Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundle bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeOct 23rd 2012
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexI'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](http://geoff.reluctantm.com/publications/sman3.pdf). 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](http://golem.ph.utexas.edu/category/2012/10/tangency.html#c042316) 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: 1. *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. 2. *preservation of pullbacks* $T^n$ preserves $T_k$ 3. *vertical lift* $(l,0)$ is a natural transformation of additive bundles $p \colon T \to Id$ to $T p \colon T^2 \to T$. 4. *switcheroo* $c$ is a natural transformation of additive bundles $p_T \colon T^2 \to T$ to $T p \colon T^2 \to T$. 5. *various coherences* including $c^2 = 1$ and $l c = l$ 6. *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).

   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:

   1. 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.

   2. preservation of pullbacks $T^n$ preserves $T_k$

   3. vertical lift $(l,0)$ is a natural transformation of additive bundles $p \colon T \to Id$ to $T p \colon T^2 \to T$.

   4. switcheroo $c$ is a natural transformation of additive bundles $p_T \colon T^2 \to T$ to $T p \colon T^2 \to T$.

   5. various coherences including $c^2 = 1$ and $l c = l$

   6. 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).