  The algebraic zigzag construction has recently been introduced as a
combinatorial foundation for a higher dimensional notion of string diagram. For
use in a proof assistant, a layout algorithm is required to determine the
optimal rendering coordinates, across multiple projection schemes including 2D,
3D, and 4D. For construction of these layouts, a key requirement is to
determine the linear constraints which the geometrical elements must satisfy in
each dimension. Here we introduce a new categorical tool called
injectification, which lifts a functorial factorization system on a category to
diagrams over that category, and we show that implementing this recursively in
the category of finite posets allows us to systematically generate the
necessary constraints. These ideas have been implemented as the layout engine
of the proof assistant homotopy.io, enabling attractive and practical
visualisations of complex higher categorical objects.
