Torsors (2010) | Hacker News Reader

Torsors (2010)

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52 pointskortex14 comments

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.

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....

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_...

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.

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

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.

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.

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

A principal bundle is not itself a torsor (unless the base space is a point); only the fiber over each point is.

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)

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)

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)

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.

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.

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