The n-Category Café

Last time # I talked about how the category of $n$-paths (I consider $n=1$ and $n=2$ only) in a space $X$ and that of $n$-paths in a regular surjection

give rise to the universal local transition $P_n(Y^{•})$ of $n$-transport # on $X$; and that this is nothing but the category of $n$-paths in $Y$ which may “jump” between different lifts along $p$.

Moreover, from any $p$-local transition data of $n$-transport (trivial transport on single patches, transitions $g$ of that on double intersections, transitions $f$ of these on triple intersections, and so on) one obtains a 2-transport

Clearly, this wants to descend to $X$. The descent is manifest if

For general $Y$ the constructions of this equivalence that I have managed to come up with (e.g. section 3. here) are a little unwieldy. But with a certain assumption on $Y$ (which in common applications is always possible) it looks much better:

$$descent of the universal transition.