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If $A$ is a cocomplete category and $C$ is small, we can say that for all functor $C \stackrel{l}{\to} A$ there exists the Kan Extension Lan$_{y_C}(l): \hat{C} \to A$. Moreover, the Kan extension is pointwise.

Is it known if the other implication is true? Namely:

Conj: If for all functor $C \stackrel{l}{\to} A$ there exists the Kan Extension Lan$_{y_C}(l): \hat{C} \to A$ then $A$ is cocomplete.

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Then answer is yes if you assume the extensions are pointwise.

Let $\ast$ be the terminal presheaf on $\hat C$. Because the extension is poinwise, we have $Lan_{y_C}(l)(\ast) = \varinjlim_{c \in C} l(c)$. This can be seen, for example, by calculating the colimit using the category of elements of $\ast$, which is just $C$ itself.

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  • $\begingroup$ Since I am not assuming that A has colimits, how can I say that the extension is pointwise? $\endgroup$
    – Ivan Di Liberti
    Commented Mar 31, 2018 at 15:36
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    $\begingroup$ There are many ways to say that an extension is poinwise. One way is to say that the Kan extension is preserved by representable functors. But now I see I did not read carefully enough -- although you mention that the extension is pointwise in the cocomplete case, you didn't assume it in your question statement. $\endgroup$
    – Tim Campion
    Commented Mar 31, 2018 at 15:39
  • $\begingroup$ This is because I didn't know how to say that there are pointwise without colimits. Thanks for your answer. $\endgroup$
    – Ivan Di Liberti
    Commented Mar 31, 2018 at 15:40
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    $\begingroup$ Actually, it's natural to guess that the answer is no if you don't assume pointwiseness. This might be a candidate example statement for the point of pointwise Kan extensions... $\endgroup$
    – Tim Campion
    Commented Mar 31, 2018 at 15:41

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