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I’m putting this in Preprints & Publications because my initial plan now is to take a good look at Robin Cockett and Geoff Cruttwell’s paper Differential structure, tangent structure, and SDG. It might develop into something else. It’s not well-formed yet, hence nForum rather than nLab as yet. If I get anywhere, I’ll send Geoff an email pointing him to this. As a focussing point, I’ll see if I can figure out how the kinematic tangent space fits in with the axiomatic situation.
Geoff said on the nCafé that it was the vertical lift (and universality of thus) which, they found, was the key property of tangent structure. So that’s obviously something to pay particular attention to.
(Note In the paper maps are written on the right. Took me a while to spot that.)
I’ll repeat the definition. A tangent structure on a category $X$ consists of a functor $T \colon X \to X$ and natural transformations $(p,0,+,l,c)$ with $p \colon T \to Id$, $0 \colon Id \to T$, $+ \colon T_2 \to T$, $l \colon T \to T^2$, $c \colon T^2 \to T^2$ (here, $T_2$ is the pullback of $p$ over itself - extremely minor typo in the paper on this). These satisfy:
additivity $(p,0,+)$ makes $T M$ into an additive bundle over $M$. Note that “bundle over” is used simply to mean “object over”, there is none of the baggage that comes in from, say, bundle theory in differential topology.
preservation of pullbacks $T^n$ preserves $T_k$
vertical lift $(l,0)$ is a natural transformation of additive bundles $p \colon T \to Id$ to $T p \colon T^2 \to T$.
switcheroo $c$ is a natural transformation of additive bundles $p_T \colon T^2 \to T$ to $T p \colon T^2 \to T$.
various coherences including $c^2 = 1$ and $l c = l$
universality of vertical lift define $v \colon T_2 \to T^2$ by $v = (\pi_1 l, \pi_2 0_T) T(+)$. Then $v$ is the equaliser of $T(p), T(p) p 0 \colon T^2 \to T$.
Right, so now I’ll go away and stare at these definitions for a bit (and wonder why \xrightarrow
isn’t working on the forum).
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