… homotopy category of finitely generated right A-modules has Auslander–Reiten triangles. …
over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein …

… in the second case we have triangles, which can be … homotopy category of a closed model
category has a rich structure and in particular is in a natural way a pretriangulated category, …

… We denote by K~' (a), K+' (o), and K (a) the homotopy categories of the categories of
complexes introduced above. Recall that two morphisms f*, g* : X* -> Y* are called homotopic if …

… triangle (3), the second row is the AuslanderReiten triangle (2), and each row and each column
is a distinguished triangle… Since the morphisms in C are homotopy classes of chain maps…

… Our aim is to construct a triangle equivalence C→CT such that the triangle … where HE
denotes the homotopy category of complexes over E. To compute this space, let G1 be the functor …

… abelian category modΛ and those of the triangulated category … category, Jørgensen described
Auslander-Reiten triangles in … classes of the morphisms in C(Mod Λ) modulo homotopy, …

… which occur in practice are compactly generated in the sense of Neeman [64], for instance
the stable homotopy category of spectra, the derived category of quasi-coherent sheaves over …

… Using the grading on B and the Calabi–Yau property we construct a triangle in the category
… show in this section that the stable category E is triangle equivalent to a generalized cluster …

… –Reiten triangles, and obtain a description of the extent to which the bilinear form is non-degenerate
in circumstances when Auslander–Reiten triangles … case: the homotopy category of …

… categories, such as Auslander–Reiten theory. This motivates Happel to introduce the notion
of Auslander–Reiten triangles for triangulated categories in [8… between homotopy categories …