Science > Outer Space and "Inner Space" Sciences
Ark - where are you headed?
eoste:
What a strange "incorporeal god" with so many appearances !
It looks like the realm of fundamentals is a (non) place to meet such weird beings or concepts.
Static Time and Dynamic Necessity... Straight coming from the Virtual (or Full Void) and generating the plurality of a world where time seems dynamic and where necessity is mechanical (static somehow).
A virtual information generating some strangely mirrored world ?
As it depends on the level of the observer's perception, it's little wonder that one would see a freaked god and another a non-trivial groupoïd (with a funny look also if you ask me) ;)
Bluelamp:
--- Quote from: ark on March 22, 2012, 05:01:47 PM ---Two static points and four arrows. Or: two static points, and two transformations: one "do nothing" and one "change the point to the other one". One passive, and one active. But still you have three loops - see the picture.
--- End quote ---
That's a very cool law of 3/Greek triad!!! Does it relate to a single bit, Cl(1), and a circle (Poincare disk) being representations of complex numbers/time/space? Is this groupoid a group also?
Patience:
--- Quote from: Bluelamp on March 22, 2012, 07:27:04 PM ---
--- Quote from: ark on March 22, 2012, 05:01:47 PM ---Two static points and four arrows. Or: two static points, and two transformations: one "do nothing" and one "change the point to the other one". One passive, and one active. But still you have three loops - see the picture.
--- End quote ---
That's a very cool law of 3/Greek triad!!! Does it relate to a single bit, Cl(1), and a circle (Poincare disk) being representations of complex numbers/time/space? Is this groupoid a group also?
--- End quote ---
Not to get bogged down in details as I think Ark wants to focus on some fundamental ideas and not so much the math itself (but I could be wrong), a groupoid is a generalisation of a group. Thus, a group is a groupoid, but there are groupoids that are not groups. A groupoid G is a set that has a unary operation f : G -> G and a binary operation * : GxG -> G that IS NOT necessarily defined for every pair of elements from G. Probably the most immediate example of what I mean by an unary operation is the association of an element in G with its inverse.
So... Bluelamp... You have always seemed mathy, so I will say that the setting in which the notion of groupoid came about is category theory. You may have seen groupoids without knowing it in studying topology. The mathematically rigorous transition from general topology to algebraic topology (mapping a topological space to its fundamental group) is done via category theory one of whose notions is the groupoid. Category theory is a sort of math that allows us to map large collection of mathematical objects to another collection in a well defined manner. In the example above, we map a member of the collection of things called "topological spaces" (the collection is called a category) to the collection of things called "groupoids." The map that does this in a well defined way is called a functor. A good basic example of such a process is associating a group to the set of elements the group contains but without the group operations. This is a functor that maps an element from the category of groups to an element in the category of sets. Notice that there is a loss of information in this particular example as we go from groups to sets.
Not to derail the discussion, but for you Bluelamp and anyone else curious, this gives you some key words for any supplementary web searches that you might do... Cheers...
dant:
Looks to me, that the groupoid diagram appears
as prime matter: water (H2O) :)
Anyway, I will just and wait until Ark provides more
material so as not to add to pure speculation.
Bluelamp:
--- Quote from: Patience on March 23, 2012, 11:02:19 AM ---Thus, a group is a groupoid, but there are groupoids that are not groups. A groupoid G is a set that has a unary operation f : G -> G and a binary operation * : GxG -> G that IS NOT necessarily defined for every pair of elements from G. Probably the most immediate example of what I mean by an unary operation is the association of an element in G with its inverse.
So... Bluelamp... You have always seemed mathy, so I will say that the setting in which the notion of groupoid came about is category theory.
--- End quote ---
The test I've tried to use for group vs. groupoid is this one:
http://cornellmath.wordpress.com/2008/01/27/puzzles-groups-and-groupoids
--- Quote ---Roughly speaking, a group is any collection of transformations of an object, subject to the following requirements:
1.If you compose two transformations (i.e. perform one and then the other), the result is also a transformation.
2.Any transformation can be undone by some other transformation.
--- End quote ---
Thus the multiples of 45 degree rotations for an octagon would be a group since any two rotations can be done as one (and you can undo your rotations) but a groupoid for one of those 4 x 4 square puzzles with one square missing where you try to move the pieces back to their original order is not a group since you can't always do the two transformations (blocked by a square).
But yes all groups are groupoids and if you are dealing with transformations that are trying to describe something that gets blocked or can't be undone (which kind of sounds arrow of time-like) then you can certainly call it all a groupoid even if some subsets of what you are doing are actual groups.
Category theory I've seen mentioned as the ultimate thing to do once you get all you can get out of lattice-like ideas but I haven't looked at it yet to see how much of it I can understand. What little I remember seeing almost reminded me of object oriented programming.
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