An infinity-gon is a disc with infinitely many marked points in its boundary, with conditions on their accumulation points, which are unmarked. The cluster category of an infinity-gon was introduced by Igusa and Todorov, and has the property that its weak cluster-tilting subcategories are in natural bijection with the triangulations of the infinity-gon. Restricting to cluster-tilting subcategories, which must be functorially finite, requires some extra restrictions on the triangulation, given by Gratz, Holm and Jørgensen.

For complete infinity-gons, in which the accumulation points are marked, a corresponding cluster category was described by Paquette and Yıldırım (see also Cummings and Gratz), but very strong restrictions are needed on a triangulation of the completed infinity-gon for it to correspond to even a weak cluster-tilting subcategory. In this talk, based on joint work with İlke Çanakçı and Martin Kalck, I will explain how to resolve this problem, using extriangulated substructures.