Diagrammatics for $F_4$

We define a diagrammatic monoidal category, together with a full and essentially surjective monoidal functor from this category to the category of modules over the exceptional Lie algebra of type $F_4$. In this way, we obtain a set of diagrammatic tools for studying type $F_4$ representation theory that are analogous to those of the oriented and unoriented Brauer categories in classical type.Comment: 20 pages. v2: Minor corrections, references added, proof of Prop. 5.2 modified; v3: Small improvement