It is well known to all that we define the differential of the map $f:R^n\to R^m$ at $x$ to be the linear map $Df(x)$, which satisfies that $$ f(x+h)-f(x)= Df(x)h+o(h) $$ Now I think linear maps as morphisms from $R^n\to R^m$, in the category of topological groups. So I think it OK to define the differential of maps between topological groups, i.e.
Definition:
Suppose $G,H$ are topological groups, $f$ be a map from $G$ to $H$, we say $f$ is differentiable(maybe plus the phrase 'on the right') at $g\in G$ if there is a map $Df(g)\in Hom(G,H)$ satisfies $$ f(gr)=f(g)[Df(g) r] o(r) $$ where $o(r)$ still means $o(r)$ passes to $e_{H}$ when $r$ passes to $e_{G}$, in their corresponding topology.
Although this definition is at least formally natural, I can not judge whether it is meaningful. Indeed, I came up with clues that in general case this definition leads to too much limits.
Firstly, It is natural to consider things(I call them group-folds?) like manifolds, if we care about things locally like Lie groups, their underlying topology are compatible with their manifold structure, so the lie-group-folds are still manifolds, nothing new happens.
As to the maps, take torus as an example, their automorphisms are discrete, as a result, the differentiable maps played between themselves have to be group homomorphisms. This is not interesting too.
Moreover, I do not know whether there is natural topology on Homomorphisms, this is also a trouble preventing us from defining $C1$ maps.
But I also feel that maps like this behave somewhat properly to illustrate the symmetry of topological groups.
So I wonder under which case this definition may be interesting? Or it is just a boring generalization?