If $\mathcal{C}$ is a category and $\lambda:\Delta_D \to id_{\mathcal{C}}$ is a cone for the identity functor, and $F:J \to \mathcal{C}$ is a functor such that $F\lambda:\Delta_D \to F$ is a limiting cone, then it follows that $D$ is an initial object of $\mathcal{C}$. This is Lemma 1 on page 230 in Maclane's Categories for the Working Mathematician. The proof given however is pretty down to earth and does not generalize immediately to higher category theory. Is there a more high-browed way of proving this that works in a more general context than 1-categories?
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1$\begingroup$ It's page 234 in the second edition. $\endgroup$– Mike ShulmanCommented May 21, 2012 at 17:16
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$\begingroup$ A while back I read something like this for the proof that "every diagram commutes" in a monoidal category, i.e. a way to soup up the proof so it's less down-to-earth and more high-browed. It may have been on nCat. I'm super busy this week, but I'll try to look up the reference later if no one else posts an answer before say Thursday. $\endgroup$– David WhiteCommented May 21, 2012 at 17:44
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$\begingroup$ Should $F\lambda$ maybe be $\lambda F$? $\endgroup$– Dylan WilsonCommented May 21, 2012 at 21:54
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The definition of object being initial is obviously equivalent to the statement "the inclusion of one-point category is final". Here by final functor I mean $p: I\to C$, such that for any $F: C\to D$ we have $$\mathrm{Lim}\ Fp \simeq \mathrm{Lim}\ F$$
The criterion of being final is proved in MacLane 9.3 (there such functors are called "initial", and "final" is used for colimits). Then $p: \ast \to C$ is the initial object iff $$\mathrm{Lim}\ 1_C \simeq \mathrm{Lim}\ 1_C\circ p = \mathrm{Lim} p = p$$
This is the theorem you mention. In higher category theory the concepts of limits and final functors work just the same, so we have a similar theorem. Note that the criterion of finalness differs: all corresponding overcategories must be contractible.
answered May 21, 2012 at 23:18
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$\begingroup$ Thanks for the answer Anton. Could you help me see how this the same as what I asked? You never talk about cones, nor any functor playing the role of $F$ in my question. $\endgroup$ Commented May 22, 2012 at 1:30
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$\begingroup$ To be more specific, first of all, I really want the dual statement for final objects, but I asked it the way I did because of what I found in Maclane. Anyhow, in my situation, I have a functor $i:C \to D$ into a cocomplete (infinity) category $D$ with $C$ small, and I know that the left Kan extension of $i$ along itself is the identify functor. I want to show that the colimit of $i$ is terminal. I have a proof using global Kan extensions, but it's flawed (I think) since there's no reason the identity functor should admit these for large diagram categories. $\endgroup$ Commented May 22, 2012 at 1:34
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$\begingroup$ (If it matters at all, $i$ is fully faithful) $\endgroup$ Commented May 22, 2012 at 1:34