advanced_tools:category_theory
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advanced_tools:category_theory [2017/11/10 10:40] jakobadmin [Why is it interesting?] |
advanced_tools:category_theory [2017/11/10 11:16] jakobadmin [Student] |
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| + | <blockquote>‘Categorification’ is the process of replacing equations by isomorphisms.<cite>[[https://arxiv.org/pdf/math/0004133.pdf|From Finite Sets to Feynman Diagrams]] | ||
| + | by John C. Baez and James Dolan</cite></blockquote> | ||
| <blockquote>So: in contrast to a set, which consists of a static collection of "things", a category consists not only of objects or "things" but also morphisms which can viewed as "processes" transforming one thing into another. Similarly, in a 2-category, the 2-morphisms can be regarded as "processes between processes", and so on. The eventual goal of basing mathematics upon ω-categories is thus to allow us the freedom to think of any process as the sort of thing higher-level processes can go between. By the way, it should also be very interesting to consider "Z-categories" (where Z denotes the integers), having j-morphisms not only for j = 0,1,2,... but also for negative j. Then we may also think of any thing as a kind of process.<cite>http://math.ucr.edu/home/baez/week74.html</cite></blockquote> | <blockquote>So: in contrast to a set, which consists of a static collection of "things", a category consists not only of objects or "things" but also morphisms which can viewed as "processes" transforming one thing into another. Similarly, in a 2-category, the 2-morphisms can be regarded as "processes between processes", and so on. The eventual goal of basing mathematics upon ω-categories is thus to allow us the freedom to think of any process as the sort of thing higher-level processes can go between. By the way, it should also be very interesting to consider "Z-categories" (where Z denotes the integers), having j-morphisms not only for j = 0,1,2,... but also for negative j. Then we may also think of any thing as a kind of process.<cite>http://math.ucr.edu/home/baez/week74.html</cite></blockquote> | ||
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| * "Sets for Mathematics" by Lawvere and Rosebrugh | * "Sets for Mathematics" by Lawvere and Rosebrugh | ||
| * [[https://arxiv.org/abs/0905.3010|Categories for the practising physicist]] by Bob Coecke, Eric Oliver Paquette | * [[https://arxiv.org/abs/0905.3010|Categories for the practising physicist]] by Bob Coecke, Eric Oliver Paquette | ||
| + | * [[https://arxiv.org/abs/math/0004133|From Finite Sets to Feynman Diagrams]] by John C. Baez, James Dolan | ||
advanced_tools/category_theory.txt · Last modified: 2018/10/11 14:14 by jakobadmin
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