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--- advanced_tools:category_theory [2017/11/09 09:15]
jakobadmin
+++ advanced_tools:category_theory [2018/10/11 14:14] (current)
jakobadmin [Layman]
@@ Line 6: / Line 6: @@
 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]].
@@ Line 15: / Line 15: @@
 Category theory is an approach to understand these connections and categorise them. In this sense, category theory is a meta-theory of mathematics.
-For further motivation, see: [[http://math.ucr.edu/home/baez/diary/fqxi_narrative.pdf|Categorifying Fundamental Physics]] by John Baez
+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:**
@@ Line 25: / Line 71: @@
 <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>
+<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
+  * "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
-  *  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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