  In this paper, we define a generalization of indexed categories and
contextual categories which we call contextually indexed (contextual)
categories. While contextual categories are models of ordinary type theories,
contextually indexed (contextual) categories are models of indexed type
theories. We also define type-theoretic semi-fibration categories which
generalize type-theoretic fibration categories. Every model category in which
cofibrations are stable under pullbacks is a type-theoretic semi-fibration
category. We show that type-theoretic semi-fibration category gives rise to a
contextually indexed contextual category with finite limits. Finally, we prove
that the category of simplicial sets with the Joyal model structure gives rise
to a locally small Cartesian closed contextually indexed contextual category
with indexed limits and colimits.
