advanced_tools:category_theory
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advanced_tools:category_theory [2017/11/09 09:27] jakobadmin [Why is it interesting?] |
advanced_tools:category_theory [2018/10/11 14:14] (current) jakobadmin [Layman] |
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| This helps to understand the connections between different branches of mathematics and helps generalizing them. This helps especially when we try to put our current physical theories on a firmer ground and want to find better theories. | This helps to understand the connections between different branches of mathematics and helps generalizing them. This helps especially when we try to put our current physical theories on a firmer ground and want to find better theories. | ||
| - | In some sense, category theory is the relational mathematics, comparable to [[advanced_notions:relational_physics|relational physics]]. | + | In some sense, category theory is relational mathematics, comparable to [[advanced_notions:relational_physics|relational physics]]. |
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| This suggests that a truly relational theory of physics should | This suggests that a truly relational theory of physics should | ||
| take advantage of category theory. <cite>[[https://groups.google.com/forum/#!msg/sci.physics.research/6cET8VmcUZU/IH9Tpq6NLTsJ|John Baez]]</cite></blockquote> | take advantage of category theory. <cite>[[https://groups.google.com/forum/#!msg/sci.physics.research/6cET8VmcUZU/IH9Tpq6NLTsJ|John Baez]]</cite></blockquote> | ||
| + | |||
| + | <blockquote>Now, given a category C, we may ‘decategorify’ it by forgetting about | ||
| + | the morphisms and pretending that isomorphic objects are equal. We are | ||
| + | left with a set (or class) whose elements are isomorphism classes of objects | ||
| + | of C. This process is dangerous, because it destroys useful information. It | ||
| + | amounts to forgetting which road we took from x to y, and just remembering | ||
| + | that we got there. Sometimes this is actually useful, but most of the time | ||
| + | people do it unconsciously, out of mathematical naivete. We write equations, | ||
| + | when we really should specify isomorphisms. ‘Categorification’ is the attempt | ||
| + | to undo this mistake. Like any attempt to restore lost information, it not | ||
| + | a completely systematic process. Its importance is that it brings to light | ||
| + | previously hidden mathematical structures, and clarifies things that would | ||
| + | otherwise remain mysterious. It seems strange and complicated at first, but | ||
| + | ultimately the goal is to make things simpler.<cite>[[https://arxiv.org/abs/math/0004133|From Finite Sets to Feynman Diagrams]] | ||
| + | by John C. Baez, James Dolan</cite></blockquote> | ||
| For further motivation, see: [[http://math.ucr.edu/home/baez/diary/fqxi_narrative.pdf|Categorifying Fundamental Physics]] by John Baez and https://math.stackexchange.com/questions/312605/what-is-category-theory-useful-for | For further motivation, see: [[http://math.ucr.edu/home/baez/diary/fqxi_narrative.pdf|Categorifying Fundamental Physics]] by John Baez and https://math.stackexchange.com/questions/312605/what-is-category-theory-useful-for | ||
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| <tabbox Layman> | <tabbox Layman> | ||
| - | <note tip> | + | * [[http://math.ucr.edu/home/baez/rosetta.pdf|Physics Topology and Computation a Rosetta Stone]] by Baez and Stay |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
| - | </note> | + | |
| | | ||
| <tabbox Student> | <tabbox Student> | ||
| + | <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>Categorification is best understood as the reverse of “decategorification”. This | ||
| + | is a process which begins with some category, and produces a structure for which | ||
| + | isomorphisms in the original category appear as equations between objects instead. | ||
| + | Categorification is the reverse process, replacing equations in some mathematical | ||
| + | setting with isomorphisms in some category in a consistent - but possibly nonunique | ||
| + | - way<cite>[[https://arxiv.org/abs/math/0601458|CATEGORIFIED ALGEBRA AND QUANTUM MECHANICS]] by JEFFREY MORTON</cite></blockquote> | ||
| * http://math.ucr.edu/home/baez/categories.html | * http://math.ucr.edu/home/baez/categories.html | ||
| + | * "Conceptual Mathematics: A First Introduction" to Categories by Lawvere | ||
| + | * "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 | ||
| - | * Conceptual Mathematics: A First Introduction to Categories by Lawvere | + | * [[https://arxiv.org/abs/math/0004133|From Finite Sets to Feynman Diagrams]] by John C. Baez, James Dolan |
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