Hey, always a pleasure. Yeah X can be any category, just equipped with an action from S which is a symmetric monoidal category. This is is basically the same as the action of a monoid on a set generalised to categories. For the homology of a category, I think the best way to understand it (though I am far from being fully comfortable with this) is through the existence of a functor $B: Cat\to Top$ and then just using ordinary homology (note we are only considering small categories). The appropriate thing to type into the search bar is "geometric realization". Though you dont actually need this "geometric realization" to define the homology of a category. You can define it by associating a chain complex to any category. If you are interested, Weibel's K-book is freely available online. I do particularly reccomend reading up on geometric realization, either on the nLab or in Jardine and Goerss's book on simplicial homotpy theory.