I started writing [nLab:Theta space](http://ncatlab.org/nlab/show/Theta+space). Not done yet, but while I am working on it:


is there a good proposal for what the "$(n+1,r+1)$-$\Theta$-space of all $(n,r)$-$\Theta$-spaces" would be?



(n+1,r+1)-Theta space of (n,r)-Theta spaces?
Let me assume $n=\infty$, to make things easier to write, so "$(\infty,r)$-$\Theta$-space" equals "$r$-$\Theta$-space".


The totality of $r$-$\Theta$-spaces forms a (large) category enriched over $r$-$\Theta$-spaces, which I'll call $C$. Given this, you can form a presheaf of spaces $X$ on the category $\Theta\_{r+1}$, where


$$X[0] = \text{class of objects of $C\_r$},$$


and 


$$X(\[m\](\theta\_1,\dots,\theta\_m)) = \coprod\_{a\_0,\dots,a\_m} C(a\_0,a\_1)(\theta\_1)\times \cdots \times C(a\_{m-1},a\_m)(\theta\_m).$$


Here "$\[m\](\theta\_1,\dots,\theta\_m)$" represents a typical object in $\Theta\_{r+1}$ (so each $\theta\_i\in \Theta\_r$). The coproduct is over tuples of objects of $C$.
The structure maps in the presheaf use the fact that $C$ is a category object. (It's like the way you get a Segal category from a category enriched over spaces.)


The gadget $X$ is *almost* an $(r+1)$-$\Theta$-space. It satisfies all the "Segal" conditions, and also all the completeness conditions *except* for the one in bottom dimension. You get an honest $(r+1)$-$\Theta$-space $X'$ from $X$ by applying a suitable localization. 


The gadget $X'$ should be the thing you want. (None of the proofs involved here have been written up, or at least not by me, though we're working on it.)


