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advanced_tools:category_theory [2017/11/09 09:01] jakobadmin [Why is it interesting?] |
advanced_tools:category_theory [2018/10/11 14:14] (current) jakobadmin [Layman] |
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Category theory is a mathematical theory of mathematical theories. This means, category theory is all about the relationship between different mathematical theories. | Category theory is a mathematical theory of mathematical theories. This means, category theory is all about the relationship between different mathematical theories. | ||
- | This helps do understand the connections between different branches of mathematics and helps generalizing them. | + | 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 relational mathematics, comparable to [[advanced_notions:relational_physics|relational physics]]. | ||
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Category theory is an approach to understand these connections and categorise them. In this sense, category theory is a meta-theory of mathematics. | Category theory is an approach to understand these connections and categorise them. In this sense, category theory is a meta-theory of mathematics. | ||
+ | One of the most important results of category theory is the [[advanced_tools:category_theory:yoneda_lemma|Yoneda lemma]], which basically tells us that an "an object is completely determined by its relationships to other objects." This is surprisingly analogous to the basic idea of [[advanced_notions:relational_physics|relational physics]]. | ||
+ | |||
+ | <blockquote>The importance of all this for physics is as follows. Lots of | ||
+ | people working on quantum gravity like to stress the importance | ||
+ | of "relationalism" - the idea that physical things only have properties | ||
+ | by virtue of their relation with other physical things. For example, | ||
+ | it only makes sense to say how something is moving *relative* to | ||
+ | other things. This idea is an old one, going back at least to | ||
+ | Leibniz, and attaining a certain prominence with Mach (who primarily | ||
+ | applied it to position and velocity, rather than other properties). | ||
+ | |||
+ | Relationalism is appealing, at least to certain kinds of people, but | ||
+ | it's a bit dizzying: if all properties of a thing make sense only in | ||
+ | relation to other things, how do we get started in the job of describing | ||
+ | anything at all? The danger of "infinite regress" has traditionally | ||
+ | made certain other kinds of people recoil from relationalism; they | ||
+ | urge that one posit of something "absolute" to get started. | ||
+ | |||
+ | Category theory provides a nice simple context to see relationalism | ||
+ | in action, in a completely rigorous and precise form. In a category, | ||
+ | objects do not have "innards" - viewed in isolation, they are all | ||
+ | just featureless dots. It's only by virtue of their morphisms to and | ||
+ | from other objects (and themselves) that they acquire distinct | ||
+ | personalities. This is why an isomorphism between objects allows | ||
+ | us to treat them as "the same": it establishes a 1-1 correspondence | ||
+ | between their morphisms to, or from, other objects. (Moreover, this | ||
+ | correspondence preserves the extra structure described above.) | ||
+ | |||
+ | 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> | ||
+ | |||
+ | <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 | ||
+ | |||
+ | ** Important Concepts:** | ||
+ | |||
+ | <nspages advanced_tools:category_theory -h1 -textPages=""> | ||
- | See also: [[http://math.ucr.edu/home/baez/diary/fqxi_narrative.pdf|Categorifying Fundamental Physics]] by John Baez | ||
<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 |