To make it simple, take the spin foam formalism of ($SU(2)$) 3D gravity. My question is about the choice of the data that will replace the (smoothly defined) fields $e$ (the triad) and $\omega$ (the connection) on the disretized version of spacetime $\mathcal{M}$: the 2complex $\Delta$: why choose to replace $e$ by the assignment of elemnts $e\in su(2)$ to each 1cell of $\Delta$, and elements $g_{e}\in SU(2)$ to each edge in the dual 2complex $\mathcal{J}_{\Delta}$? I mean, these are both $su(2)$valued 1forms, thus, roughly speaking, assigning elements of $su(2)$ to vectors of the tangent bundle $\mathcal{TM}$, in other terms assigning elements of $su(2)$ to infinitesimal displacements represented by the 1cells of $\Delta$. I can understand the choise for $e$, but not for $\omega$, why this is so? Why it is not the inverse choice?
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