough s the category of abstract sets and arbitrary mappings and we will begin to go through the definition of that category by exploring the objects and arrows and s and then build our own category based on s music in video three we will discuss composition in detail with abstract examples and examples from music s the topic of video four is the associative law the identity arrow and the identity laws in video five we will learn how to calculate the number of maps between sets and s and get our first introduction to finite inverse limits and Col limits Universal mapping properties as we will call them in s we will introduce them by discussing the terminal and initial object this will lead us to discuss in the nature of maps and s up the opposite or mirror category as we see to understand duality armed with the powerful terminal object we will rid ourselves of the set theory baggage that we've been carrying around by redefining familiar concepts about sets in the language of category theory and we will formulate our first proofs by constructing commutative diagrams how much we will be covering in category theory the quick answer is a lot but slowly my goal is to help you form an intuition for all the elementary concepts before we use them to build higher-order concepts so for example it makes no sense formulating with natural transformation before you have a good intuition for founder's and white formulated foundries if you don't first have a fundamental understanding of a few concrete categories this method is especially necessary since we will be applying everything we learn we will start first with elementary concrete category theory by describing s we will introduce and explain many of the popular finite inverse limits and co limits such as the terminal and initial object monic and epic arrows pullbacks and push outs equalizers and co equalizers and products and koch brothers we will also discuss in detail impart and constructs such as a truth value object and its relation to sub objects and map objects etc at the end we will discuss the notion of the limit more formally and begin to define our own limits in music s discussing s in detail prepare us for discussing other categories as we shall see many categories of structure sets can be derived by considering limits and Coulomb X as objects for example the category of variable sets dynamical systems permutations graphs various types groups of various types pull sets bouquets by sets pointer sets and many more can be derived from the axioms or the limits and set we will in turn go through the limits and colormix in these categories explain their use and share how we can form even more structured categories by objectifying their limits the pace at which we move depends on the group lessons will ideally be produced at a rate of one every 1 to 1.5 weeks at the end of each lesson I will listen to your feedback upload supplemental videos if anything needs further explanation and after this I will upload the problems that video and also post a PDF of the problem set so that you can have some interesting problems to work on while I make the next lesson every concept in category theory is connected in interesting and complex ways and often I will be unable to resist the temptation to point out these connections like for example when discussing the identity laws we see that the category of sets of an morphism and in general any category see if an Animorph ism is logically implied by the axioms of a category itself whenever I give one of these a size things will go green to indicate it you can follow along if you like but if you don't fully understand don't worry because we will be discussing that concept in detail in the future category theory is very powerful I can give a complete introduction I can give an accessible introduction but I can't also make it a quick introduction it will take about five or six lessons before we have covered enough for you to get a glimpse of the full potential of the theory and I will take another few months for us to cover most of our ground these videos are intended to support discussion in the Facebook group product of category theory music and computer science for now this is where we will be discussing and applying the content in these videos so let's get started by discussing the format of our this cycle we will follow will allow us to learn a little then apply it in the beginning it will take a while to complete one cycle but as we progress the pace will pick up first we'll have a category theory overview where we introduce a theoretical concepts we will use in our formulation there will be some examples from music but just simple examples used to illustrate a certain point we will not do any detail formulation at this stage to category theory deep dive here we will take each concept introduced in the overview and catalog the many ways we could apply these concepts to music theory this will give you a better understanding both of the concepts and the ways that they can be applied we will use music theory informally during this phase as well but we will learn more about it as we progress three music theory formulation here we will actually begin formulating the music theory we will choose the best tools from our knowledge of category theory so far and use these tools to model and to understand certain concepts in music theory we will then identify the questions we would like to ask with these formulations or how these formulations can help us further advance our knowledge of music theory for is a software implementation here we will design the implementation that most closely mirrors the formulations which will allow us to create ways to answer the questions that we have formulated in the previous step let me illustrate by giving an example of the first cycle don't worry if you don't understand any of this this is not the actual formulation we will use and we will go through all these concepts in great detail to another discussion stage one I will introduce the category of abstract sets and arbitrary muffins in a series of lessons in stage two we will start by representing the 12 pitch classes as a set X we can then define an isomorphism n into a set C of pitch class names and we will see how the finite inverse limits in column sets will allow us to represent some more useful musical in state 3 after we have explored all the possible ways we could represent certain concepts we will form a list of the questions we would like to ask and figure out how best to do this within the theory for example ask in which classes the major minor mode have in common is best represented as a question about sub objects given to sub objects mi MJ of X representing the major minor mode respectively can you find a torus about 2a of s that is also sub object of mi MJ and moreover that a is the largest set they satisfy these conditions in stage 4 using prime injection and fundamental theorem of arithmetic we can design an algorithm that closely mirrors the formulation for example we can define a Mon at map from s to and natural numbers that associates the nth prime of the F element of X so that the major mode for example can be represented by the unique integer 29 million 150 3410 and the minor mode by 14 million seven hundred and thirteen thousand seven hundred and ninety at similar all the pitch classes they have in common is equivalent to ax and identity of the set a which is equivalent to a sand in this formulation what is the greatest common divisor of twenty nine million one hundred and fifty three thousand four hundred and ten and fourteen million seven hundred and thirteen thousand seven hundred and ninety designing algorithms in this way will allow for better understand of the concepts and will allow non programmers actually design the outline of computer algorithms without knowing any computer science or if all having any practical experience with programming after completing all four stages of a cycle we will repeat the entire process with some new collection of theoretical concepts maybe will define and explore three or four new categories in one go then repeat the process I haven't decided yet let's see how things go for the first cycle but in real life you really ever work in just one category but rather you'll find yourself moving between ten to fifteen categories in one formulation well at least is how I work so the more quickly we can learn a few different categories and discuss filters in more detail the happier lives will be for the music prerequisites you'll just need to understand the concept of the 12 pitch classes if you look at a keyboard you will notice that there are groups of two black notes and groups of three consider only the groups of two black notes the white note to the left of the first black note in that group is a C since this is a repeated pattern on the keyboard we can pick one C then take all the notes between dot C and the next C but not including the next and studied these separately now we will call these pitch classes instead of notes and technically there aren't notes until they've been played different C's on the keyboard or different pitches but the same pitch class any mode or scale can be formed from these pitch classes we say that a mode is a subset of these pitch classes to begin with you'll define each more starting on C and as we move further along we learn how to transpose from C major to F major for example but for now when we talk of major or minor we're referring to C major and C minor respectively the major scale for example is made up of all the white pitch classes in this collection notice that each Platinum can have one of two names this is because each black note lies between two white notes one to left let's call it L I want to write less color R so that the black note can be called L sharp because it's higher or sharper than L or R flat because it's lower or flatter than R so this first black note can be called C sharp or D flat because it's between C and D in some cases in music one name must be used instead of another because of conventions for example the minor mode is made up of the following pitches it would be incorrect according to a traditional music theory to call the E flat D sharp or the a flat G sharp the reason for this is that these modes are built with the I data they must contain one of each pitch name so for example if we construct C minor we have C D we already have a kind of D so next nut must be a kind of e e flat so we have E flat F and G now we need some kind of a so a flat instead of G sharp and B not almost are constructed like this some modes can have an E flat in the e for example but many of the most used in early music are constructed this way the chromatic scale is made up of all these pitches we can list the pitches of the chromatic scale like this while these things imply we'll explore later but for now this is all you need to understand from music to get started I was born in Barbados and in 2005 I completed my Bachelors of Science in chemistry from Morgan State University I received my PhD from Texas A&M University in 2012 my dissertation research was focused on the analysis of nuclear collisions with the solenoidal tracker a trick star at the relativistic heavy ion collider Rick our Haven National Laboratories in Long Island New York I started exploring category theory in 2011 I was disenchanted with current data analysis methodologies and techniques and wide to find better I was also considering switching to theoretical nuclear physics but didn't think that group Theory was sufficiently powerful as a mathematical base my initial plan was to spend a few weeks exploring category theory but a few years later I had completed designing a new data analysis methodology and a metal language that would allow me to easily design software to implement this methodology by 2014 I was hooked I switched to category theory and I am applying it to data analysis music theory linguistics software design and number theory my first love so since I'm no longer doing nuclear physics I am applying specifically toppest theory to currency trading and similar markets in six to eight months I hope to have completed the development and testing of the first iteration of my trading platform and trading algorithms this is what I will be doing in parallel to making these videos I currently live in estelí Nicaragua best known for cigars where I am continuing to develop my theoretical methods compose and do photography both everyday snapshots and art photography that's enough about me it's time to begin exploring category theory in the next video we will look at the definition of a category in general and the category of abstract sets and arbitrary mappings specifically so see