Jason (jcreed) wrote,

Kock's book on Synthetic Differential Geometry is completely blowing my mind. Thanks to krasnoludek for alerting me to the existence of SDG at all. I'm sort of surprised I haven't heard of it, since it is basically Freyd and Kock and friends trying to make calculus all categorifed and constructive and shit, kind of by doing things like nonstandard analysis, but more so.

Also reading this paper by John Bell, titled "Two Approaches to Modelling the Universe". He makes the interesting claim that

The whole point of synthetic differential geometry is to render the tangent bundle functor representable (p.5)

because what you get from taking the idea of infinitesimals seriously is a space Δ that contains them all. Then the tangent bundle over a manifold M simply is the set of all mappings from Δ into M! The fact that we have an object Δ such that the concept we're interested in (i.e. tangent vectors tangent to some point of M) is isomorphic to the set of functions from some object (the "representation" Δ) is exactly what representability means. It's a category-theoretic concept I hadn't given much respect to until lately; the other case where representability popped up just yesterday was finding out from CWM that monoid objects in a strict monoidal category are representable as functors out of a category also called Δ (no coincidence of choice of notation, I think! it suggests simplices, too...) whose objects are finite ordinals and whose arrows are monotone maps, whose monoidal structure is just ordinal addition.
Tags: infinitesimals, math
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