This is magical!!! The word dual has been used a lot in mathematics, I don't think I have seen as clear an illustration of its meaning as this. I will be listening to this lecture over and over until it is ALL absorbed. I appreciate the detailed demonstration of Sop 3->2, beautiful application to queuing theory. Just thinking, I can utilise this to distribute notes to different instrumental strata!! This is just so wonderful and I have not even created a melody yet!!! Loving this and thank you so much, Doctor Codrington!!!
Programmers are already familiar with how computer memory represents 0 for false and 1 for true. Likewise, initial is 0 and terminal (or "co-initial") is 1. Coupled with the Curry-Howard-Lambek isomorphism, they can now understand how true is co-false and false is co-true.
Can someone explain how you can reverse a function where there is an isolated element in the co-domain? When the function is reversed, this isolated element is not pointing to anything. Therefore, the reverse is not a function!
HitomiAyumu Firstly it's not reversing the function, it's reversing the arrow and requiring that they describe the same information. It is the opposite concept of a function: You used the word function twice above and they both mean very different things. Look at the definition of an arrow in Sop again, it says that each element of the domain is assigned a subset of the elements of the codomain (since the empty set is a subset of all sets then this is a valid assignment), and that these subsets are disjoint and their union is isomorphic to the codomain. In general in CT the best word to use is arrow, arrows can be anything as long as they follow the rules. A "function" is just a special case of an arrow.
+Dr. Martin J.M. Codrington I remember reading norman biggs' "discrete mathematics" and getting a formal definition of sets functions for the first time. it all seemed so abstract and general. finding out about category theory and how you can even generalize the concepts of sets and functions, it's all so beautiful. I hope my brain stays with me, because I wanna see how deep the rabbit hole goes!
This is magical!!! The word dual has been used a lot in mathematics, I don't think I have seen as clear an illustration of its meaning as this. I will be listening to this lecture over and over until it is ALL absorbed. I appreciate the detailed demonstration of Sop 3->2, beautiful application to queuing theory. Just thinking, I can utilise this to distribute notes to different instrumental strata!! This is just so wonderful and I have not even created a melody yet!!! Loving this and thank you so much, Doctor Codrington!!!
Leibniz's monads are points arranged according to category theory.
God is the initial object, which is appropriate, since there are 0 instances of him :P
"You know you've found a good mathematical theory, when it allows you to perform algebra on your definitions to produce new ones"
Programmers are already familiar with how computer memory represents 0 for false and 1 for true. Likewise, initial is 0 and terminal (or "co-initial") is 1. Coupled with the Curry-Howard-Lambek isomorphism, they can now understand how true is co-false and false is co-true.
Sir....can u plzz tell how you calculated maps in the opposite category......
Can someone explain how you can reverse a function where there is an isolated element in the co-domain? When the function is reversed, this isolated element is not pointing to anything. Therefore, the reverse is not a function!
HitomiAyumu Firstly it's not reversing the function, it's reversing the arrow and requiring that they describe the same information. It is the opposite concept of a function: You used the word function twice above and they both mean very different things. Look at the definition of an arrow in Sop again, it says that each element of the domain is assigned a subset of the elements of the codomain (since the empty set is a subset of all sets then this is a valid assignment), and that these subsets are disjoint and their union is isomorphic to the codomain.
In general in CT the best word to use is arrow, arrows can be anything as long as they follow the rules. A "function" is just a special case of an arrow.
Dr. Martin J.M. Codrington Thank you so much! :)
HitomiAyumu
No problem man!
+Dr. Martin J.M. Codrington I remember reading norman biggs' "discrete mathematics" and getting a formal definition of sets functions for the first time. it all seemed so abstract and general. finding out about category theory and how you can even generalize the concepts of sets and functions, it's all so beautiful. I hope my brain stays with me, because I wanna see how deep the rabbit hole goes!