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Let $D: I \to \mathcal C$ be a diagram, and suppose we have a colimit decomposition $I = \varinjlim_{j \in J} I_j$ in $Cat$. Then under certain conditions, we can decompose the colimit of $D$ as $\varinjlim_{i \in I} D_i = \varinjlim_{j \in J} \varinjlim_{i \in I_j} D_i$. But I've never seen general conditions along these lines spelled out for 1-categories.

Question 1: Is there some place where conditions making the above true are given in the 1-categorical setting?

For $\infty$-categories, there is Corollary 4.2.3.10 of Higher Topos Theory. Unfortunately, the formulation of the result is somewhat abstruse, being expressed in terms of the bespoke simplicial set denoted $K_F$ there (defined using 4 conditions in Notation 4.2.3.1).

As a result, I'm having the following problem: it seems to me that for any cocone of $\infty$-categories $(I_j \to I)_{j \in J}$, one should be able to construct a natural map $\varinjlim_{j \in J} \varinjlim_{i \in I_j} D_i \to \varinjlim_{i \in I} D_i$, and one would expect HTT 4.2.3.10 to imply that under the appropriate conditions, this map is an equivalence. But the formulation doesn't seem to easily lend itself to confirming this.

Question 2: Is the natural map $\varinjlim_{j \in J} \varinjlim_{i \in I_j} D_i \to \varinjlim_{i \in I} D_i$ constructed somewhere in reasonable generality? (Or else is it easy to construct from general machinery given somewhere?)

Question 3: Is there written somewhere an account of conditions (perhaps analogous to those of HTT 4.2.3.10) which ensure that this map is an equivalence?

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    $\begingroup$ In his answer to the question you reference, Peter seems to claim this is always true for $1$-categories. $\endgroup$ Sep 3 '20 at 21:55
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    $\begingroup$ But he doesn't give a reference, and the claim doesn't seem that obvious to me (it seems a bit subtle, because of the possibly weird behaviour of colimits in Cat, as you point out) , so I guess it'd be good to have some clarification (should it only be to see if the proof goes through in more generality, which you seem to seek). $\endgroup$ Sep 3 '20 at 22:13
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    $\begingroup$ (Actually, it might not be that subtle in the $1$-categorical case; I think it mostly relies on the fact that Cat is cartesian closed, and on the analysis of hom-sets in a limit of categories) $\endgroup$ Sep 3 '20 at 22:21
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    $\begingroup$ There is theorem 7.3.16 in my book on higher categories (in the spirit of the result you quote from HTT but a little bit more usable). This is what explains decompositions of diagrams with Reedy-like considerations, as explained in corollary 7.4.4 proposition 7.4.5 of loc. cit. for instance. $\endgroup$ Sep 3 '20 at 22:36
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    $\begingroup$ @TimCampion you might want to check out example 2.5 here: arxiv.org/pdf/1705.04933.pdf. Asaf Horev and I prove here what you ask for. $\endgroup$
    – KotelKanim
    Nov 15 '20 at 11:53
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Let $p \colon E \to J$ be the cocartesian fibration for the diagram $j \mapsto I_j$. Then the colimit over $E$ of $F \colon E \to C$ can always (assuming the appropriate colimits exist in $C$) be written as an iterated colimit: $$ \mathrm{colim}_E \, F \simeq \mathrm{colim}_J \, p_! F \simeq \mathrm{colim}_{j \in J} \, \mathrm{colim}_{I_j} \, F|_{I_j} $$ by first doing the colimit in two steps using the left Kan extension along $p$ and then that the inclusion $E_j \to E \times_J J_{/j}$ is cofinal since $p$ is cocartesian.

Now the colimit $I$ can be described as the localization of $E$ at the cocartesian morphisms. Since any localization is cofinal, this means there is a cofinal functor $q \colon E \to I$. For a functor $D \colon I \to C$, this means we have equivalences $$ \mathrm{colim}_I \, D \simeq \mathrm{colim}_E \, Dq \simeq \mathrm{colim}_{j \in J} \, \mathrm{colim}_{I_j} \,D|_{I_j}. $$

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  • $\begingroup$ Nice! Thanks, Rune. I'll probably accept this one -- though I also like Dylan's presentation of essentially the same argument in his answer below. And of course, it would be nice to work out an analog of Zhen Lin's argument in a higher setting -- those types of arguments in terms of homsets tend to generalize more easily to the enriched setting. $\endgroup$
    – Tim Campion
    Sep 3 '20 at 23:22
  • $\begingroup$ I wonder if the proofs in HTT of the relevant statements actually depend on the more technical statement of 4.2.3.10. It would be nice if they didnt'! $\endgroup$
    – Tim Campion
    Sep 3 '20 at 23:26
  • $\begingroup$ I may be misinterpreting your statement/making a mistake, but I don't think that $I$ is always the localization of $E$. For instance consider the case where $J$ is a one object groupoid $BG$ and $J \to Cat$ is the constant diagram with value the trivial category. Then the colimit is the trivial category, but $E$ is $BG$, and all localizations of $E$ are isomorphic to $E$. $\endgroup$ Sep 4 '20 at 1:01
  • $\begingroup$ @PhilTosteson The colimit of the constant diagram in the $\infty$-category of $\infty$-categories is $BG$, not the point. This is just another way in which colimits are better behaved in $\infty$-categories :). $\endgroup$ Sep 4 '20 at 5:15
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    $\begingroup$ @PhilTosteson You are right. The localization of E at cocartesian morphism is literally the 2-colimit (i.e. the colimit in the $\infty$-category obtained by inverting equivalences of categories in the $1$-category of small categories). This does not coincide with the colimit (in the $1$-category of small categories) in general. Rune's answer together with Zhen Lin's show that the comparison map from the $2$-colimit to the $1$-colimit, although not an equivalence, is colimit-final. I also thought that the question was about $1$-colimits though. $\endgroup$ Sep 4 '20 at 9:00
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The way I always remember this stuff is as follows:

  • Given a map $J \to \mathsf{Cat}$ form the associated cocartesian fibration $E \to J$.
  • By assumption, $I$ is the actual colimit (as opposed to the left lax one) so we have a (weak) localization $E \to I$. Weak localizations are final (and initial, in fact) so, to compute the colimit over $I$ is the same as computing it over $E$.
  • To compute the colimit over $E$ we may first left Kan extend to $J$.
  • Since $E \to J$ is cocartesian, the map $E_x \to E_{/x}$ is final, and we may replace $E_{/x}$ with $E_x=I_x$ in the formula for left Kan extensions.

That gives the result.

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  • $\begingroup$ ah! Rune beat me to it $\endgroup$ Sep 3 '20 at 23:15
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I assume $\varinjlim_{j : \mathcal{J}} \mathcal{I}_j = \mathcal{I}$ is meant in the strict sense of 1-categories. Since $\textbf{Cat}$ is cartesian closed, $$\textstyle [\mathcal{I}, \mathcal{C}] \cong \varprojlim_{j : \mathcal{J}} [\mathcal{I}_j, \mathcal{C}]$$ where the limit on the RHS is also meant in the strict sense of 1-categories. Let $\lambda_j : \mathcal{I} j \to \mathcal{I}$ be the component of the colimit cocone in $\textbf{Cat}$. Then, we also get a limit formula for the hom-sets of $[\mathcal{I}, \mathcal{C}]$, namely, $$\textstyle [\mathcal{I}, \mathcal{C}](D, \Delta T) \cong \varprojlim_{j : \mathcal{J}} [\mathcal{I}_j, \mathcal{C}](D \lambda_j, \Delta T)$$ so if the relevant colimits exist in $\mathcal{C}$, $$\textstyle \mathcal{C} \left( \varinjlim_\mathcal{I} D, T \right) \cong \varprojlim_{j : \mathcal{J}} \mathcal{C} \left( \varinjlim_{\mathcal{I}_j} D \lambda_j, T \right) \cong \mathcal{C} \left( \varinjlim_\mathcal{J} \varinjlim_{\mathcal{I}_j} D \lambda_j, T \right)$$ as desired.

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  • $\begingroup$ I think the main point of contension would be with regards to the first isomorphism you wrote in the second line. Your answer would probably benefit from an additional step, where you use that hom-sets in limits of categories are the limits of hom-sets in the categories. $\endgroup$ Sep 3 '20 at 22:28
  • $\begingroup$ Right. Actually I think it might be more important to emphasise that I am working with strict limits of categories rather than pseudolimits. $\endgroup$
    – Zhen Lin
    Sep 3 '20 at 22:43
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    $\begingroup$ Note to self: the $\infty$-categorical result doesn't quite specialize to the strict result that Zhen Lin gives. Instead it should specialize to the analogous version with pseudocolimits. $\endgroup$
    – Tim Campion
    Sep 3 '20 at 23:33
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    $\begingroup$ I think this proof also goes through for $\infty$-categories, which suggests it should also work for pseudo-colimits in Cat - but then I'm confused about where the difference between colimits and pseudo-colimits in Cat shows up. Is it just that the canonical map between them is cofinal for 1-categories, so in $\mathcal{C}$ you just can't tell the difference (and both give the same isomorphisms of Hom-sets)? $\endgroup$ Sep 4 '20 at 9:31
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    $\begingroup$ I had the same doubt, which is why I edited to say my proof is for strict colimits. $[-, \mathcal{C}]$ will send pseudocolimits to pseudolimits but I don't know how (well, haven't tried) to calculate hom-sets of the pseudolimit. $\endgroup$
    – Zhen Lin
    Sep 4 '20 at 11:40

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