one could say that category would be then (∞,∞)

No, I think the “$\infty$-” to be dropped is that which is equivalent to “homotopy-“. So in 50 years we’ll say “category” for $(\infty,1)$-category and “$n$-category” for $(\infty,n)$-category and “$\infty$-category” for “$(\infty,\infty)$-category”.

Apart from this, I am not sure what you would like me to change. If the context is clear, then we can drop the “$\infty$-“. But every $n$Lab entry is its own context, and so we can’t drop it in the title of the entry *∞-module*.

No, I am not that radical (one could say that category would be then $(\infty,\infty)$ if we take that every notion should be by deafult taken absolutely most generally). I mean CONTEXT should always be specified precisely. Once the context is known then no modifiers are needed. For example, one traditionally meant by a ball, certain kind of geometrical bodies in Euclidean space. Once one takes their equation one can have a version in metric space. Usually, if one just say a ball, one will hardly mean that this is in the generality of metric space. But in the context of metric spaces one does not need to say metric ball, just ball is enough. So never mentioning infinity and just saying category is misleading. But once we say we work with infinity categories then module is just module in this context. This is my opinion.

]]>One day als “category” will be the preferred term for “$(\infty,1)$-category”. But we are not quite at the point yet that it would be useful to enforce the implicit infinity-category theory convention globally on the $n$Lab. I’d think.

]]>Isn’t just “module” the preferred term ?

]]>I noticed that in

the *∞-module* was kind of missing (we had module over an algebra over an (∞,1)-operad). So I created something stubby.