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!
