El Mehdi Cherradi
We introduce a general notion of $J$-tribe, and construct the $J$-tribe of $J$-frames in a given tribe $\mathcal{T}$, where $J$ a suitable generalized direct category. This construction applies to semi-cubical diagrams for a category of semi-cubes with symmetries and reversals.
This paper contains 5 sections, 21 theorems, 24 equations.
Lemma 1.1
The category of presheaves $\mathbf{Set}^{J^{op}}$ admits a cofibrantly generated weak factorization system whose left class is the class of monomorphisms. Moreover, this class is generated by the border inclusions where $J^x$ is the representable presheaf represented by $x$ and $\partial J^x$ its subpresheaf given as the "latching" colimit (i.e, $\partial J^x \to J^x$ is the latching map at $x$
