Khovanov (2014) gave a categorification of the Fock space representation of the Heisenberg algebra via induction and restriction functors between characteristic zero representations of the symmetric group. By considering endofunctors of strict polynomial functors, Hong and Yacobi (2014) made some advances towards a characteristic-free version of this theory. In this talk we will discuss some progress made (and obstacles faced) in building a 'nice' characteristic-free categorification of Fock space.