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Reading the article on Quillen bifunctors on the Nlab (https://ncatlab.org/nlab/show/Quillen+bifunctor) I stumbled upon the following claim: Let $\mathcal{C}$ be a combinatorial simplicial model category and let $\mathcal{D}$ be a small category. Furthermore, let $F \colon \mathcal{D}\to \mathcal{C}$ be a functor and $\star \colon \mathcal{C}^\text{op} \to \text{sSet}$ be the constant functor on the terminal object. Denote by $\text{sSet}^{\mathcal{D}^\text{op}}_\text{proj}$ and $\mathcal{C}^{\mathcal{D}}_\text{inj}$ the obvious categories endowed with the projective and injective model structures, respectively. The claim made is then that the homotopy colimit of $F$ may be given by a coend formula: $$\mathop{\text{hocolim}}\limits_{\mathcal{D}} F = \int\limits^\mathcal{D} Q_\text{proj}(\star) \cdot Q_\text{inj}(F)$$ where $Q_\text{proj}$ and $Q_\text{inj}$ are the respective cofibrant replacement functors. Apparently this formula follows from the fact that $$\int\limits^\mathcal{D}(-\cdot -): \text{sSet}^{\mathcal{D}^\text{op}}_\text{proj} \times \mathcal{C}^{\mathcal{D}}_\text{inj}\to \mathcal{C}$$ where $\cdot \colon \text{sSet} \times \mathcal{C} \to \mathcal{C}$ denotes the copower, is a left Quillen bifunctor.

How exactly does one deduce the homotopy colimit formula from the fact that the above is a Quillen bifunctor, assuming that the homotopy colimit is defined to be the left derived functor of $\text{colim}$?

h3fr43nd
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  • It might be helpful to learn about weighted colimits and observe that the colimit of $F$ weighted by the constant $\Delta^0$ is isomorphic to the ordinary colimit. Then you can use the properties of Quillen bifunctors to deduce that the formula, considered as a functor in $F$, is a left derived functor for the ordinary colimit functor. – Zhen Lin Dec 14 '22 at 06:17
  • @ZhenLin But this is just the same as considering $\Delta^0 \cdot_\mathcal{D} - =\int\limits^\mathcal{D} \Delta^0 \cdot - \cong \text{colim}$, right? This functor is a left Quillen functor and therefore we may derive it. This then yields $\mathbb{L} \int\limits^\mathcal{D} = \int\limits^{\mathcal{D}} Q(-) = \text{hocolim}$. This is almost the formula we want, but I don't see how you get to $Q_\text{proj}(\star) \cdot Q_\text{inj}(-)$ inside the coend. – h3fr43nd Dec 14 '22 at 11:59
  • Derived functors are defined up to weak equivalence, not isomorphism. It suffices to check that the two formulae are related by a weak equivalence. (There is an obvious comparison coming from the weak equivalence $\Delta^0 \to Q_\textrm{proj} (\Delta^0)$.) – Zhen Lin Dec 14 '22 at 12:14
  • @ZhenLin Thanks, I think I got it :) I will write up an answer myself later. – h3fr43nd Dec 14 '22 at 13:37
  • @ZhenLin Is it really always true that $\Delta^0\cdot_\mathcal{D}(-)$ is equal to $\mathop{\text{colim}}\limits_\mathcal{D}$? Or put differently, is it true that $\Delta^0 \cdot X \cong X$ for all $X \in \mathcal{C}$? – h3fr43nd Dec 14 '22 at 17:16
  • Yes, because $\Delta^0$ is the unit of the monoidal structure on $\textbf{sSet}$. – Zhen Lin Dec 14 '22 at 22:08
  • @ZhenLin I would be very grateful if you could briefly look over my answer so as to ensure I didn't overlook anything :) – h3fr43nd Dec 15 '22 at 00:55

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Following the hints in the comments, I have come up with the following: Denote by $\odot \colon \text{sSet} \times \mathcal{C}\to \mathcal{C}$ the tensoring of $\mathcal{C}$ over $\text{sSet}$. We then have $$\mathcal{C}(\Delta^0 \odot X, Y) \cong \text{Map}(\Delta^0,\mathcal{C}(X,Y))\cong \mathcal{C}(X,Y)$$ for all $X,Y \in \mathcal{C}$ (where $\text{Map}(-,-)$ is the internal hom in $\text{sSet}$) and hence $\Delta^0 \odot - \cong 1_\mathcal{C}$. This then implies $\int\limits^\mathcal{D} \Delta^0 \odot (-) \cong \mathop{\text{colim}}\limits_\mathcal{D}$. This is a left Quillen functor, so we may derive it to find: $$\mathop{\text{hocolim}}\limits_\mathcal{D}=\mathbb{L}\int\limits^\mathcal{D} \Delta^0 \odot (-) =\int\limits^\mathcal{D} \Delta^0 \odot Q_\text{inj}(-)$$ But then since $$\int\limits^\mathcal{D} (-) \odot (-)$$ is a left Quillen bifunctor, the weak equivalence $Q_\text{proj}(\Delta^0) \overset{\simeq}{\to} \Delta^0$ induces a natural weak equivalence $$\int\limits^\mathcal{D} Q_\text{proj}(\Delta^0)\odot Q_\text{inj}(-) \simeq \int\limits^\mathcal{D} \Delta^0 \odot Q_\text{inj}(-)$$ Hence indeed, $$\mathop{\text{hocolim}}\limits_\mathcal{D} \simeq \int\limits^\mathcal{D} Q_\text{proj}(\Delta^0)\odot Q_\text{inj}(-)$$

h3fr43nd
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