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Let ${\bf Set}$ be the category of sets with cartesian product denoted $\times$, and let Sets be the corresponding multi-category of sets, where $$Hom_{\bf Sets}(A_1,\ldots,A_n;B)=Hom_{\bf Set}(A_1\times\cdots\times A_n,B).$$

Suppose that $M$ and $M'$ are multi-categories, and that $F\colon M\to M'$ is a multi-functor. Let $P\colon M\to{\bf Sets}$ be a set-valued multi-functor.

Q: Is there in general a left Kan extension $Lan_F(P)\colon M'\to{\bf Sets}$?

Q: Is there some kind of point-wise formula for it?

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  • $\begingroup$ For definiteness, are your multi-categories symmetric? $\endgroup$ Mar 31 '14 at 20:36
  • $\begingroup$ @RicardoAndrade: I'd like to know the general case if possible, or just the symmetric case if someone can help with that. $\endgroup$ Mar 31 '14 at 22:09
  • $\begingroup$ This question was cross-posted on Math.StackExchange. Dear @David, please do not simultaneously post a question on both sites. This leads to unnecessary duplication of effort, and is frowned upon by both communities. Please post in only one site and wait for an answer for at least a few days before reposting elsewhere. Thank you. $\endgroup$ Mar 31 '14 at 23:39
  • $\begingroup$ Ok, I deleted it there. Is there a good way to decide which of the two sites a given question (say, this one) is appropriate for? $\endgroup$ Mar 31 '14 at 23:55
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    $\begingroup$ I think if you have "multi-categorical" in the title it's safe to assume it's for MO. $\endgroup$ Apr 1 '14 at 18:28
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I only have an answer when $M'$ is the terminal multicategory. This can be helpful however for the general solution too. See the considerations below.

The left Kan extension when $M'$ is the terminal multicategory $1$ (this has one object, and one multimorphism for each $n$) can be constructed as follows.

A functor $1 \to \mathbf{Sets}$ is just a monoid $A$ in $\mathbf{Sets}$. The 2-cell which will exhibit it as the left extension of $P : M \to \mathbf{Sets}$ along the unique $M \to 1$ consists of maps $$d_X : P(X) \to A,$$ for each object $X$ of $M$, such that for any multimorphism $f\colon (X_1,...X_n) \to X$ $$P(X_1)\times...\times P(X_n) \xrightarrow{P(f)} P(X) \xrightarrow{d_x} A$$ $$=$$ $$P(X_1)\times...\times P(X_n) \xrightarrow{d_{x_1}\times...\times d_{x_n}} A\times... \times A \xrightarrow{\text{mult.}} A.$$

Given $P$ one constructs $A$ as follows. Take a coproduct $\amalg_X P(X)$. Take the free monoid on this $\mathcal{Fr}(\amalg_X P(X))$, and factor this out by the relations $$P(f)(x_1, x_2,\ldots, x_n)\sim x_1x_2...x_n.$$

The $d_X$ will be defined as the canonical maps.

For the general case, one could follow Kelly, who defines point-wise Kan extensions withing a general bicategory with comma objects. (I'll insert a reference for this.) Using this for the 2-category of categories, the usual formula for $Lan_F(P)(X)$ is obtained by finding the left Kan extension of the composite $$P\downarrow X \to M \to Sets$$

along the functor $P\downarrow X \to 1$ to the terminal category. This happens to be just the colimit. So for categories, finding point-wise formulas are reduced to finding Kan extension along the functors to the terminal category. This however relies one the fact that an object $X$ of a category can be given by a functor $1 \to M'$.

I believe that the 2-category of multicategories has comma objects. Unfortunately, the multifunctors $1 \to M'$ are monoids in $M'$ rather than objects.

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  • $\begingroup$ This paper arxiv.org/ftp/math/papers/0303/0303175.pdf talks about the concept of point wise Kan extension within a bicategoy (page 16). Can't find the original definition. It was by Kelly I think. $\endgroup$ Apr 1 '14 at 22:21
  • $\begingroup$ I assume you mean take the free monoid $\mathcal{Fr}(\amalg_X P(X))$? $\endgroup$ Apr 3 '14 at 23:50
  • $\begingroup$ Yes, corrected. $\endgroup$ Apr 3 '14 at 23:59

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