With all due respect with your brand as LessWrong’s ornamental hermeneutic I’m afraid I’ll need some clarification.
What is the monad of 1 exactly? A monad is a functor—what category are we talking about here?
In particular—what are the unit and multiplication maps?
(my guess: 1+kϵ↦k for the unit and (1+kϵ)(1+mϵ)↦1+(k+m)ϵ+k⋅mϵ2=1+(k+m)ϵ+0 but now I’m using nilsquare infinitesimals instead of invertible infinitesimals.)
I’m not sure what tangent space we are talking about—but I assume it’s a Lie group (hyperfinite graph?) and we are looking at the tangent space of the identity element. In this case—what is the clifford algebra of the tangent space? Construction of a clifford algebra needs a choice of inner product (or more generally a quadratic form) - what are we picking here?
-”On any finite dim space we have a canon inner product by taking the positive definite one.”
What? A finite dimensional space has more than one positive definite inner product (well, unless it is zero-dimensional), this choice is certainly not canonical. For example in R^2 any ellipse centered at the origin corresponds to a positive definite inner product.
With all due respect with your brand as LessWrong’s ornamental hermeneutic I’m afraid I’ll need some clarification.
What is the monad of 1 exactly? A monad is a functor—what category are we talking about here?
In particular—what are the unit and multiplication maps?
(my guess: 1+kϵ↦k for the unit and (1+kϵ)(1+mϵ)↦1+(k+m)ϵ+k⋅mϵ2=1+(k+m)ϵ+0 but now I’m using nilsquare infinitesimals instead of invertible infinitesimals.)
I’m not sure what tangent space we are talking about—but I assume it’s a Lie group (hyperfinite graph?) and we are looking at the tangent space of the identity element. In this case—what is the clifford algebra of the tangent space? Construction of a clifford algebra needs a choice of inner product (or more generally a quadratic form) - what are we picking here?
On any finite dim space we have a canon inner product by taking the positive definite one.
Monad is a synonym for infinitesimal neighborhood, common on the literature. Not the category theory monad.
Also hermeneutic lmfao
-”On any finite dim space we have a canon inner product by taking the positive definite one.”
What? A finite dimensional space has more than one positive definite inner product (well, unless it is zero-dimensional), this choice is certainly not canonical. For example in R^2 any ellipse centered at the origin corresponds to a positive definite inner product.
I was thinking the one corresponding to a unit circle, just the ordinary dot product.
Canon is probably the wrong word in a mathy context.
Also yes the infinitesimal neighborhood of the identity.