Video Infoblog - Category Theory: The Beginner’s Introduction (Lesson 1 Video 1)

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. These videos will be discussed in the Facebook group: http://is.gd/SJlp7I Follow me on Twitter: @mjmcodrington g+: google.com/+MJMCodrington The Facebook Group is: http://is.gd/SJlp7I Martin Codrington

Transcript

0:10

welcome to category theory the beginners

0:14

introduction I am Martin Cottington and

0:16

this is lesson 1 video 1 of 6 welcome to

0:21

the first video of the first lesson of

0:22

what I hope will be a complete

0:24

accessible and excitin aspiration of

0:26

category theory

0:27

I am dr. Martin Codrington and I'll tell

0:30

you a little bit about me at the end of

0:31

this video let's first discuss the

0:33

intent and the intended audience of this

0:35

video series intend to provide a

0:37

self-contained introduction to category

0:39

theory no mathematics prerequisites it

0:41

will focus on the ways in which it can

0:42

be practically applied to solve in many

0:44

different types of real-world problems

0:46

for this reason our exploration be

0:49

project-based it will be divided into

0:51

sections that will consist of five to

0:52

ten lessons at the end of each section

0:54

we will apply what we have learned to

0:56

design a software that implements it

0:59

for the first few sections the area

1:01

focus will be music theory there are two

1:03

reasons for choosing music rather than

1:04

say data analysis the area to which I've

1:06

done most of my category theory work the

1:09

first is because I wanted to reach the

1:10

widest possible audience category theory

1:13

can be applied to everything from pure

1:14

and applied mathematics anthropology

1:16

economics financial engineering

1:18

philosophy and linguistics just to name

1:21

a few

1:21

I would like anyone who is interested to

1:23

be able to follow along not just in the

1:25

theoretical discussions but in the

1:27

application phase as well also many of

1:30

the sets we'll be dealing with in the

1:31

beginning with music are very small on

1:33

the order of seven twelve seventeen

1:36

elements so we can easily visualize the

1:38

internal diagrams with the maps we

1:40

discuss and with music you can hear the

1:43

results of our theoretical aspirations

1:45

in a few weeks when we were able to

1:47

write software that creates a melody and

1:49

harmonizes it beautifully and takes that

1:51

melody and translates it to another mode

1:54

and modifies it reharmonization

1:56

satellite what it did use a normal

1:59

language that you can understand you

2:00

will quickly see the power of your

2:02

theory as this sounds like a lot are

2:04

difficult to design in program stick

2:06

around for the first seven lessons where

2:08

we will be exploring the category of

2:10

abstract sets and arbitrary mappings and

2:12

introduced a category of sets of an

2:14

endomorphism commonly interpreted as

2:16

dynamical systems or tone of tones and

2:18

it's subcategory the category of sets of

2:21

a permutation with these categories

2:24

alone we can do all these things rather

2:25

easily this first lesson is intended to

2:29

give you an overview of the category

2:31

theory way of thinking we will go

2:33

through the definition of a category and

2:34

get used to constructing and reasoning

2:36

with commutative diagrams in this video

2:39

you will go over the plan structure of

2:41

our exploration in category theory and I

2:43

will discuss the one music prerequisite

2:45

you need understanding of the 12 pitch

2:47

classes

2:49

in video 2 I will give the definition of

2:51

a general category enough s the category

2:54

of abstract sets and arbitrary mappings

2:55

and we will begin to go through the

2:57

definition of that category by exploring

2:59

the objects and arrows and s and then

3:01

build our own category based on s music

3:04

s

3:07

in video three we will discuss

3:09

composition in detail with abstract

3:11

examples and examples from music s the

3:15

topic of video four is the associative

3:17

law the identity arrow and the identity

3:20

laws in video five we will learn how to

3:23

calculate the number of maps between

3:24

sets and s and get our first

3:26

introduction to finite inverse limits

3:28

and Col limits Universal mapping

3:31

properties as we will call them in s we

3:34

will introduce them by discussing the

3:35

terminal and initial object this will

3:37

lead us to discuss in the nature of maps

3:39

and s up the opposite or mirror category

3:41

as we see to understand duality armed

3:45

with the powerful terminal object we

3:47

will rid ourselves of the set theory

3:48

baggage that we've been carrying around

3:50

by redefining familiar concepts about

3:52

sets in the language of category theory

3:54

and we will formulate our first proofs

3:57

by constructing commutative diagrams

4:02

how much we will be covering in category

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theory the quick answer is a lot but

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slowly my goal is to help you form an

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intuition for all the elementary

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concepts before we use them to build

4:12

higher-order concepts so for example it

4:15

makes no sense formulating with natural

4:17

transformation before you have a good

4:18

intuition for founder's and white

4:20

formulated foundries if you don't first

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have a fundamental understanding of a

4:24

few concrete categories this method is

4:27

especially necessary since we will be

4:28

applying everything we learn we will

4:31

start first with elementary concrete

4:33

category theory by describing s we will

4:35

introduce and explain many of the

4:36

popular finite inverse limits and co

4:38

limits such as the terminal and initial

4:40

object monic and epic arrows pullbacks

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and push outs equalizers and co

4:45

equalizers and products and koch

4:47

brothers we will also discuss in detail

4:50

impart and constructs such as a truth

4:52

value object and its relation to sub

4:54

objects and map objects etc at the end

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we will discuss the notion of the limit

4:59

more formally and begin to define our

5:01

own limits in music s discussing s in

5:04

detail prepare us for discussing other

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categories as we shall see many

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categories of structure sets can be

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derived by considering limits and

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Coulomb X as objects for example the

5:14

category of variable sets dynamical

5:16

systems permutations graphs various

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types groups of various types pull sets

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bouquets by sets pointer sets and many

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more can be derived from the axioms or

5:27

the limits and set we will in turn go

5:29

through the limits and colormix in these

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categories explain their use and share

5:33

how we can form even more structured

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categories by objectifying their limits

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the pace at which we move depends on the

5:41

group lessons will ideally be produced

5:43

at a rate of one every 1 to 1.5 weeks at

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the end of each lesson I will listen to

5:48

your feedback upload supplemental videos

5:50

if anything needs further explanation

5:52

and after this I will upload the

5:54

problems that video and also post a PDF

5:56

of the problem set so that you can have

5:59

some interesting problems to work on

6:00

while I make the next lesson every

6:02

concept in category theory is connected

6:04

in interesting and complex ways and

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often I will be unable to resist the

6:08

temptation to point out these

6:09

connections like for example when

6:12

discussing the identity laws we see that

6:14

the category of sets of an

6:15

morphism and in general any category see

6:18

if an Animorph ism is logically implied

6:20

by the axioms of a category itself

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whenever I give one of these a size

6:24

things will go green to indicate it you

6:26

can follow along if you like

6:27

but if you don't fully understand don't

6:29

worry because we will be discussing that

6:30

concept in detail in the future category

6:34

theory is very powerful I can give a

6:36

complete introduction I can give an

6:37

accessible introduction but I can't also

6:39

make it a quick introduction it will

6:41

take about five or six lessons before we

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have covered enough for you to get a

6:45

glimpse of the full potential of the

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theory and I will take another few

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months for us to cover most of our

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ground these videos are intended to

6:52

support discussion in the Facebook group

6:54

product of category theory music and

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computer science for now this is where

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we will be discussing and applying the

7:01

content in these videos so let's get

7:03

started by discussing the format of our

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explorations

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this cycle we will follow will allow us

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to learn a little then apply it in the

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beginning it will take a while to

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complete one cycle but as we progress

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the pace will pick up first we'll have a

7:18

category theory overview where we

7:20

introduce a theoretical concepts we will

7:21

use in our formulation there will be

7:23

some examples from music but just simple

7:25

examples used to illustrate a certain

7:27

point we will not do any detail

7:28

formulation at this stage to category

7:32

theory deep dive here we will take each

7:34

concept introduced in the overview and

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catalog the many ways we could apply

7:38

these concepts to music theory this will

7:40

give you a better understanding both of

7:42

the concepts and the ways that they can

7:43

be applied we will use music theory

7:46

informally during this phase as well but

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we will learn more about it as we

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progress three music theory formulation

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here we will actually begin formulating

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the music theory we will choose the best

7:57

tools from our knowledge of category

7:58

theory so far and use these tools to

8:00

model and to understand certain concepts

8:02

in music theory we will then identify

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the questions we would like to ask with

8:06

these formulations or how these

8:08

formulations can help us further advance

8:10

our knowledge of music theory for is a

8:13

software implementation here we will

8:15

design the implementation that most

8:17

closely mirrors the formulations which

8:19

will allow us to create ways to answer

8:20

the questions that we have formulated in

8:22

the previous step let me illustrate by

8:25

giving an example of the first cycle

8:27

don't worry if you don't understand any

8:29

of this this is not the actual

8:30

formulation we will use and we will go

8:32

through all these concepts in great

8:33

detail to another discussion stage one I

8:37

will introduce the category of abstract

8:39

sets and arbitrary muffins in a series

8:41

of lessons in stage two we will start by

8:44

representing the 12 pitch classes as a

8:46

set X we can then define an isomorphism

8:49

n into a set C of pitch class names and

8:53

we will see how the finite inverse

8:56

limits in column sets will allow us to

8:58

represent some more useful musical

9:00

concepts

9:01

in state 3 after we have explored all

9:04

the possible ways we could represent

9:05

certain concepts we will form a list of

9:08

the questions we would like to ask and

9:09

figure out how best to do this within

9:11

the theory for example ask in which

9:13

classes the major minor mode have

9:15

in common is best represented as a

9:17

question about sub objects given to sub

9:20

objects mi MJ of X representing the

9:23

major minor mode respectively can you

9:25

find a torus about 2a of s that is also

9:27

sub object of mi MJ and moreover that a

9:30

is the largest set they satisfy these

9:32

conditions in stage 4 using prime

9:35

injection and fundamental theorem of

9:36

arithmetic we can design an algorithm

9:38

that closely mirrors the formulation for

9:40

example we can define a Mon at map from

9:42

s to and natural numbers that associates

9:45

the nth prime of the F element of X so

9:47

that the major mode for example can be

9:49

represented by the unique integer 29

9:52

million 150 3410 and the minor mode by

9:57

14 million seven hundred and thirteen

9:59

thousand seven hundred and ninety at

10:02

similar all the pitch classes they have

10:03

in common is equivalent to ax and

10:05

identity of the set a which is

10:07

equivalent to a sand in this formulation

10:09

what is the greatest common divisor of

10:11

twenty nine million one hundred and

10:13

fifty three thousand four hundred and

10:14

ten and fourteen million seven hundred

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and thirteen thousand seven hundred and

10:18

ninety designing algorithms in this way

10:20

will allow for better understand of the

10:22

concepts and will allow non programmers

10:24

actually design the outline of computer

10:26

algorithms without knowing any computer

10:28

science or if all having any practical

10:29

experience with programming after

10:32

completing all four stages of a cycle we

10:34

will repeat the entire process with some

10:36

new collection of theoretical concepts

10:37

maybe will define and explore three or

10:40

four new categories in one go then

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repeat the process I haven't decided yet

10:43

let's see how things go for the first

10:45

cycle but in real life you really ever

10:47

work in just one category but rather

10:49

you'll find yourself moving between ten

10:51

to fifteen categories in one formulation

10:53

well at least is how I work so the more

10:57

quickly we can learn a few different

10:58

categories and discuss filters in more

11:00

detail the happier lives will be for the

11:04

music prerequisites you'll just need to

11:06

understand the concept of the 12 pitch

11:08

classes if you look at a keyboard you

11:10

will notice that there are groups of two

11:12

black notes and groups of three consider

11:15

only the groups of two black notes the

11:18

white note to the left of the first

11:19

black note in that group is a C since

11:23

this is a repeated pattern on the

11:25

keyboard we can pick one C then take all

11:28

the notes between dot C and the next C

11:30

but not including the next and studied

11:33

these separately now we will call these

11:35

pitch classes instead of notes and

11:38

technically there aren't notes until

11:39

they've been played different C's on the

11:42

keyboard or different pitches but the

11:43

same pitch class any mode or scale can

11:46

be formed from these pitch classes we

11:48

say that a mode is a subset of these

11:50

pitch classes to begin with you'll

11:52

define each more starting on C and as we

11:54

move further along we learn how to

11:55

transpose from C major to F major for

11:58

example but for now when we talk of

12:00

major or minor we're referring to C

12:02

major and C minor respectively

12:06

the major scale for example is made up

12:08

of all the white pitch classes in this

12:10

collection notice that each Platinum can

12:19

have one of two names this is because

12:21

each black note lies between two white

12:23

notes one to left let's call it L I want

12:26

to write less color R so that the black

12:28

note can be called L sharp because it's

12:31

higher or sharper than L or R flat

12:33

because it's lower or flatter than R so

12:36

this first black note can be called C

12:38

sharp or D flat because it's between C

12:41

and D in some cases in music one name

12:45

must be used instead of another because

12:46

of conventions for example the minor

12:50

mode is made up of the following pitches

12:57

it would be incorrect according to a

13:00

traditional music theory to call the E

13:01

flat D sharp or the a flat G sharp the

13:04

reason for this is that these modes are

13:06

built with the I data they must contain

13:07

one of each pitch name so for example if

13:10

we construct C minor we have C D we

13:14

already have a kind of D so next nut

13:16

must be a kind of e e flat so we have E

13:19

flat F and G now we need some kind of a

13:22

so a flat instead of G sharp and B not

13:26

almost are constructed like this some

13:28

modes can have an E flat in the e for

13:30

example but many of the most used in

13:32

early music are constructed this way the

13:36

chromatic scale is made up of all these

13:37

pitches we can list the pitches of the

13:39

chromatic scale like this

13:49

while these things imply we'll explore

13:52

later but for now this is all you need

13:54

to understand from music to get started

13:56

before expiration

13:59

I was born in Barbados and in 2005 I

14:03

completed my Bachelors of Science in

14:05

chemistry from Morgan State University I

14:07

received my PhD from Texas A&M

14:09

University in 2012 my dissertation

14:12

research was focused on the analysis of

14:15

nuclear collisions with the solenoidal

14:17

tracker a trick star at the relativistic

14:19

heavy ion collider Rick our Haven

14:22

National Laboratories in Long Island New

14:24

York I started exploring category theory

14:27

in 2011 I was disenchanted with current

14:32

data analysis methodologies and

14:34

techniques and wide to find better

14:35

solutions

14:36

I was also considering switching to

14:38

theoretical nuclear physics but didn't

14:40

think that group Theory was sufficiently

14:42

powerful as a mathematical base my

14:44

initial plan was to spend a few weeks

14:46

exploring category theory but a few

14:48

years later I had completed designing a

14:50

new data analysis methodology and a

14:52

metal language that would allow me to

14:54

easily design software to implement this

14:56

methodology by 2014 I was hooked I

15:00

switched to category theory and I am

15:02

applying it to data analysis music

15:04

theory linguistics software design and

15:06

number theory my first love

15:09

so since I'm no longer doing nuclear

15:11

physics I am applying specifically

15:14

toppest theory to currency trading and

15:16

similar markets in six to eight months I

15:19

hope to have completed the development

15:21

and testing of the first iteration of my

15:23

trading platform and trading algorithms

15:25

this is what I will be doing in parallel

15:27

to making these videos I currently live

15:29

in estelí Nicaragua best known for

15:31

cigars where I am continuing to develop

15:33

my theoretical methods compose and do

15:36

photography both everyday snapshots and

15:40

art photography that's enough about me

15:43

it's time to begin exploring category

15:45

theory in the next video we will look at

15:47

the definition of a category in general

15:49

and the category of abstract sets and

15:51

arbitrary mappings specifically so see

15:54

you next time

English (auto-generated)