There is a general process given a site (C,J) for a topos T and an internal site (D,K) in T to build an external site for the category of internal sheaves. the category is the Grothendieck construction for D seen as a functor C^op \to cat, and the topology is generated from K and J. I'm sure this is in the literature, very likely in the elephant, but I don't remember exactly where, but I'd start looking around where fibration of sites are discussed.

@HenrikRüping That way I think about it is that because of Stone duality, $C(\text{LimProj} A_i, 2) = \text{LimInd} C(A_i,2)$ (where on the left we have a projective limite of stone space and the right an inductive limit of boolean algebras) so if $X$ is pro-finite, them $C(X,2)$ is "Ind-finite" (so, discrete). ( of course, we need to check that the above is indeed a homomorphism and not just a bijection, which might be the part you find unintuitive).

Sorry, I misread, I thought you were assuming $X$ was compact.

Related comment: Badzioch has shown in arxiv.org/abs/math/0110101 that weak model of a (1-categorical) Lawvere theory in Spaces are all equivalent to strict ones. In short, you don't want to take a Lawvere theory which is a 1-category: maps in the theory will always corresponds to maps between free models and free rigs are not set truncated.

@JohnBaez higher categorical Lawvere theory works the same as 1-categorical ones: if you define a higher lawvere theory as say an $(\infty,1)$-category $C$ with finite products ans whose objects are all the finite power $O^n$ for some fixed $O$, a Model of $C$ to be a (weakly) product preserving functor $C \to Spaces$, then the initial model of $C$ is the representable at $O^0$ (and more generally the free model on n-generator is the representable at $O^n$). In particular if $C$ happen to be a $1$-category, its free models are set-truncated.

The correct conjecture probably involves a "2-"Lawvere theory whose morphisms are polynomial functor (and 2-cells natural isomorphism). But maybe this is what you mean when you talk about rigs in Cat?

Unless I'm missing something, That conjecture is false. The initial weak model is just the initial rig, so N, but I assume the initial symmetric rig category is the category of finite sets

You'll find a general overview on theta spaces and lots of references here : ncatlab.org/nlab/show/Theta-space in short they are a generalisation of Segal spaces that provide a model for $(\infty,n)$-categories.

Segal spaces are another model for $(\infty,1)$-category, this time using simplicial spaces instead of simplicial sets ( so bi-simplicial sets). Not a model of $(\infty,n)$ or $(\infty,\infty)$. They are however easier to generalise to $(\infty,n)$ or $(\infty,\infty)$ than quasicategories (using either $\Theta$-spaces, or iterated Segal spaces).