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Is there a known definition of vector fields on a simplicial manifold?

For me, it seems natural that the definition should be something along the lines: Let $M_{\bullet}$ be a simplicial manifold with degeneracy maps $s_i: M_n \rightarrow M_{n+1}$ and face maps $\partial_i: M_{n+1} \rightarrow M_n$. A vector field on $M_{\bullet}$ is a vector field $X_n$ on each $M_n$ such that $X_n$ is $s_i$-related to $X_{n+1}$ plus some compatibility condition with $\partial_i$ (e.g. the Lie derivatives $\mathcal{L}_{X_n}$ commute with the Bott-Shulman differential $\delta = \sum (-1)^i \partial_i^*$).

However, I can't find any reference with a definition close to that.

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What kind of properties do you want your "vector fields" to have? There can't be anything close to a literal vector field, since the tangent (micro)bundle does not have the structure of a vector space. But for various purposes there are approximations to the idea. –  Ryan Budney May 17 '13 at 21:07
    
I don't know if that's what you are looking for, but the realization of the simplicial manifold ($\sqcup \Delta^n \times M_n/ \sim$) carries a differentiable space structure (e.g. Gajer "Geom. of Deligne cohomology", arxiv.org/abs/alg-geom/9601025, section 1). Starting from that you talk about differentiable forms (e.g. op. cit., section 4.1) and probably just the same way also about vector fields. –  Jakob May 17 '13 at 21:29
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1 Answer

up vote 2 down vote accepted

How about this? Apply the tangent functor $T$ to $M_\bullet$ to get a new simplicial manifold $TM_\bullet,$ that is take the composite

$$\Delta^{op} \stackrel{M_\bullet}{\longrightarrow} Mfd \stackrel{T}{\longrightarrow} VectBun \to Mfd,$$ where the last functor is the forgetful functor. There is an obvious map $\pi_\bullet:TM_\bullet \to M_\bullet.$ Say a vectorfield on $M_\bullet$ is a section $X_\bullet$ of $\pi_\bullet$ in the category of simplicial manifolds.

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With this definition, the vector fields $X_n$ will be both $s_i$ and $\partial_i$ related to each other. This makes sense. –  Thiago Drummond May 17 '13 at 21:54
    
You may also be interested in this: arxiv.org/abs/0810.0979 –  David Carchedi May 17 '13 at 22:08
    
P.S. depending on your motivation, you may want to take a section over a hypercover of $M_\bullet,$ e.g. if you are trying to model a vector field on the associated higher differentiable stack. –  David Carchedi May 17 '13 at 22:55
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