  We generalize a framework of list colouring results to correspondence
colouring. Correspondence colouring is a generalization of list colouring
wherein we localize the meaning of the colours available to each vertex. As
pointed out by Dvo\v{r}\'ak and Postle, both of Thomassen's theorems on the
5-choosability of planar graphs and 3-choosability of planar graphs of girth at
least five carry over to the correspondence colouring setting. In this paper,
we show that the family of graphs that are critical for 5-correspondence
colouring as well as the family of graphs of girth at least five that are
critical for 3-correspondence colouring form hyperbolic families. Analogous
results for list colouring were shown by Postle and Thomas and by Dvo\v{r}\'ak
and Kawarabayashi, respectively. Using results on hyperbolic families due to
Postle and Thomas, we show further that this implies that locally planar graphs
are 5-correspondence colourable; and, using results of Dvo\v{r}\'ak and
Kawarabayashi, that there exist linear-time algorithms for the decidability of
5-correspondence colouring for embedded graphs. We show analogous results for
3-correspondence colouring graphs of girth at least five.
