Let $\mathscr{M}$ be a homotopical category, i.e. endowed with a subcategory $\mathscr{W}$ of weak equivalences satisfying 2-of-6. Let $\mathbb{A}$ be a small category, and note $\mathscr{M}^\mathbb{A}$ is again homotopical by taking weak equivalences pointwise. Write $\delta$ for the localization $\mathscr{M} \to \text{Ho}(\mathscr{M})$ (see Riehl - Categorical homotopy theory)

One can define the homotopy limit as the right derived functor of $\lim$, i.e. a homotopical functor $\mathbb{R}\lim :\mathscr{M}^\mathbb{A} \to \mathscr{M}$ together with a natural transformation $\rho: \lim \Rightarrow \mathbb{R}\lim$ such that $(\delta \mathbb{R}\lim, \delta \rho)$ is initial in the category of pairs $(H,\beta)$ with $H:\mathscr{M}^\mathbb{A} \to \text{Ho}(\mathscr{M})$ sending weak equivalences to isomorphisms and $\beta: \delta \lim \Rightarrow H$. One can show for example the classical notion of homotopy limits in simplicial model categories is of this form.

Now my question is: What happens when you take the left derived functor of $\lim$? My guess is you get something uninformative. I tried to show that if a left deformation of $\lim$ exists, then $\lim$ must already be homotopical, which would justify my guess. I can show my work towards this guess if anybody is interested. Mainly I want to know why one should or should not take a left derived functor of $\lim$.

  • $\begingroup$ My guess would be that you use the right adjointness of $\lim$ for the existence of the right derived functor. $\endgroup$ – user60589 Dec 1 '16 at 18:11

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