category with duals (list of them)
dualizable object (what they have)
Two important periodic tables are the table of -tuply monoidal -categories and the table of -categories. These can actually be combined into a single 3D table, which surprisingly also includes -tuply groupal -groupoids.
A -tuply monoidal -category is a pointed -category (which you may interpret as weakly or strictly as you like) such that: * any two parallel -morphisms are equivalent, for ; * any -morphism is an equivalence, for ; * any two parallel -morphisms are equivalent, for .
Keep in mind that one usually relabels the -morphisms as -morphisms, which explains the usage of and instead of and . As explained below, we may assume that , , , and (if convenient) .
To interpret this correctly for low values of , assume that all objects (-morphisms) in a given -category are parallel, which leads one to speak of the two -morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any -morphism is an equivalence for , so if , then the condition is satisfied for any smaller value of . Thus, we may assume that . Similarly, since there is a chosen object (the basepoint), any parallel -morphisms are equivalent for ,
The conditions that and that will overlap if , so we don't use such values of . In other words, any -tuply monoidal -category is also a -tuply monoidal -category for any .
If any two parallel -morphisms are equivalent, then any -morphism between equivalent -morphisms is an equivalence (being parallel to an equivalence for and automatically for ). Accordingly, any -tuply monoidal -category is automatically also a -tuply monoidal -category for any , and any -tuply monoidal -category for is also a -tuply monoidal -category. Thus, we don't need or .
According to the stabilisation hypothesis, every -tuply monoidal -category for may be reinterpreted as an -tuply monoidal -category. Unlike the other restrictions on values of , this one is not trivial.
A -tuply monoidal -category is simply a pointed -category. The restriction that becomes that . This is why -categories use rather than the restriction on given before.
A -tuply monoidal -category is a -tuply monoidal -groupoid. A -tuply monoidal -category is a -tuply groupal -groupoid. This is why groupal categories? don't come up much; the progression from monoidal categories to monoidal groupoids? to groupal groupoids? is a straight line up one column of the periodic table of monoidal? -categories. (But if we moved to a 4D table that required all -morphisms to be equivalences for sufficiently low values of , then groupal categories would appear there.)
A -tuply monoidal -category is simply a -tuply monoidal -category. A -tuply monoidal -category is a -tuply monoidal -poset. Note that a -tuply monoidal -category and a -tuply monoidal -poset are the same thing.
A stably monoidal -category, or symmetric monoidal -category, is an -tuply monoidal -category. Although the general definition above won't give it, there is a notion of stably monoidal -category, basically an -category that can be made -tuply monoidal for any value of in a consistent way.