In this thesis we work with combinatorial and homological aspects of Igusa–Todorov discrete cluster categories Cm. Fix a positive integer m. The category Cm is an infinite discrete version of the classical cluster category of type An. This is a 2-Calabi–Yau triangulated category with cluster-tilting subcategories. The category Cm has a nice geometric model in terms of an ∞-gon, Zm, having m two-sided accumulation points, in which the indecomposable objects of Cm are in bijection with the arcs of Zm.

The Paquette–Yıldırım completion, Cm, of Cm has a geometric model where the indecomposable objects can be regarded as “limits of arcs” of Zm. The arc combinatorics of Cm allows us to classify the torsion pairs, t-structures, co-t-structures, and recollements of Cm. We observe that the categories Cm and Cm, despite having similar combinatorics, have some relevant homological differences.

We also work on defining different Calabi–Yau versions of Cm. We provide a candidate w-Calabi–Yau version for w ≥ 2, Cw,m, by taking the subcategory of w-admissible objects and morphisms of Cm. We expect that, by restricting the triangulated structure of Cm, we obtain a triangulated structure for Cw,m. Under the assumption that Cw,m is triangulated, we classify its w-cluster tilting subcategories and torsion pairs.

We also define the category C−1,m, the (−1)-Calabi–Yau version of Cm. To do so, we define an infinite discrete version of symmetric Nakayama representations using techniques from persistence theory. We obtain an abelian category which is Frobenius, uniserial, and symmetric. After stabilising, we obtain our desired (−1)-Calabi–Yau triangulated category. The category C−1,1 is additive equivalent to the Holm–Jørgensen category having (−1)-Calabi–Yau dimension, and we conjecture that the two categories are triangle equivalent.