6 ms·

Torsors (2010)

52 points by kortex 4y ago

kortex 4y ago
Torsors are algebraic structures that basically answer the question "why can I subtract two points in space / dates in time, but not add them?". Sure, you can pick an origin (space/time), but that's exactly because they are an affine space, and hence act like torsors.
- markisus 4y ago
  I think C++'s (std::chrono::time_point, std::chrono::time_delta) together make a torsor. The group G is std::chrono::time_delta and the set is std::chrono::time_point. Both the group action (time_point + time_delta) and the group operation (time_delta + time_delta) are declared here https://en.cppreference.com/w/cpp/chrono/time_point/operator_arith2 https://en.cppreference.com/w/cpp/chrono/time_point/operator....
  - endgame 4y ago
    Quite likely - see this discussion on whether to reify an explicit torsor concept in the design of Haskell's time libraries: https://old.reddit.com/r/haskell/comments/j9hfnd/torsors_in_the_time_library/ https://old.reddit.com/r/haskell/comments/j9hfnd/torsors_in_...
- pgorczak 4y ago
  An example that really stuck with me is that you can draw an arrow on a blank piece of paper and measure its length and direction without needing to define an x/y coordinate system on the paper. There is something there (the manifold and the action?) that is independent from the ways we can choose to describe the space.
082349872349872 4y ago
also: the difference between two memory addresses is often useful, but their sum almost never
plaguepilled 4y ago
Never heard of a principal bundle being referred to as a "torsor" - but Wikipedia informs me they're synonyms. Very cool!
- C-x_C-f 4y ago
  A principal bundle is not itself a torsor (unless the base space is a point); only the fiber over each point is.
  - red_trumpet 4y ago
    Algebraic geometers use both terms for the same thing. From [1]: > In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. So I wouldn't be too surprised if other people confuse this too. [1] https://en.wikipedia.org/wiki/Torsor_(algebraic_geometry) https://en.wikipedia.org/wiki/Torsor_(algebraic_geometry)
fatneckbeardz 4y ago
politics. on any political spectrum there is no origin point, there are just differences. for example in the UK they have socialized medicine and conservative politicians run campaigns on defending that socialized institution because its popular with the public. in america if you propose socialized medicine you would be considered a radical communist by half the population. no conservative would ever propose it. (except for military, then its ok)
- nh23423fefe 4y ago
  Show me your embedding from high-dimensional political belief to 1d left-right spectrum. What you really seem to be commenting on is contextuality not the absence of a center.
  - itishappy 4y ago
    Embedded or not, I still feel the absence of a center is true. The fact that "center" requires context to make sense is exactly the point. There's no clear "identity ideology," it requires an arbitrary choice. Put another way, you can subtract/compare ideologies: "X is a bit to the left of Y" (or "X is more in favor of policy P than Y" for any P in your high-dimensional space), but adding them doesn't really make sense.
dreamcompiler 4y ago
There's clearly a connection here to geometric [Clifford] algebra but I don't quite know how to state it other than "vector algebra is messy because it confuses groups and torsors while geometric algebra deconfuses the two." Also it appears that recognizing torsors is analogous to being rigorous with types. Or perhaps typeclasses.
- chombier 4y ago
  If anything, I would say that geometric algebra (with its vector division) confuses the group/torsor situation even more since they're both part of the same algebra.
hoosieree 4y ago
Is there any correlation between this affine property and the fact that you can emulate addition with subtraction (but not the other way around) and that you can emulate multiplication with division (but not the other way around). e.g. x+y == x-(0-y) x*y == x/(1/y)
- deleted 4y ago
  [deleted]
- deleted 4y ago
  [deleted]

Torsors (2010)

52 points by kortex 4y ago

Torsors are algebraic structures that basically answer the question "why can I subtract two points in space / dates in time, but not add them?". Sure, you can pick an origin (space/time), but that's exactly because they are an affine space, and hence act like torsors.

markisus 4y ago
I think C++'s (std::chrono::time_point, std::chrono::time_delta) together make a torsor. The group G is std::chrono::time_delta and the set is std::chrono::time_point. Both the group action (time_point + time_delta) and the group operation (time_delta + time_delta) are declared here https://en.cppreference.com/w/cpp/chrono/time_point/operator_arith2 https://en.cppreference.com/w/cpp/chrono/time_point/operator....
- endgame 4y ago
  Quite likely - see this discussion on whether to reify an explicit torsor concept in the design of Haskell's time libraries: https://old.reddit.com/r/haskell/comments/j9hfnd/torsors_in_the_time_library/ https://old.reddit.com/r/haskell/comments/j9hfnd/torsors_in_...
pgorczak 4y ago
An example that really stuck with me is that you can draw an arrow on a blank piece of paper and measure its length and direction without needing to define an x/y coordinate system on the paper. There is something there (the manifold and the action?) that is independent from the ways we can choose to describe the space.

082349872349872 4y ago

also: the difference between two memory addresses is often useful, but their sum almost never

plaguepilled 4y ago

Never heard of a principal bundle being referred to as a "torsor" - but Wikipedia informs me they're synonyms. Very cool!

C-x_C-f 4y ago
A principal bundle is not itself a torsor (unless the base space is a point); only the fiber over each point is.
- red_trumpet 4y ago
  Algebraic geometers use both terms for the same thing. From [1]: > In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. So I wouldn't be too surprised if other people confuse this too. [1] https://en.wikipedia.org/wiki/Torsor_(algebraic_geometry) https://en.wikipedia.org/wiki/Torsor_(algebraic_geometry)

fatneckbeardz 4y ago

politics. on any political spectrum there is no origin point, there are just differences. for example in the UK they have socialized medicine and conservative politicians run campaigns on defending that socialized institution because its popular with the public. in america if you propose socialized medicine you would be considered a radical communist by half the population. no conservative would ever propose it. (except for military, then its ok)

nh23423fefe 4y ago
Show me your embedding from high-dimensional political belief to 1d left-right spectrum. What you really seem to be commenting on is contextuality not the absence of a center.
- itishappy 4y ago
  Embedded or not, I still feel the absence of a center is true. The fact that "center" requires context to make sense is exactly the point. There's no clear "identity ideology," it requires an arbitrary choice. Put another way, you can subtract/compare ideologies: "X is a bit to the left of Y" (or "X is more in favor of policy P than Y" for any P in your high-dimensional space), but adding them doesn't really make sense.

dreamcompiler 4y ago

There's clearly a connection here to geometric [Clifford] algebra but I don't quite know how to state it other than "vector algebra is messy because it confuses groups and torsors while geometric algebra deconfuses the two." Also it appears that recognizing torsors is analogous to being rigorous with types. Or perhaps typeclasses.

chombier 4y ago
If anything, I would say that geometric algebra (with its vector division) confuses the group/torsor situation even more since they're both part of the same algebra.

hoosieree 4y ago

Is there any correlation between this affine property and the fact that you can emulate addition with subtraction (but not the other way around) and that you can emulate multiplication with division (but not the other way around). e.g. x+y == x-(0-y) x*y == x/(1/y)

deleted 4y ago
[deleted]
deleted 4y ago
[deleted]