On a notion of oplax 3-functor

Abstract

We introduce a notion of normalised oplax $3$-functor suitable for the elementary homotopy theory of strict $3$-categories, following the combinatorics of orientals. We show that any such morphism induces a morphism of simplicial sets between the Street nerves and we characterise those morphisms of simplicial sets coming from normalised oplax $3$-functors. This allows us to prove that normalised oplax $3$-functors compose. Finally we construct a strictification for normalised oplax $3$-functors whose source is a $1$-category without split-monos or split-epis.