Difference between revisions of "Categories for the Working Mathematician"
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=Kan Extensions= | =Kan Extensions= | ||
According to Saunders Mac Lane: | According to Saunders Mac Lane: | ||
This chapter<ref>{{:Book/Categories for the Working Mathematician}}, 233-250</ref> begins by expressing adjoints as limits and ends by expressing "everything" as Kan extensions. | This chapter<ref>{{:Book/Categories for the Working Mathematician}}, 233-250</ref> begins by expressing adjoints as limits and ends by expressing "[[everything]]" as Kan extensions. | ||
This simple statement reveals the fact that all data structures can be represented and transformed to others by using this Kan extension as a primitive or universal abstraction. At the same time, according to the lecturer at [https://campus.swarma.org/course/2723 Swarma.org]<ref>https://campus.swarma.org/course/2723</ref>, the investigative topic in Category Theory is represent-ability. More elaborately, it is about using a universal object, namely the arrow, as a single representational device to encode information of any kind. | This simple statement reveals the fact that all data structures can be represented and transformed to others by using this Kan extension as a primitive or universal abstraction. At the same time, according to the lecturer at [https://campus.swarma.org/course/2723 Swarma.org]<ref>https://campus.swarma.org/course/2723</ref>, the investigative topic in Category Theory is represent-ability. More elaborately, it is about using a universal object, namely the arrow, as a single representational device to encode information of any kind. | ||
Revision as of 14:09, 18 September 2021
The book:Categories for the Working Mathematician[1], is a must have reference for people who studies Category Theory. It has an entire chapter on Kan Extensions.
Kan Extensions
According to Saunders Mac Lane:
This chapter[2] begins by expressing adjoints as limits and ends by expressing "everything" as Kan extensions.
This simple statement reveals the fact that all data structures can be represented and transformed to others by using this Kan extension as a primitive or universal abstraction. At the same time, according to the lecturer at Swarma.org[3], the investigative topic in Category Theory is represent-ability. More elaborately, it is about using a universal object, namely the arrow, as a single representational device to encode information of any kind.
References
- ↑ Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. 5 (2nd ed.). local page: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.
- ↑ Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. 5 (2nd ed.). local page: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001. , 233-250
- ↑ https://campus.swarma.org/course/2723