Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a smooth and proper dg- (or $A\_\infty$-) category which admits a faithful action 
$$ MCG(\Sigma) \to Auteq(\mathcal{C})$$
by the mapping class group by auto-equivalences. 


1. What good properties about $\mathcal{C}$ does that imply? (or
: why can representation theory tell me about the category or its auto-equivalences)
2. Are there any known cases where such an action arises which do not come from symplectic topology/mirror symmetry? The cases I know of so far come from [Seidel-Thomas](https://arxiv.org/abs/math/0001043) spherical twists or Ivan's [Floer cohomology and pencils of quadrics](https://arxiv.org/abs/1006.1099) paper. I assume there must be some examples from geometric representation theory or categorification and alike...
3. What happens if I replace the mapping class group with the Torelli group $\mathcal{I}\_g$ (or more generally, level $k$ in the Johnson filtration)?



Categorical mapping class group action
[This is an elaboration of parts of Mark Penney's answer]


A natural source of categorical actions of the mapping class group is the category assigned by any 4d TFT to a surface. Such categories are often written as Fukaya categories - eg Donaldson theory and Seiberg-Witten theory will attach Fukaya categories of moduli of G-bundles or symmetric products of the curve - but there are other examples, coming e.g. from factorization homology constructions as Mark wrote.


Here's what seems like the most natural example: the MCG acts on moduli spaces of G local systems ("character varieties" - or really character derived stacks) for G a complex reductive group say. So it acts on any category of sheaves on these moduli spaces. One natural answer is to look at quasicoherent sheaves - i.e. to any $\Sigma$ we look at $QC(Loc\_G\Sigma)$. This is neither smooth nor proper as dg category, but it's faithful and interesting.. It's also the output of factorization homology, when the input is the braided (in fact symmetric) monoidal category $Rep(G)$ of representations of $G$. A more interesting answer coming from [Betti version of] geometric Langlands is to replace $QC$ by $IndCoh$ (possibly with fixed singular support). This is supposed to indeed be the value of an interesting 4d TFT on $\Sigma$ - namely the Kapustin-Witten B-twist of $\mathcal N=4$ super Yang-Mills. [An even more interesting (but conjectural) answer is the A-side of geometric Langlands.]


The character varieties have a symplectic form preserved by $MCG(\Sigma)$, so one can try to quantize them and get more interesting categorical actions. That's what is done e.g. in my work with Brochier and Jordan that Mark referenced, using factorization homology. 


