Sequential colim vs sequential hocolim

Question

Suppose we have some homotopical setting in which we can speak of homotopy colimits. The setting I have in mind at the moment is that of a compactly generated triangulated category with a model, but we could look at any model category or some other homotopical context. Let $A_0 \to A_1 \to A_2 \to \cdots$ be a system of objects indexed by the natural numbers. My question is if there are theorems of the form "under such and such hypotheses, hocolim $(A_i)$ agrees with colim $(A_i)$".

Answer 1

Suppose that we are in one of the two settings:

1. C is the abelian category of (unbounded) complexes in a certain abelian category A satisfying Ab5, and D is the derived category of A; or
2. C is the abelian category of DG-modules over a DG-ring or DG-category A (with closed morphisms between the DG-modules) and D is the derived category of DG-modules.

Then for any inductive system of objects indexed by natural numbers in C, the image in D of their colimit in C coincides with their homotopy colimit in D. But one needs Ab5 for this to hold in the case 1, and if one replaces colimits with limits in either case 1 or 2, the assertion will no longer hold.

The proof is that in both cases 1 and 2 for any inductive system $X_i$ in C indexed by the natural numbers the telescope sequence $0\to \bigoplus_i X_i\to \bigoplus_i X_i\to \varinjlim X_i\to 0$ is exact in C, hence the cone of the left arrow is quasi-isomorphic to the right term. (In fact, this argument applies already in the coderived category of complexes or DG-modules, so the assertions are true for the coderived categories. This allows to replace a DG-ring with a CDG-ring.)

Answer 2

The obvious answer is: if all the maps are cofibrations (this might be your definition of hocolim). Or, slightly more generally, if a cofinal subsystem consists of cofibrations. It's still not an if and only if --- are you looking for something stronger than that?

Answer 3

Not an answer, but a clarifying comment. Language gets a little tricky here. For the sake of any beginners, and to try to standardize terms, let me say this.

Recall that a diagram $\cal I\to \cal M$ of shape $\cal I$ in a model category $\cal M$ has a hocolim, well-defined up to weak equivalence. Better, there is a functor "hocolim" from the diagram category $\cal M^{\cal I}$ to $\cal M$, well-defined up to natural weak equivalence. And that this respects weak equivalences, meaning that if a map $X\to Y$ of diagrams is an objectwise weak equivalence (meaning that $X(i)\to Y(i)$ is an equivalence for all $i$) then the resulting map from $hocolim X$ to $hocolim Y$ is an equivalence. So this yields a functor from $w^{-1}(\cal M^{\cal I})$ to $w^{-1}\cal M$ that can also be called hocolim. It is a common misunderstanding to think that there in fact a related functor from $(w^{-1}\cal M)^{\cal I}$ to $w^{-1}\cal M$, but in general it's not true. You can have two diagrams $\cal I\to \cal M$ such that they yield isomorphic diagrams in $w^{-1}M$ but have inequivalent hocolims. In particular, hocolim is not colim in $w^{-1}\cal M$. Diagrams in $w^{-1}\cal M$ don't always have colimits. When a diagram in $\cal M$ is such that as a diagram in $w^{-1}\cal M$ it has a colim, then the latter must be a retract of the hocolim.

In the special case where $\cal I$ is the ordered set of natural numbers, I find that I don't have anything very definitive to say. Certainly there are many examples of spaces or spectra where (sequential) hocolim is not colim in the homotopy category.