We initiate a systematic study of lattices thick subcategories for arbitrary essentially small triangulated categories. To this end we give several examples illustrating the various properties these may, or may not, have and show that as soon lattice is distributive it automatically spatial frame. then construct two non-commutative spectra, one functorial with restricted functoriality, universa...

In previous work, based on the work of Zwara and Yoshino, we defined studied degenerations objects in triangulated categories analogous to degeneration modules. ${\mathcal{T}}$ , it is surprising that zero object may degenerate. We show subcategory generated by are coincides with consisting a vanishing image Grothendieck group $K_{0}({\mathcal{T}})$ .

Abstract We introduce the notion of a rank function on triangulated category