Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As part of the proof, we also establish combinatorially the invariance of the linear-categorical representation of the strongly based mapping class groupoid given by the same theory.
"Combinatorial proofs in bordered Heegaard Floer homology." Algebr. Geom. Topol. 16 (5) 2571 - 2636, 2016. https://doi.org/10.2140/agt.2016.16.2571