The naive approach to deriving profunctors - What's wrong with it?

Question

Let $(\mathcal{C,W})$ be relative category (equipped with a wide subcategory of weak equivalences satisfying 2 out of 3 property). Consider a profunctor $F: \mathcal{C \times D^{op}} \to \mathsf{Set}$ where $\mathcal{D}$ is an ordinary category. We can define right "derived" functor of $F$ by the following procedure (the left derivation is completely analogous):

• The right derived functor of $F$ is defined by $RF(X)= \lim_{{X \to Z} \in \mathcal{W}} F(Z)$ for $X\in \mathcal{C}$ where the limit is taken in the presheaf category $[\mathcal{D}^{op},\mathsf{Set}]$.

If we're lucky and $RF(X)$ is isomorphic to a representable presheaf for every $X \in \mathcal{C}$ then we have the actual derived functor. Otherwise we can still define it as a profunctor by the above procedure.

I assume that to actually compute this guy we need a more structured situation for the localization (on both the target and the source categories) to be tractable enough (e.g. model categories / triangulated categories). But as a definition I don't see anything wrong with it.

What's wrong with this naive approach? Why haven't I found it anywhere in the literature? (Is this the same as a Kan extension and I'm just being dumb?)

Answer 1

The problem with this definition is that the formula $\mathrm{colim}_{X \to Z \in \mathcal{W}} F(Z)$ does not, in general, depend functorially on $X$. For it to depend functorially on $X$ you need a way to push forward weak equivalences along arbitrary maps. This would work, for example, if $\mathcal{C}$ is a cofibration category and you take the colimit indexed only by trivial cofibrations out of $X$. Of course, taking the actual colimit may be too strict, but if you take the homotopy colimit $$\displaystyle\mathop{\mathrm{hocolim}}_{X \to Z \in \mathcal{W} \cap \mathcal{C}of} F(Z)$$ (computed, say inside the simplicial presheaf category $\mathrm{Fun}(\mathcal{D}^{op}, \mathrm{Set}_{\Delta})$), then you do get something sensible. For example, if $\mathcal{C}$ is a cofibration category and $X,Y$ are cofibrant objects then $$\displaystyle\mathop{\mathrm{hocolim}}_{X \to Z \in \mathcal{W} \cap \mathcal{C}of} \mathrm{Hom}(Y,Z)$$ is a model for the derived mapping space $\mathrm{Map}(Y,X) = \mathbb{R}\mathrm{Hom}(Y,X)$.