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It may be silly, I know, but I wound up writing Pythagorean theorem. It’s not exactly “nPOV”. I did it to un-gray a link.
And now Pythagorean triple. (There is no end to gray links, is there?)
And better with redirects :-)
I’m not sure what the nPOV on the Pythagorean Theorem would be, but to perhaps aid in finding it, here’s how I think of the theorem:
WLOG, we may consider the context of a 2-dimensional Euclidean space (the theorem being trivial in lower dimensions, and Euclidean structure on a higher-dimensional space just amounting to compatible Euclidean structure on each of its 2-dimensional subspaces). Over such a space, we consider the symmetric bilinear product taking two vectors to the oriented area of the parallelogram formed by the first and the quarter-turn of the second. (The quarter-turn operator is only canonical up to negation, but it doesn’t matter for the canonicality of this product, interpreting its codomain suitably). Vectors are perpendicular just in case this product of them is zero; furthermore, the quadratic form corresponding to this product takes a vector to the area of the square formed upon it, which is to say, its squared length. Thus, we are in fact using the notions of perpendicularity and length which arise from an inner product space, and the Pythagorean Theorem amounts to a special case of the simple algebraic identity $(a + b)^2 = a^2 + b^2 + 2 a b$ (which more generally is, of course, the Law of Cosines).
That the Pythagorean Theorem holds in an inner product space is trivial; all that matters is establishing that one is, in fact, working in an inner product space. What it takes to establish this depends on what one is starting from (e.g., one might just as well axiomatize Euclidean geometry as the study of inner product spaces of particular dimension…), but I feel like the above approach might fairly be construed as teasing out the inner product structure inherent in geometry as an interested amateur would already be inclined to think of it.
Please feel free to add such remarks, Sridhar!
I suppose the nPOV would look to categorify, e.g., inner products in 2-Hilbert spaces, such as here.
There is the concept of Calabi-Yau object which is one $n$POV on non-degenerate bilinear forms.
I have added a few more hyperlinks to the entry and cross-linked a bit more. (Euclid, Elements, triangle, parallel postulate, length…)
Sridhar, I second Todd: what you say is a good point that absolutely deserves to be added to the entry. Please do so!
While trying to make sense of #7, filled in good monoidal (∞,1)-category. Why use such a bland word? Supposedly, the terminology is explained is Higher Topos Theory.
Back to the original point, do we ever see orthogonality appear in categorified inner products?
Yes, orthogonality is a fundamental phenomenon that is discussed in the context of Calabi-Yau categories. A random reference would be arXiv:1208.4046 (search the text for “orthogonal”).
This would be nice to expand on in some $n$Lab entry. Myself, I don’t have time for more than this telegraphic comment at the moment.
I’ve added to the article basically the same remarks I made in this thread.
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