Homotopy limits commute

Question

I'm looking for a reference for the fact that homotopy limits commute. That is

$\mathrm{holim}_U \mathrm{holim}_W E = \mathrm{holim}_W\mathrm{holim}_U E $.

Preferably looking for a result that define homotopy limit as the right derived functor of $\lim$.

Answer 1

I couldn't find a reference for this either, so came up with this argument of the cuff.

We can define homotopy limits as the right derived functor of $\lim$, and recall that this is a right adjoint - this is really the crucial thing to note.

Then $\mathrm{holim}_U \mathrm{holim}_W E = \mathrm{holim}_U \lim_W RE$, where $R$ denotes fibrant replacement.

(Note that I am implicitly assuming that $E$ is in a nice model category with a functorial factorisation, so the fibrant replacement functor exists)

Now $\mathrm{holim}$ is a right adjoint functor hence commutes with limits, it follows that

$\mathrm{holim}_U \lim_W RE = \lim_W \mathrm{holim}_U RE$

Now since $RE$ is fibrant already, this last is, $$ \lim_W \lim_U RE$$.

Now if you take the other side of the equation, and follow the same process we get $$ \mathrm{holim}_W \mathrm{holim}_U E = \mathrm{holim}_W \lim_U RE = \lim_U \mathrm{holim}_W RE = \lim_U \lim_W RE = \lim_W \lim_U RE$$

with the last equality following by the fact limits commute [ this is just a special case of right adjoints preserving limits]