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Let $ C $ be a small category and $ \mathfrak{X} $ be complete and cocomplete category. The coend may be seen as a functor

$ \int ^C : Fun (C\times C^{op}, \mathfrak{X} )\to \mathfrak{X} $

Indeed, $ \int ^C = colim\circ T $, in which $ T $ is a functor which associates each object of $ Fun (C\times C^{op} , \mathfrak{X}) $ to the appropriate coequalizer diagram.

So, I wonder if this functor $T$ has a right adjoint. I tried to find such a right adjoint, but I couldn't. Unfortunately, I couldn't use the Freyd adjoint theorem: despite the fact that $ \int ^C $ preserves colimits.

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1 Answer 1

up vote 2 down vote accepted

The coend is a special case of the notion of a weighted colimit. Indeed, let $H : \mathcal{C}^\mathrm{op} \times \mathcal{C} \to [(\mathcal{C}^\mathrm{op} \times \mathcal{C})^\mathrm{op}, \mathbf{Set}]$ be the Yoneda embedding and let $W = \int^\mathcal{C} H$. It can be shown (using the Yoneda lemma for ends/coends) that $W (c, c') \cong \mathcal{C} (c', c)$, and for any $\mathcal{X}$ where coends exist, there is the following natural bijection: $$\mathcal{X} (\int^\mathcal{C} F, T) \cong [(\mathcal{C}^\mathrm{op} \times \mathcal{C})^\mathrm{op}, \mathbf{Set}](W, \mathcal{X}(F, T))$$ Thus, $\int^\mathcal{C} F$ is the weighted colimit $W \star F$. Thus, it suffices to show that $W \star {-}$ has a right adjoint for a general weight $W$.

Let $\mathcal{J}$ be any small category and let $W : \mathcal{J}^\mathrm{op} \to \mathbf{Set}$ be a weight. Suppose $\mathcal{X}$ is complete and cocomplete. Then, for any diagram $F : \mathcal{J} \to \mathcal{X}$, \begin{align} \mathcal{X} (W \star F, T) & \cong [\mathcal{J}^\mathrm{op}, \mathbf{Set}] (W, \mathcal{X} (F, T)) \\ & \cong \int_{j : \mathcal{J}} \mathbf{Set} (W j, \mathcal{X} (F j, T)) \\ & \cong \int_{j : \mathcal{J}} \mathcal{X} (F j, W j \pitchfork T) \\ & \cong [\mathcal{J}, \mathcal{X}] (F, W \pitchfork T) \end{align} where $W j \pitchfork T$ denotes the $W j$-fold cartesian product of $T$. Thus, $$W \star {-} \dashv W \pitchfork {-} : \mathcal{X} \to [\mathcal{J}, \mathcal{X}]$$ In the case where $W$ is the terminal weight, this recovers a well-known adjunction: $$\varinjlim_\mathcal{J} \dashv \Delta : \mathcal{X} \to [\mathcal{J}, \mathcal{X}]$$

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