Periodic homotopy categories are triangulated #
The componentwise split exact category of cyclic cochain complexes is Frobenius, and its stable category is equivalent to Mathlib's homotopy category. The equivalence commutes with all integral shifts, retaining the signed cyclic shift on the homotopy category. Transporting Happel's triangulation along it makes the periodic homotopy category triangulated.
Its distinguished triangles are precisely the triangles isomorphic to images of distinguished
stable triangles. This description supplies the categorical structure for comparison with
explicit mapping-cone triangles. mappingCone_mem_distTriang proves the mapping-cone triangle
is distinguished, and mem_distTriang_iff shows every distinguished triangle is isomorphic to one.
No abelianity or positive-period assumption is needed; period
zero gives integer indexing.
homotopyCategory_mem_distTriang_iff characterizes distinguished triangles as images of
standard stable triangles of componentwise split conflations, up to isomorphism.
homotopyCategory_map_stableConflationTriangle_mem_distTriang supplies the corresponding
introduction rule.
The transport follows TauCeti.CommutativeAlgebra.MatrixFactorization.Triangulated, using
Mathlib's localization at isomorphisms.
References #
- D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, Chapter I, Section 2, Theorem 2.6.
- B. Keller, Chain complexes and stable categories, Manuscripta Mathematica 67 (1990), 379–417, Section 1.
- T. Stai, The triangulated hull of periodic complexes, Mathematical Research Letters 25 (2018), 199–236, Section 3.
The periodic homotopy category is pretriangulated, with distinguished triangles transported from the componentwise split stable category and the existing signed cyclic shift.
The canonical equivalence from the split stable category to the periodic homotopy category is a triangle functor.
The periodic homotopy category with its signed cyclic shift is triangulated.
The image of the standard stable triangle of a componentwise split conflation is distinguished in the periodic homotopy category.
A triangle in the periodic homotopy category is distinguished exactly when it is isomorphic to the image of the standard stable triangle of a componentwise split conflation.
The mapping-cone triangle of a periodic chain map in the homotopy category. Its last morphism is the negative cone projection, followed by the quotient's shift comparison.
Instances For
The first vertex of the mapping-cone triangle is the source's homotopy class.
The second vertex of the mapping-cone triangle is the target's homotopy class.
The third vertex of the mapping-cone triangle is the cone's homotopy class.
The first arrow of the mapping-cone triangle is the original chain map's homotopy class.
The second arrow of the mapping-cone triangle is the cone inclusion's homotopy class.
The third arrow of the mapping-cone triangle is minus the cone projection, with the quotient's shift comparison.
Mapping-cone triangles are distinguished. For a morphism f : X ⟶ Y of periodic
complexes, the image in the homotopy category of the triangle
X ⟶ Y ⟶ cone(f) ⟶ X⟦1⟧, whose last map is minus the cone projection, is distinguished.
The triangulation of the periodic homotopy category is the mapping-cone triangulation.
A triangle is distinguished exactly when it is isomorphic to the image of a mapping-cone triangle
X ⟶ Y ⟶ cone(f) ⟶ X⟦1⟧ of periodic complexes, whose last map is minus the cone projection.