Relative Cluster Categories And Higgs Categories With Infinite-Dimensional Morphism Spaces

This Ph.D.thesis has two parts.In the first part,we show that a short exact sequence of abelian categories gives rise to short exact sequences of abelian categories of complexes,homotopy categories,and unbounded derived categories,refining a result of J.Miyachi.The second part focuses on the categorification of cluster algebras with coefficients by using the relative Calabi–Yau formalism which was developed by To(?)n and Brav–Dyckerhoff.Cluster categories were introduced in 2006 by Buan-Marsh-Reineke-Reiten-Todorov in order to categorify acyclic cluster algebras without coefficients.Their construction was generalized by Amiot and Plamondon to arbitrary cluster algebras associated with quivers(2009 and 2011).A higher dimensional generalization is due to Lingyan Guo(2011).Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians,double Bruhat cells,unipotent cells,....The work of GeissLeclerc-Schr(?)er often yields Frobenius exact categories which allow to categorify such cluster algebras.We give the construction of the Higgs category(generalizing GLS'Frobenius categories E)and of the relative cluster category(generalizing the derived category of E).The Higgs category is no longer exact but still extriangulated in the sense of Nakaoka-Palu(2019).We also give the construction of a cluster character in this setting under suitable hypotheses.