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?

enrichedin orientals here? $\endgroup$ – Manuel Bärenz Aug 8 '16 at 8:02