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How is the weight $J(-)$ determined in this characterization of the category of element of a copresheaf as a coend? http://ncatlab.org/nlab/show/category+of+elements#properties_11

The functor el(-) seems to admit a right adjoint given by $D \to K(D)$ : $c\mapsto Ob([Jc, D])$, which would imply cocontinuity; am I right in deducing this from a purely formal-nonsense argument?

I'm trying to unravel the coend definition in the simple example of the action groupoid of a G-set, to see if I'm able to find what $J$ has to be at least in that case. Maybe this is linked to the presence of an adjoint for the functor $X\mapsto X//G$ from G-sets to groupoids. Does anybody know a reference for that?

Thanks.

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  • $\begingroup$ @tetrapharmakon, the category of elements is obtained by a special case of the Grothendieck construction. In light of this comment, the first part of your question has been answered in: mathoverflow.net/questions/128393/… If you take a careful glimpse into mentioned Ross Street's paper, you'll find the answer to your second question as well. $\endgroup$ May 24, 2013 at 11:43
  • $\begingroup$ I hoped for a more explicit answer (for example clarifying how is $J$ obtained starting from the functor $F$), but in any case, thank you... $\endgroup$
    – fosco
    May 24, 2013 at 14:14
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    $\begingroup$ Actually, in the literature, there are two concepts of categories of elements --- one for covariant functors $F \colon \mathbb{C} \rightarrow \mathbf{Set}$, and the other for contravariant functors $G \colon \mathbb{C}^{op} \rightarrow \mathbf{Set}$. These categories are defined as follows: $$\begin{array}{lcl} \mathit{elt}(F) &=& \int^{C \in \mathbb{C}} C/\mathbb{C} \times F(C)\newline \mathit{elt}^{op}(G) &=& \int^{C \in \mathbb{C}} \mathbb{C}/C \times G(C)\newline \end{array}$$ $\endgroup$ May 24, 2013 at 14:58
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    $\begingroup$ where the functor: $$\mathbb{C}/(-) \colon \mathbb{C} \rightarrow \mathbf{Cat}$$ is defined on arrows by: $$\mathbb{C}/(C \overset{f}\rightarrow D) \mapsto \lambda [X \overset{h}\rightarrow C]. f \circ h$$ and similarly the functor: $$(-)/\mathbb{C} \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$$ is defined by: $$(D \overset{g}\rightarrow C)/\mathbb{C} \mapsto \lambda [C \overset{k}\rightarrow X]. k \circ g$$ $\endgroup$ May 24, 2013 at 14:59
  • $\begingroup$ Definitely more clear! :) So if I understood you are weighting with the functor $\mathbf C\to \bf Cat$ which sends $c$ either to the category of arrows above or below $c\in \bf C$? Is there a hands-on way to show that elt(F) really is the cat of elements (e.g. appealing to the definition or the characterization as comma category)? At this point, I'm planning to add a couple of row in the nlab entry. $\endgroup$
    – fosco
    May 24, 2013 at 15:30

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