If a group acts on a category (in some sense), sometimes the phrases "weak action" and "strong action" come up. I don't know what these mean though. Could someone provide an appropriate definition? (Or is it really context dependent?)
Note if someone finds a link to the answer in the top ten results in a search engine... please tell me which phrase you typed... I couldn't seem to get anything.
On
The notion of weak vs strong categorical actions in the context you are interested in (i.e. Ben Webster's paper on knot homologies) is defined in this paper. A weak categorification of an irreducible $\mathfrak{sl}_2$ representation consists of the data of categories $V_\lambda$, $\lambda\in\mathbb{Z}$, together with functors $$E:V_{\lambda}\to V_{\lambda+2},\;\;\;F:V_{\lambda}\to V_{\lambda-2}$$ which satisfy isomorphisms $$EF\cong FE\oplus Id^{\oplus\lambda}$$ if $\lambda\geq 0$ and $$FE\cong EF\oplus Id^{\oplus(-\lambda)}$$ if $\lambda<0$.
A weak categorification becomes a (strong) categorification if you additionally have the data of natural transformations $$X:E\to E\;\;\;\mbox{and}\;\;\; T:E^2\to E^2$$ which satisfy Hecke algebra type relations $TX-XT=Id$ (though one can certainly generalize this using formal group laws). These allow you to obtain a categorification of the divided powers $E^{(n)}$ and $F^{(n)}$. All this can be generalized to categorifications of arbitrary Lie algebras (see here and here and here).
I recommend reading the following paper by Lauda before trying to tackle Ben's paper.
My impression, and I'm not sure about this at all, is that there are two things people mean by "weak action," one of which is bad. One is weak-as-opposed-to-strict, meaning that for every element of the group $g \in G$ you have a functor $F(g): C \to C$, and instead of requiring that $F(gh)$ is literally equal to $F(g) F(h)$ you require that there is a natural isomorphism between them, which is further subject to an associativity condition (see e.g. this blog post). This is the correct notion of a group action on a category.
The bad definition involves not requiring the associativity condition, as Tobias says in the comments. I think the only reason this definition appears in the literature is that it's the best people can do in some situations.
One reason the bad definition is bad is that typically in category theory the "weak" thing is the thing you actually want, but that's not the case here.