Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?

What I have in mind is, for example, the simplicial set $BG$ (whose n-simplices are $G^{n+1}$, for some group $G$). For example, $B \mathbf Z$ is well-known to be homotopic to the circle, which has a tangent space etc. I would like to know whether it is possible to view these data "directly" (and somehow, simplicially) on $B \mathbf Z$ or similar simplicial sets. I presume it is possible to endow the standard topological realization with a structure of a differentiable space and talk about its geometric features. However, if possible, I would prefer a more direct approach. For example, for the purposes of rational homotopy theory, Sullivan defined, for a simplicial set $M$, the (piecewise linear) differential forms as

$$Hom(M, \Omega_*)$$

where the $n$-simplices in $\Omega_*$ are $\Lambda (x_0, \dots, x_n, dx_i) / (\sum x_i-1, \sum dx_i)$.

Thank you!

  • $\begingroup$ Some people use the notation BG for just the homotopy type of the thing you mention. One version to ask your question would be: Is there always a space with the homotopy type of BG such that one can define a sensible tangent space (with metrics)? $\endgroup$ – Konrad Voelkel Feb 20 '13 at 14:57
  • $\begingroup$ Yes, but I am looking for an answer not just for BG. Of course, in general the "tangent space" things will probably fail to be meaningful, but I just want a general definition that depends directly on the simplicial set $M$, not its homotopy type. $\endgroup$ – Jakob Feb 20 '13 at 15:16
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    $\begingroup$ You can define tangent cones using geometric realization, but for $B\mathbf{Z}$ you'll get something infinite dimensional. $\endgroup$ – S. Carnahan Feb 21 '13 at 4:56
  • $\begingroup$ OK, thanks. How exactly would this go? (Sure, the resulting tangent space/cone/"bundle" will be huge in any case...) $\endgroup$ – Jakob Feb 21 '13 at 10:21

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