Section: New Results

Path Functors in the Category of Small Categories

Participant : François Lamarche.

In [31] François Lamarche gives a detailed description of two path functors in the category of small categories, which he calls $𝐏𝐞$ and $𝐏$, and proves some of their important properties. The second of these is the functor which is used to model the Martin-Löf identity type in  [47] ; it associates to every small category $X$ an internal category structure whose object of objects is $X$; one important theorem which is proved in [31] is that the category of internal (co- or contravariant) presheaves on $𝐏X$ coincides with the category of Grothendieck bifibrations over the base $X$. Thus, through a trivial use of monadic abstract nonsense, we can say that $𝐏X$ is the free bifribration over $X$. The category $𝐏X$ is obtained by taking the bigger $𝐏𝐞X$, which is a little more than just a category, being poset-enriched, and getting rid of the order enrichment by quotienting. $𝐏𝐞X$ is a more general kind of bifibration than an ordinary Grothendieck bifibration, and the enrichment is necessary to describe its properties, thus taking us outside of the theory 1-categores.