Holmstrom Pre-triangulated category

See Hovey p. 170. Basicly, it is an abstraction of the homotopy category of a pointed model category, with its cofiber and fiber sequences. Hovey calls it “the unstable analogue of a triangulated category”.

Consider a (right) closed $Ho \ Sset_*$-module $S$. We have functors:

• $\wedge : S \times Ho \ Sset_* \to S$
• $Hom : Ho \ Sset_* \times S \to S$
• $Map: S \times S \to S$
• Suspension functor: $\Sigma A = A \wedge S^1$
• Loop functor: $\Omega A = Hom (S^1, A)$

The last two are cogroup and group objects as expected.

A pretriangulation on $S$ is now a collection of cofiber sequences (or left triangles) and a collection of fiber sequences (or right triangles) satisfying nine axioms. To define a pretriangulated category, Hovey also requires existence of small products and coproducts.

Can define exact adjunction between pretriangulated categories, as a pair preserving cofiber and fiber sequences, respectively.

A summarizing theorem: The homotopy pseudo-2-functor of Hovey’s Theorem 5.7.3 lifts to a pseudo-2-functor from pointed model cats to pre-triangulated cats, which commutes with the duality 2-functor.

Can also define closed (central, symmetric) monoidal pre-triangulated category.

Hovey defines a triangulated category as a pre-triangulated category in which the suspension functor is an equivalence. Every triangulated category in the sense of Hovey is a triangulated category in the classical sense, but not the other way around (as shown by Muro et al). Hovey claims that “every triangulated category in nature is the homotopy category of a stable model category”.

nLab page on Pre-triangulated category