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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",, 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
up vote 3 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: – 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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