It seems to be that in the context of rigid body dynamics, the moment of inertia is introduced as the quantity that maps the components of the angular velocity into the components of angular momentum.
Well the angular velocity is a generalised velocity, therefore it ought to be an element of the tangent space at some point on the manifold and the angular momentum lives in the cotangent space whose components are defined as gradient components of the Lagrangian i.e.
$$ p_i = \dfrac{\partial\mathcal{L}(q,\dot{q},t)}{\partial\dot{q}^i}$$
Here $p_i$ are the components of the angular momentum and $q^i$ are components of angular velocity. Now is it the case that we assume that these components are related by linear transformation, namely the moment of inertia? If so then it's not at all obvious to me why $L$ ought to depend linearly with $\omega$. Moreover even if it did, how exactly does the moment of inertia qualify as a $(1,1)$ tensor? It seems to be mapping a vector to a covector.