nLab S-category

Depending what $S$ stands for various things can be called $S$-category. E.g. if $S$ is a category then we talk about categories over $S$, for them see overcategory.

This entry is rather about another notion of $S$-category introduced in

• Tomasz Brzeziński, Notes on formal smoothness, in: Modules and Comodules (series Trends in Mathematics). T Brzeziński, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkhäuser, Basel, 2008, pp. 113-124 (doi, arXiv:0710.5527)

which is similar to the notion of $Q$-category of Rosenberg. It is called $S$-category as it is suitable context for a generalization of a separable functor.

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A little info is found in Toen: Homotopical and higher categorical structures in algebraic geometry. File Toen web unpubl hab.pdf. S-cats are closely related to Segal cats.

nLab page on S-category