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1. • CommentRowNumber1.
   • CommentAuthorSridharRamesh
   • CommentTimeFeb 7th 2012
   • (edited Feb 7th 2012)
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   Author: SridharRamesh
   Format: MarkdownItexI grok as of yet very little about higher categorical limits; undeterred by that fact, I have a question regarding them. Suppose $x, y : 1 \rightarrow C$ are parallel global sections in the $\infty$-category of $\infty$-categories. Then there ought to be another object $Hom_C(x, y)$ in the $\infty$-category of $\infty$-categories representing, well, exactly what the name suggests: the $\infty$-category whose 0-cells are the 1-cells in $C$ between $x$ and $y$, whose 1-cells between those are the suitable 2-cells in $C$, and so on. It seems to me intuitive that $Hom_C(x, y)$ can be obtained abstractly in some fashion by limits from $x$ and $y$; I am curious as to whether this is true, and if so, what the appropriate kind of limit is. For example, if instead of using the $\infty$-category of $\infty$-categories, we were discussing only the category of sets, then $Hom_C(x, y)$ could be obtained as the equalizer of $x$ and $y$. If we were discussing the 2-category of categories, then $Hom_C(x, y)$ could be obtained, I believe, as the inserter of $x$ and $y$. Most generally, I imagine, the Hom-functor on $C$ could be represented as a two-sided fibration, which is a certain sort of morphism into $C \times C$ (I'm guessing its domain will be the power of $C$ by the walking arrow); this could then be pulled back in some sense along $x \times y$ to produce $Hom_C(x, y)$. (But everything in the last sentence would have to be $\infty$-fied.) Are my thoughts here reasonable? Can $Hom_C(x, y)$ be obtained from $x$ and $y$ by limits, and if so, how?

   I grok as of yet very little about higher categorical limits; undeterred by that fact, I have a question regarding them.

   Suppose $x, y : 1 \rightarrow C$ are parallel global sections in the $\infty$-category of $\infty$-categories. Then there ought to be another object $Hom_C(x, y)$ in the $\infty$-category of $\infty$-categories representing, well, exactly what the name suggests: the $\infty$-category whose 0-cells are the 1-cells in $C$ between $x$ and $y$, whose 1-cells between those are the suitable 2-cells in $C$, and so on.

   It seems to me intuitive that $Hom_C(x, y)$ can be obtained abstractly in some fashion by limits from $x$ and $y$; I am curious as to whether this is true, and if so, what the appropriate kind of limit is.

   For example, if instead of using the $\infty$-category of $\infty$-categories, we were discussing only the category of sets, then $Hom_C(x, y)$ could be obtained as the equalizer of $x$ and $y$. If we were discussing the 2-category of categories, then $Hom_C(x, y)$ could be obtained, I believe, as the inserter of $x$ and $y$. Most generally, I imagine, the Hom-functor on $C$ could be represented as a two-sided fibration, which is a certain sort of morphism into $C \times C$ (I’m guessing its domain will be the power of $C$ by the walking arrow); this could then be pulled back in some sense along $x \times y$ to produce $Hom_C(x, y)$. (But everything in the last sentence would have to be $\infty$-fied.)

   Are my thoughts here reasonable? Can $Hom_C(x, y)$ be obtained from $x$ and $y$ by limits, and if so, how?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 7th 2012
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   Author: Urs
   Format: MarkdownItexThere is, as far as I am aware, essentially no theory of $(\infty,2)$-categorical limits, which you would need here. Of course one could argue that any sensible definition of those should make the evident sentence with "$\infty$-inserter" true, I suppose. You could look at the special case where $C$ happens to be an $\infty$-groupoid. Then of course everything is well understood.

   There is, as far as I am aware, essentially no theory of $(\infty,2)$-categorical limits, which you would need here. Of course one could argue that any sensible definition of those should make the evident sentence with “$\infty$-inserter” true, I suppose.

   You could look at the special case where $C$ happens to be an $\infty$-groupoid. Then of course everything is well understood.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 7th 2012
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   Author: Mike Shulman
   Format: MarkdownItexSurprisingly, the evident sentence is already false when $C$ is a (2,2)-category. The "2-inserter" of $x,y\colon 1\to C$ in the 3-category $2Cat$ is the category whose objects are morphisms $x\to y$ but whose morphisms are the *isomorphisms* between these in $C$. As far as I can tell, this is an inevitable consequence of the fact that the 2-cells in $2Cat$ are pseudo (rather than lax or oplax) natural transformations. I recorded some (rather melodramatic) thoughts about this problem [here](http://ncatlab.org/michaelshulman/show/n-topos+for+large+n). (That page talks about [[comma objects]], which are what you need to use instead of inserters when the domain of $x$ and $y$ is not $1$ — they can always be constructed as a pullback of the cotensor-with-2 two-sided fibration, as you suggest.)

   Surprisingly, the evident sentence is already false when $C$ is a (2,2)-category. The “2-inserter” of $x,y\colon 1\to C$ in the 3-category $2Cat$ is the category whose objects are morphisms $x\to y$ but whose morphisms are the isomorphisms between these in $C$. As far as I can tell, this is an inevitable consequence of the fact that the 2-cells in $2Cat$ are pseudo (rather than lax or oplax) natural transformations. I recorded some (rather melodramatic) thoughts about this problem here. (That page talks about comma objects, which are what you need to use instead of inserters when the domain of $x$ and $y$ is not $1$ — they can always be constructed as a pullback of the cotensor-with-2 two-sided fibration, as you suggest.)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 7th 2012
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   Author: Urs
   Format: MarkdownItexRight. "The evident statement involving the right terms.", then. :-)

   Right. “The evident statement involving the right terms.”, then. :-)

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 7th 2012
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   Author: Mike Shulman
   Format: MarkdownItexAre you referring to the statement using lax transformations as "evident"? I don't think it's particularly evident; first of all we'd need to define the right notion of "lax transformation" for $(\infty,\infty)$-categories, then come up with some way to talk about "limits" of such things, which is already hard enough for (2,2)-categories.

   Are you referring to the statement using lax transformations as “evident”? I don’t think it’s particularly evident; first of all we’d need to define the right notion of “lax transformation” for $(\infty,\infty)$-categories, then come up with some way to talk about “limits” of such things, which is already hard enough for (2,2)-categories.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeFeb 7th 2012
   • (edited Feb 7th 2012)
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   Author: Urs
   Format: MarkdownItexAll I am saying is the tautology that with the right notions, there will be $(\infty,2)$-category in complete analogy to 2-category theory, just as there is now $(\infty,1)$-category in complete analogy to 1-category theory.

   All I am saying is the tautology that with the right notions, there will be $(\infty,2)$-category in complete analogy to 2-category theory, just as there is now $(\infty,1)$-category in complete analogy to 1-category theory.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 8th 2012
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   Author: Mike Shulman
   Format: MarkdownItexWell, firstly that's not a tautology; it's just a reasonable expectation and hope. Also, not everything in $(\infty,1)$-category theory is a "complete analogy" to 1-category theory; some things behave a bit differently. Finally, as I said, the "evident statement" is already false for 2-categories, so it will presumably also be false for $(\infty,2)$-categories.

   Well, firstly that’s not a tautology; it’s just a reasonable expectation and hope. Also, not everything in $(\infty,1)$-category theory is a “complete analogy” to 1-category theory; some things behave a bit differently. Finally, as I said, the “evident statement” is already false for 2-categories, so it will presumably also be false for $(\infty,2)$-categories.

8. • CommentRowNumber8.
   • CommentAuthorSridharRamesh
   • CommentTimeFeb 9th 2012
   • (edited Feb 9th 2012)
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   Author: SridharRamesh
   Format: MarkdownItexDamn, that's a shame. In the years since writing your original melodramatic thoughts, have you had any further ideas on how to deal with the issue, Mike? Do you still think we need "lax n-categories", or have you had a change of heart?

   Damn, that’s a shame. In the years since writing your original melodramatic thoughts, have you had any further ideas on how to deal with the issue, Mike? Do you still think we need “lax n-categories”, or have you had a change of heart?

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 9th 2012
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   Author: Mike Shulman
   Format: MarkdownItexI still don't know what the right answer is, if there is one right answer. Recently I've been focused more on $(\infty,1)$-logic (homotopy type theory) than $n$-logic for $n\gt 2$. I have had some recent thoughts about a way to find $n$-logic inside of $(\infty,1)$-logic, which I might get around to posting about one day, but I don't think that will solve this problem.

   I still don’t know what the right answer is, if there is one right answer. Recently I’ve been focused more on $(\infty,1)$-logic (homotopy type theory) than $n$-logic for $n\gt 2$. I have had some recent thoughts about a way to find $n$-logic inside of $(\infty,1)$-logic, which I might get around to posting about one day, but I don’t think that will solve this problem.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 9th 2012
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    Author: Urs
    Format: MarkdownItexMike, I can't understand why you make a simple evident remark into such a problem. But let's leave it at that.

    Mike,

    I can’t understand why you make a simple evident remark into such a problem. But let’s leave it at that.

11. • CommentRowNumber11.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 11th 2012
    • (edited Feb 11th 2012)
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    Author: Harry Gindi
    Format: TextThe (ho)limit idea works if you use one of the model structures that I'm working on, or using (a slight modification of) Rezk's model structure on Θ_∞-spaces. There's also some relevant stuff in some of the recent work by Batanin, Cisinski, and Weber on multitensors.

    The (ho)limit idea works if you use one of the model structures that I'm working on, or using (a slight modification of) Rezk's model structure on Θ_∞-spaces.
    
    There's also some relevant stuff in some of the recent work by Batanin, Cisinski, and Weber on multitensors.
12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 11th 2012
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    Author: Mike Shulman
    Format: MarkdownItexHarry, can you explain? How does that work around the problem of only seeing the invertible higher cells?

    Harry, can you explain? How does that work around the problem of only seeing the invertible higher cells?

13. • CommentRowNumber13.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 12th 2012
    • (edited Feb 12th 2012)
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    Author: Harry Gindi
    Format: TextSince the 2-point suspension is a parametric left adjoint, you have a right adjoint, and these are quillen functors. Then you can compute the hom-object by applying this right adjoint to a bipointed object. Then Rezk shows that this can be computed as a limit in his paper (Specifically, he works with the case of ΘA=Δ\wr A, where A is some small category), which gives a simplicial presheaf on A. However, in the case of Θ_∞, this actually gives us back a simplicial presheaf on Θ_∞. When applied in the strict case, this recovers the strict ω-category that Sridhar asked for. I also do this in my model, where things work out because things can be resolved in a nice way (as opposed to making them work using the additional simplicial structure like Rezk). Since these are homotopy limits and colimits in the associated model categories, we expect that these are computing the appropriate (pseudo)-limits and (pseudo)-colimits in the weak ω-category of weak ω-categories. However, neither I nor Rezk appear to have figured out how to construct the "ω-category of ω-categories" to prove the final assertion, but that's one of the directions that my work will head into. In fact, this hom-object can always be constructed for any cellular set or cellular space. However, the fibrant objects allow us to actually compute the Hom-object through a sequence of m+1 vertices, because the "m-spine suspension" is a parametric right adjoint while the "m-simplex suspension" is not. But the Segal lifting conditions imply that the m-spine suspension is weakly equivalent to the m-simplex suspension, so its right adjoint is a homotopy right-adjoint for the m-simplex suspension, which we call Hom[m](x_0,...,x_m). Oh, sorry, something I left out that actually doesn't change the answer: I just explained the case where the x_i are 0-cells. In general, If the x_i are parallel n-cells, you just take the nth power of the suspension functor, which is still a PLA and do everything else the same. If I remember correctly, the problem where you only get the invertible higher cells comes from taking a limit over the internal function object (in cellular sets/spaces). Rezk's paper shows the right way to do it. Edit: Yes, Rezk does it in a way that makes essential use of the simplicial structure. My way is complicated, but it involves the fact that you can build a canonical cosimplicial resolution for the suspension functor and take a bunch of adjoints and get the right thing. This extra effort is only necessary when you're trying to exhibit it as a homotopy limit. Then the last step using my version is to pass back to an ordinary presheaf along the Quillen equivalence between the simplicial completion and the original homotopy structure.

    Since the 2-point suspension is a parametric left adjoint, you have a right adjoint, and these are quillen functors. Then you can compute the hom-object by applying this right adjoint to a bipointed object. Then Rezk shows that this can be computed as a limit in his paper (Specifically, he works with the case of ΘA=Δ\wr A, where A is some small category), which gives a simplicial presheaf on A. However, in the case of Θ_∞, this actually gives us back a simplicial presheaf on Θ_∞. When applied in the strict case, this recovers the strict ω-category that Sridhar asked for.
    
    I also do this in my model, where things work out because things can be resolved in a nice way (as opposed to making them work using the additional simplicial structure like Rezk). Since these are homotopy limits and colimits in the associated model categories, we expect that these are computing the appropriate (pseudo)-limits and (pseudo)-colimits in the weak ω-category of weak ω-categories. However, neither I nor Rezk appear to have figured out how to construct the "ω-category of ω-categories" to prove the final assertion, but that's one of the directions that my work will head into.
    
    In fact, this hom-object can always be constructed for any cellular set or cellular space. However, the fibrant objects allow us to actually compute the Hom-object through a sequence of m+1 vertices, because the "m-spine suspension" is a parametric right adjoint while the "m-simplex suspension" is not. But the Segal lifting conditions imply that the m-spine suspension is weakly equivalent to the m-simplex suspension, so its right adjoint is a homotopy right-adjoint for the m-simplex suspension, which we call Hom[m](x_0,...,x_m).
    
    Oh, sorry, something I left out that actually doesn't change the answer:
    
    I just explained the case where the x_i are 0-cells. In general, If the x_i are parallel n-cells, you just take the nth power of the suspension functor, which is still a PLA and do everything else the same.
    
    If I remember correctly, the problem where you only get the invertible higher cells comes from taking a limit over the internal function object (in cellular sets/spaces). Rezk's paper shows the right way to do it.
    
    Edit: Yes, Rezk does it in a way that makes essential use of the simplicial structure. My way is complicated, but it involves the fact that you can build a canonical cosimplicial resolution for the suspension functor and take a bunch of adjoints and get the right thing. This extra effort is only necessary when you're trying to exhibit it as a homotopy limit.
    
    Then the last step using my version is to pass back to an ordinary presheaf along the Quillen equivalence between the simplicial completion and the original homotopy structure.
14. • CommentRowNumber14.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 12th 2012
    • (edited Feb 12th 2012)
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    Author: Harry Gindi
    Format: MarkdownItexAnyway, the ultimate most satisfying answer will arise by using the lax Gray tensor product (and possibly the lax join). However, to show that my homotopy structure (and by extension, Rezk's, since it is the simplicial completion of mine) is compatible with the lax tensor product, there is a large amount of important work to be done. I am currently collaborating with Tim Porter on this specific piece of the puzzle. Basically, it amounts to showing that Berger's inner horn inclusions for Batanin cells (i.e. objects of Θ_∞) generate the same "weakly saturated class of maps" in the category of cellular sets as the lower righthand corner maps $\Lambda^1[2] \otimes \Theta[t] \bigcup \Delta[2] \otimes \partial\Theta[t] \hookrightarrow \Delta[2]\otimes \Theta[t]$ (where [t] is a Batanin cell). I have been working on this conjecture myself for the past few months, with very little to show for it, so I decided that it might be prudent to call in the help of a shuffle specialist.

    Anyway, the ultimate most satisfying answer will arise by using the lax Gray tensor product (and possibly the lax join). However, to show that my homotopy structure (and by extension, Rezk’s, since it is the simplicial completion of mine) is compatible with the lax tensor product, there is a large amount of important work to be done. I am currently collaborating with Tim Porter on this specific piece of the puzzle.

    Basically, it amounts to showing that Berger’s inner horn inclusions for Batanin cells (i.e. objects of Θ_∞) generate the same “weakly saturated class of maps” in the category of cellular sets as the lower righthand corner maps $\Lambda^1[2] \otimes \Theta[t] \bigcup \Delta[2] \otimes \partial\Theta[t] \hookrightarrow \Delta[2]\otimes \Theta[t]$ (where [t] is a Batanin cell). I have been working on this conjecture myself for the past few months, with very little to show for it, so I decided that it might be prudent to call in the help of a shuffle specialist.

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 12th 2012
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    Author: Mike Shulman
    Format: MarkdownItexHarry, you've completely lost me. Can you throw away the model categories and explain explicitly what happens in the case of the 3-category $2Cat$?

    Harry, you’ve completely lost me. Can you throw away the model categories and explain explicitly what happens in the case of the 3-category $2Cat$?

16. • CommentRowNumber16.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 12th 2012
    • (edited Feb 12th 2012)
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    Author: Harry Gindi
    Format: MarkdownItexDo you want me to express the Hom as a limit or do you want me to express the Hom as an adjoint? Also if you're viewing 2-cat as a 3-category, the hom between two parallel cells can be either a 2-category, a category, or a set. The whole point of the ω-case is that none of this matters in the least. They can all be treated in a uniform way. I'll first express what's going on using an adjuction: The suspension functor Δ_1[-] sending an ω-category X to the ω-category with two 0-cells and such that the Hom object between those two objects is X in one direction and empty in the other. This functor extends canonically to cellular sets, where it is a parametric left adjoint. That is, the induced functor Psh(Θ)->Δ_1[ø]/Psh(Θ) admits a right adjoint. However, Δ_1[ø] is just a disjoint union of two vertices. So we have a right adjoint Γ sending bipointed cellular sets to cellular sets, and this is the functor that assigns the hom-object between two vertices. However, given a bipointed cellular set (X,x_0,x_1), we may form the cellular set Γ(X,x_0,x_1) as follows: Γ(X,x_0,x_1)[t]= lim(e -> Hom(Δ_1[ø],X) <- Hom(Δ_1[t],X)). This ends up being exactly what we wanted. The homotopy properties are used to show that the adjunction between Δ_1 and Γ is a quillen adjunction.

    Do you want me to express the Hom as a limit or do you want me to express the Hom as an adjoint?

    Also if you’re viewing 2-cat as a 3-category, the hom between two parallel cells can be either a 2-category, a category, or a set. The whole point of the ω-case is that none of this matters in the least. They can all be treated in a uniform way.

    I’ll first express what’s going on using an adjuction: The suspension functor Δ_1[-] sending an ω-category X to the ω-category with two 0-cells and such that the Hom object between those two objects is X in one direction and empty in the other.

    This functor extends canonically to cellular sets, where it is a parametric left adjoint. That is, the induced functor Psh(Θ)->Δ_1[ø]/Psh(Θ) admits a right adjoint. However, Δ_1[ø] is just a disjoint union of two vertices. So we have a right adjoint Γ sending bipointed cellular sets to cellular sets, and this is the functor that assigns the hom-object between two vertices.

    However, given a bipointed cellular set (X,x_0,x_1), we may form the cellular set Γ(X,x_0,x_1) as follows:

    Γ(X,x_0,x_1)[t]= lim(e -> Hom(Δ_1[ø],X) <- Hom(Δ_1[t],X)).

    This ends up being exactly what we wanted.

    The homotopy properties are used to show that the adjunction between Δ_1 and Γ is a quillen adjunction.

17. • CommentRowNumber17.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 12th 2012
    • (edited Feb 12th 2012)
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    Author: Harry Gindi
    Format: MarkdownItexOhh.. I misread.. Can you do it with a "lax 2-inserter"? Also, it seems like the question of what the ambient morphisms are is not relevant, since we can define the lax and oplax limit and colimit of a higher diagram entirely with the machinery of the lax join and ordinary join. Let me give an example: If we want to define a laxly initial vertex x of a cellular set C, we require that the canonical map from the lax slice over C/_{lax}x to C be a trivial fibration of cellular sets. If we want a vertex to be pseudo-initial, we replace the lax join wth an appropriate version of the cellular join. Taking these together, we can define pseudo and lax universal properties without ever having to worry about the ambient morphisms. By the way, a good choice for the "pseudo cone" might be Cisinski's decalage for Batanin cells.

    Ohh.. I misread..

    Can you do it with a “lax 2-inserter”?

    Also, it seems like the question of what the ambient morphisms are is not relevant, since we can define the lax and oplax limit and colimit of a higher diagram entirely with the machinery of the lax join and ordinary join.

    Let me give an example:

    If we want to define a laxly initial vertex x of a cellular set C, we require that the canonical map from the lax slice over C/_{lax}x to C be a trivial fibration of cellular sets. If we want a vertex to be pseudo-initial, we replace the lax join wth an appropriate version of the cellular join.

    Taking these together, we can define pseudo and lax universal properties without ever having to worry about the ambient morphisms.

    By the way, a good choice for the “pseudo cone” might be Cisinski’s decalage for Batanin cells.

18. • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 12th 2012
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    Author: Mike Shulman
    Format: MarkdownItexNo, a lax 2-inserter in the 3-category of 2-categories is not sufficient; as long as the 2-cells in $2Cat$ are pseudonatural rather than lax natural (and there is no real way to make a 3-category of the ordinary sort whose 2-cells are lax), the inserter will only find the invertible 2-cells. A lax limit makes the *cone* lax, but doesn't change the "internal" laxness of the cells in the ambient 3-category. Certainly, hom-$n$-categories can be extracted as a right adjoint to the "directed suspension" $n Cat \to (n+1)Cat$. But the question was about constructing them as limits inside of $(n+1) Cat$.

    No, a lax 2-inserter in the 3-category of 2-categories is not sufficient; as long as the 2-cells in $2Cat$ are pseudonatural rather than lax natural (and there is no real way to make a 3-category of the ordinary sort whose 2-cells are lax), the inserter will only find the invertible 2-cells. A lax limit makes the cone lax, but doesn’t change the “internal” laxness of the cells in the ambient 3-category.

    Certainly, hom-$n$-categories can be extracted as a right adjoint to the “directed suspension” $n Cat \to (n+1)Cat$. But the question was about constructing them as limits inside of $(n+1) Cat$.