Analogues of 'cone' distinguished triangles for pointed model categories?

Question

For an additive $A$ and any morphism $f:X\to Y$ in $C(A)$ one has the following distinguished triangle in the homotopy category $K(A)$: $X\to Y\to Cone(f)\to X[1]$.

1. What is the closest analogue of this construction for a (more or less) general pointed homotopy category? My problem here is that we do not have to put any restrictions on $f$ in $C(A)$, whereas in model categories (co)fibration sequences are defined for (co)fibrations of (co)fibrant objects only. Certainly, there are model structures for categories of complexes for which all objects are (co)fibrant; yet being a (co)fibration is surely a restriction on $f$ even in this setting. Should one 'rotate' (co)fibration sequences?

2. Under which conditions one can prove that a filtered limit (or homotopy limit) of (co)fibration sequences is a (co)fibration sequence? Note that that the distinsuished triangles for cones commute with arbitrary(?) small limits (those that exist in $C(A)$; the existence of all such limits is determined by $A$).

3. For a spectral model category this limit question seems to be related with certain 'continuity of the enrichment'. Are there any terms or papers related to this property (or does it follow from some other properties or axioms for spectral categories)?

Answer 1

1) Mikhail, mapping cones etc, are defined for arbitrary maps. The problem is that they are not homotopy invariant unless your model category is left proper. Therefore, in general you must take cofibrant replacements etc. Complexes form a left proper model category with the projective and with the injective model structures.

2) Homotopy colimits of cofiber sequences are cofiber sequences. If you want arbitrary filtered colimits of cofiber sequences to be cofiber sequences, the most usual hypotheses are set theoretical: your model category must be combinatorial. Then there is a cardinal $\alpha$ such that $\alpha$-filtered colimits of cofiber sequences are cofiber sequences. The cardinal $\alpha$ is the smallest one such that your category is locally $\alpha$-presentable and has sets of generating (trivial) cofibrations with $\alpha$-presentable sources (and maybe targets too, I don't remember). E.g. for simplicial sets you can take $\alpha=\aleph_0$.

I don't know of how being spectral may simplify my answers above. Referecens for 1): Hirschhorn's book rather than Hovey's. For 2): you have to look at papers on combinatorial model structures (Beke, Dugger, Rosický...), as books don't cover well this part of the theory.

Answer 2

Миша, привет! Вдобавок к тому, что сказал Фернандо, рекомендую статью Брауна 1973 года (найдешь), в которой излагается, как строить теорию гомотопий (и, в частности, конуса) для "категорий с кофибрантными объектами". Этот класс категориий является более общим, нежели модельные категории.