Geometric realisation of simplicial sets can be roughly thought of like this:

  • In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would choose some model of the concrete $n$-simplex, but other categories are perfectly possible.
  • For the face and degeneracy maps, we choose gluing morphisms for the objects, satisfying the simplicial identities.
  • For a given simplicial set, we replace the sets by copies of the objects we've specified and glue them together with the gluing morphisms.

(All this can of course be described abstractly and beautifully with Kan extensions.)

I'm wondering whether there is a higher categorical version of this, possibly involving orientals. While every usual simplex $[n]$ is just a category, and its nerve is therefore a 1-skeleton, the $n$-th oriental $\mathcal{O}_n$ is an $n$-category. For example, $\mathcal{O}_2$ has the same objects and 1-morphisms as $[2]$, but it also has a nontrivial $2$-morphism representing the area of the triangle.

I'm looking for the following kind of construction:

  • In some kind of weak $n$-category $\mathcal{C}$, choose a $k$-morphism for the $k$-th oriental, compatible with face maps. More specifically: Choose an object $X_0$, a 1-morphism $X_1: X_0 \to X_0$, a 2-morphism $X_2: X_1 \circ X_1 \implies X_1$, a 3-morphism $X_3: X_2 \circ \left(X_2 \cdot 1_{X_1}\right) \Rrightarrow X_2 \circ \left(1_{X_1} \cdot X_2\right)$ and so on. All of the morphisms above $n$ are being mapped on coherence axioms.
  • A (suitable) simplicial set gives rise to an $m$-morphism in $\mathcal{C}$ by taking copies of $X_m$ and composing them along the morphisms from the lower simplices. (I'm thinking of simplicial sets arising from a triangulated framed $m$-dimensional cobordism, but you don't need to.)

Note that a simplicial set with dimension $n$ (the dimension of the highest nondegenerate simplex) will give rise to a $n$-morphism. The simplicial set serves as a kind of pasting diagram.

Does such a construction exist, in some good model of higher categories? What kind of tools do I need to define it? Do higher Kan extensions or something like that exist?

Bonus: Can I use this to define extended TQFTs?

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    $\begingroup$ I don't understand the question. What do you want this construction to output? An $\infty$-category? $\endgroup$ – Qiaochu Yuan Mar 29 '15 at 17:56
  • $\begingroup$ @QiaochuYuan, an $m$-morphism for an "$m$-dimensional" simplicial set. Geometric realisation produces a topological space or in general an object in $\mathcal{C}$ for a simplicial set, so geometric realisation of the data I specified should give data in the $n$-category for every (suitable) simplicial set. $\endgroup$ – Manuel Bärenz Mar 29 '15 at 20:50
  • $\begingroup$ I don't get the question too. If you want a geometric realization giving a top. space. Just pick the nerve of an oriental and then the top. realization. If you have a category enriched in orientals the same procedure applies. $\endgroup$ – user40276 Aug 8 '16 at 1:12
  • $\begingroup$ @user40376, what category is enriched in orientals here? $\endgroup$ – Manuel Bärenz Aug 8 '16 at 8:02
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    $\begingroup$ Sorry, I meant enriched in omega groupoids. By what I understand you want a Kan extension that picks an oriental and gives you an infinity category (although I don't know which model you want). If you accept simplicially enriched categories, you can just pick the nerve and then you have a simplicial object, now you can pick the realization associated to the coherent nerve. If you want an omega category, then you can pick the nerve and then the realization to omega categories. $\endgroup$ – user40276 Aug 8 '16 at 16:45

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