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--- advanced_tools:category_theory:groupoids [2017/11/24 11:07]
jakobadmin ↷ Links adapted because of a move operation
+++ advanced_tools:category_theory:groupoids [2018/05/03 12:19] (current)
jakobadmin [Intuitive]
@@ Line 1: / Line 1: @@
 ====== Groupoids ======
-<tabbox Why is it interesting?>
-Groupoids are a natural generalization of [[advanced_tools:group_theory|groups]] and are able to describe symmetries that usual group theory cannot describe. In some sense, a groupoid is a collection of groups.
-<blockquote>Groupoids are a modern way to think about symmetries<cite>[[http://math.ucr.edu/home/baez/week249.html|John Baez]]</cite></blockquote>
-<blockquote>Mathematicians tend to think of the notion of symmetry as being virtually synonymous
-with the theory of [[advanced_tools:group_theory|groups]] . . . In fact, though groups are indeed sufficient to
-characterize homogeneous structures, there are plenty of objects which exhibit what
-we clearly recognize as symmetry, but which admit few or no nontrivial automorphisms.
-It turns out that the symmetry, and hence much of the structure, of such
-objects can be characterized algebraically if we use groupoids and not just groups.<cite>Weinstein, A. (1996) ‘Groupoids: Unifying Internal and External Symmetry</cite></blockquote>
-<blockquote>A groupoid is a generalization of a group that is particularly handy
-to express local symmetries of geometrical structure.<cite>http://philsci-archive.pitt.edu/2133/1/geometrie.pdf</cite></blockquote>
-<blockquote>The concept of groupoid is one of the means by which the twentieth century reclaims
-the original domain of application of the group concept. The modern, rigorous concept
-of group is far too restrictive for the range of geometrical application envisaged in
-the work of Lie. <cite>Mackenzie, K. (1987) Lie Groupoids and Lie Algebroids in Differential Geometry</cite></blockquote>
-<blockquote>Next to the title of this article one sees a photograph of a herd
-of zebra. No explicit explanation is offered for its presence, nor is one needed.
-The received account as to why zebras sport stripes is that when they stand in a
-herd, a charging lioness is presented with a strongly patterned visual array, making
-it very difficult for her to detect the outline of a single member of the herd. The
-rationale for the choice of this picture, in which one imagines Weinstein played a
-part, rests in his idea that groupoids are better than groups at detecting the inner
-symmetry of patterns of this kind. This idea Weinstein explicitly illustrates in the
-article itself with a discussion of the symmetries of a set of bathroom tiles. In
-contrast to this rather mundane concern of the mathematician contemplating the
-pattern of the grouting while enjoying a soak, the cover picture makes clear that
-such inner symmetry is a matter of life and death. **As any zebra will tell you,
-‘symmetry capturable by groupoids but not by groups saves lives’.** <cite>[[http://www.sciencedirect.com/science/article/pii/S0039368101000073|The importance of mathematical conceptualisation]] by David Corfield</cite></blockquote>
-<blockquote>
-There is no benefit today in arithmetic in Roman numerals. There is also no benefit today in insisting that the group concept is more fundamental than that of groupoid.
-<cite>Ronald Brown</cite>
-</blockquote>
-<blockquote>It is interesting in this respect to note the view of Connes [58] that Heisenberg discovered
-quantum mechanics by considering the groupoid of quantum transitions rather than the group of
-symmetry.<cite>[[http://intlpress.com/site/pub/files/_fulltext/journals/hha/1999/0001/0001/HHA-1999-0001-0001-a001.pdf|Groupoids and crossed objects in algebraic topology]] by Ronald Brown</cite></blockquote>
-<blockquote>The idea of making systematic use of groupoids . . . , however evident it may look
-today, is to be seen as a significant conceptual advance, which has spread into the
-most manifold areas of mathematics . . . In my own work in algebraic geometry, I
-have made extensive use of groupoids<cite>Grothendieck, as quoted in [[http://intlpress.com/site/pub/files/_fulltext/journals/hha/1999/0001/0001/HHA-1999-0001-0001-a001.pdf|Groupoids and crossed objects in algebraic topology]] by Ronald Brown </cite></blockquote>
-<blockquote>From [...] to the objects of field values for [[advanced_tools:gauge_symmetry|gauge field theory]], which need to keep track of gauge equivalences, we find we are treating geometric forms of groupoid. If in the latter case instead we take the simple quotient, the plain set of equivalences classes, which amount to taking gauge symmetries to be redundant,
- **the physics goes wrong** in some sense, in that we cannot retain a //local// quantum field theory. Furthermore, in the case of gauge equivalence, we do not just have one level of arrows between points or objects, but arrows between arrows and so on, representing equivalences between gauge equivalences. And this process continues indefinetely to "higher" groupoids, needed to capture the higher symmetries of [[theories:speculative_theories:string_theory|string theory]] (Schreiber 2013).
-<cite>[[https://books.google.de/books?id=_RguDwAAQBAJ&lpg=PA136&ots=bFQqR0RHZq&dq=groupoids%20gauge%20symmetry&hl=de&pg=PA134#v=onepage&q&f=false|page 134 in What is a Mathematical Concept?]]</cite></blockquote>
-<blockquote>. Now, copies of a non-rigid structure cannot be canonically
-identified, even if they are identical (this point has been discussed at
-length in Ref. (Catren, 2008a)). This statement presupposes a detrivialization
-of the notion of identity, in the sense that an entity can
-be identical (i.e. identified) to itself in many non-trivial ways. More
-generally, two entities can be identical in many different ways. The
-notion of groupoid is the mathematical notion that formalizes the
-situations in which one deals with multiple possible identifications
-between identical structures. In turn, the notion of group is a
-particular case of the notion of groupoid, in the sense that a group
-(understood as a category) is a groupoid with a unique object.
-Whereas groups encode the multiple identities (or, in more usual
-terms, the symmetries) of non-rigid structures, groupoids encode
-the (self- and hetero-)identifications in a family of non-rigid
-structures. The formalism of groupoids permits us to understand
-the transition between Klein's Erlangen program and Ehresmann's
-theory of fiber bundlesd made possible by Cartan's generalization
-of the former - as a transition from the group-theoretical selfidentifications
-of a single structure (e.g. Klein geometries) to the
-groupoid-theoretical (self- and hetero-)identifications in a family of
-identical structures. <cite>Klein-Weyl's program and the ontology of gauge and quantum systems by Gabriel Catren</cite></blockquote>
-<tabbox Layman>
+<tabbox Intuitive>
+[{{ :advanced_tools:category_theory:groupoids.gif?nolink|Source: https://wolfweb.unr.edu/homepage/ramazan/groupoid/}}]
 <blockquote>What is a Groupoid?
@@ Line 124: / Line 43: @@
 An example you might've played with at some point is the [[https://en.wikipedia.org/wiki/15_puzzle|15 puzzle]], which forms a groupoid where the places are the possible configurations of the puzzle and the ways are ways to slide tiles around to pass between configurations.
 <cite>[[https://www.quora.com/What-is-an-intuitive-explanation-of-a-groupoid|Qiaochu Yuan]]</cite></blockquote>
-<tabbox Student>
+<tabbox Concrete>
 <blockquote>By a groupoid we mean simply a small [[advanced_tools:category_theory|category]] in which every morphism is an isomorphism.
@@ Line 174: / Line 93: @@
 <note warning>"it's common but arguably very misleading to think of a groupoid as being just a collection of groups. The practical problem is that there are many kinds of extra structure you can place on groupoids, and structured groupoids are usually much richer than structured groups. For example, groupoids with an action of a group G, topological groupoids, and Lie groupoids are all much richer objects than unions of groups with an action of a group G, topological groups, or Lie groups. The categorical problem, which foreshadows the practical problem, is that in order to identify a groupoid with a collection of groups you need to pick a bunch of basepoints."<cite>[[https://www.quora.com/What-is-an-intuitive-explanation-of-a-groupoid|Qiaochu Yuan]]</cite></note>
-<tabbox Researcher>
+<tabbox Abstract>
   * See https://ncatlab.org/nlab/show/groupoid
+  * [[https://arxiv.org/pdf/math-ph/0506024.pdf|Lie Groupoids and Lie algebroids in physics and noncommutative geometry]] by N.P. Landsman
-<tabbox Examples>
+<tabbox Why is it interesting?>
---> Example1#
+Groupoids are a natural generalization of [[advanced_tools:group_theory|groups]] and are able to describe symmetries that usual group theory cannot describe. In some sense, a groupoid is a collection of groups.
+<blockquote>Groupoids are a modern way to think about symmetries<cite>[[http://math.ucr.edu/home/baez/week249.html|John Baez]]</cite></blockquote>
-<--
---> Example2:#
+<blockquote>Mathematicians tend to think of the notion of symmetry as being virtually synonymous
-<--
+with the theory of [[advanced_tools:group_theory|groups]] . . . In fact, though groups are indeed sufficient to
+characterize homogeneous structures, there are plenty of objects which exhibit what
+we clearly recognize as symmetry, but which admit few or no nontrivial automorphisms.
+It turns out that the symmetry, and hence much of the structure, of such
+objects can be characterized algebraically if we use groupoids and not just groups.<cite>Weinstein, A. (1996) ‘Groupoids: Unifying Internal and External Symmetry</cite></blockquote>
-<tabbox FAQ>
+<blockquote>A groupoid is a generalization of a group that is particularly handy
+to express local symmetries of geometrical structure.<cite>http://philsci-archive.pitt.edu/2133/1/geometrie.pdf</cite></blockquote>
-<tabbox History>
+<blockquote>The concept of groupoid is one of the means by which the twentieth century reclaims
+the original domain of application of the group concept. The modern, rigorous concept
+of group is far too restrictive for the range of geometrical application envisaged in
+the work of Lie. <cite>Mackenzie, K. (1987) Lie Groupoids and Lie Algebroids in Differential Geometry</cite></blockquote>
+<blockquote>Next to the title of this article one sees a photograph of a herd
+of zebra. No explicit explanation is offered for its presence, nor is one needed.
+The received account as to why zebras sport stripes is that when they stand in a
+herd, a charging lioness is presented with a strongly patterned visual array, making
+it very difficult for her to detect the outline of a single member of the herd. The
+rationale for the choice of this picture, in which one imagines Weinstein played a
+part, rests in his idea that groupoids are better than groups at detecting the inner
+symmetry of patterns of this kind. This idea Weinstein explicitly illustrates in the
+article itself with a discussion of the symmetries of a set of bathroom tiles. In
+contrast to this rather mundane concern of the mathematician contemplating the
+pattern of the grouting while enjoying a soak, the cover picture makes clear that
+such inner symmetry is a matter of life and death. **As any zebra will tell you,
+‘symmetry capturable by groupoids but not by groups saves lives’.** <cite>[[http://www.sciencedirect.com/science/article/pii/S0039368101000073|The importance of mathematical conceptualisation]] by David Corfield</cite></blockquote>
+<blockquote>
+There is no benefit today in arithmetic in Roman numerals. There is also no benefit today in insisting that the group concept is more fundamental than that of groupoid.
+<cite>Ronald Brown</cite>
+</blockquote>
+<blockquote>It is interesting in this respect to note the view of Connes [58] that Heisenberg discovered
+quantum mechanics by considering the groupoid of quantum transitions rather than the group of
+symmetry.<cite>[[http://intlpress.com/site/pub/files/_fulltext/journals/hha/1999/0001/0001/HHA-1999-0001-0001-a001.pdf|Groupoids and crossed objects in algebraic topology]] by Ronald Brown</cite></blockquote>
+<blockquote>The idea of making systematic use of groupoids . . . , however evident it may look
+today, is to be seen as a significant conceptual advance, which has spread into the
+most manifold areas of mathematics . . . In my own work in algebraic geometry, I
+have made extensive use of groupoids<cite>Grothendieck, as quoted in [[http://intlpress.com/site/pub/files/_fulltext/journals/hha/1999/0001/0001/HHA-1999-0001-0001-a001.pdf|Groupoids and crossed objects in algebraic topology]] by Ronald Brown </cite></blockquote>
+<blockquote>From [...] to the objects of field values for [[advanced_tools:gauge_symmetry|gauge field theory]], which need to keep track of gauge equivalences, we find we are treating geometric forms of groupoid. If in the latter case instead we take the simple quotient, the plain set of equivalences classes, which amount to taking gauge symmetries to be redundant,
+ **the physics goes wrong** in some sense, in that we cannot retain a //local// quantum field theory. Furthermore, in the case of gauge equivalence, we do not just have one level of arrows between points or objects, but arrows between arrows and so on, representing equivalences between gauge equivalences. And this process continues indefinetely to "higher" groupoids, needed to capture the higher symmetries of [[theories:speculative_theories:string_theory|string theory]] (Schreiber 2013).
+<cite>[[https://books.google.de/books?id=_RguDwAAQBAJ&lpg=PA136&ots=bFQqR0RHZq&dq=groupoids%20gauge%20symmetry&hl=de&pg=PA134#v=onepage&q&f=false|page 134 in What is a Mathematical Concept?]]</cite></blockquote>
+<blockquote>. Now, copies of a non-rigid structure cannot be canonically
+identified, even if they are identical (this point has been discussed at
+length in Ref. (Catren, 2008a)). This statement presupposes a detrivialization
+of the notion of identity, in the sense that an entity can
+be identical (i.e. identified) to itself in many non-trivial ways. More
+generally, two entities can be identical in many different ways. The
+notion of groupoid is the mathematical notion that formalizes the
+situations in which one deals with multiple possible identifications
+between identical structures. In turn, the notion of group is a
+particular case of the notion of groupoid, in the sense that a group
+(understood as a category) is a groupoid with a unique object.
+Whereas groups encode the multiple identities (or, in more usual
+terms, the symmetries) of non-rigid structures, groupoids encode
+the (self- and hetero-)identifications in a family of non-rigid
+structures. The formalism of groupoids permits us to understand
+the transition between Klein's Erlangen program and Ehresmann's
+theory of fiber bundlesd made possible by Cartan's generalization
+of the former - as a transition from the group-theoretical selfidentifications
+of a single structure (e.g. Klein geometries) to the
+groupoid-theoretical (self- and hetero-)identifications in a family of
+identical structures. <cite>Klein-Weyl's program and the ontology of gauge and quantum systems by Gabriel Catren</cite></blockquote>
 </tabbox>

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