Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

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: $$\varinjlim_\mathcal{J} \dashv \Delta : \mathcal{X} \to [\mathcal{J}, \mathcal{X}]$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.