Definition. Path-linearisable category [mde-HYYV]

Path-linearisable categories axiomatize categories in which strings of morphisms with equal composites have a canonical interleaving.

A category \(\mathbb {C}\) is path-linearisable if,

1. Every commutative square has at least one diagonal filler in either direction, and
2. for every

   

   where \(\alpha f = \alpha g\) and \(\beta f = \beta g\), then \(f = g\).

The second condition ensures that any two parallel fillers are equal and any two antiparallel fillers are inverses.

ULF functors over a path-linearisable category \(C\) form a sheaf topos.