On Tuesday, 26 August 2003 06:06:31 UTC+7, Jason  wrote:   
   > I just started thinking about groupoids recently and started to wonder:   
   > Why is symmetry transformations usually described by a group rather than   
   > a groupoid? The more I think about it, the more it seems the most   
   > general kind of symmetry (of the classical kind; I'm still trying to   
   > figure out just what in the world Hopf algebras "really" are and do on   
   > an intuitive and mathematical level.) ought to be a groupoid, not a   
   > group! I mean, why should a symmetry operation act upon ALL states   
   > instead of just one or a couple of states?   
   >   
   > Just to take a silly non-physics example, let's consider the 14-15   
   > puzzle. If we consider the blank square as an empty tile, then only half   
   > of all possible permutations are permissible configurations. Suppose we   
   > consider only permissible configurations as "solutions" and we wish to   
   > describe the "symmetry operation" of sliding blocks, which only take   
   > permissible solutions to permissible solutions. We can divide all   
   > permissible solutions into 16 "nodes", each describing where the empty   
   > tile is, and what the "sliding symmetry" does is to "map" one node into   
   > another. In other words, a groupoid. So, we have the groupoid generated   
   > by the following graph:   
   >   
   >   = = =   
   > | | | |   
   >   = = =   
   > | | | |   
   >   = = =   
   > | | | |   
   >   = = =   
   >   
   > with the configuration of the 14-15 puzzle acting as a representation of   
   > this groupoid.   
   >   
   > So, why groups, not groupoids?   
      
   have a look at this:   
   https://arxiv.org/abs/1508.04632   
   here they address exactly your concerns, by employing lie grupoids and their   
   linearized lie algebroids for tacking symmetries in field theory   
      
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