We can often have multiple duals for the same type of object. A simple example is the Pontryagin dual vs the Langlands dual of a group $G$. Is the use of the word "dual" here precise, i.e. every appearance of the word "dual" in mathematics, implies a specific categorical context in which we consider our initial (or primal) object, where the dual category gives the object we have called dual? Could you give some examples of these categories (e.g. start with the dual groups given above)?

  • $\begingroup$ There is duality in the sense of reversing arrows, and there is duality in the sense of linking two concepts via some isomorphism and making them dual. Categorically, the latter is sometimes expressible in terms of natural isomorphisms or equivalences of categories. But, they are different meanings of 'dual'. They might intersect occasionally - I don't know - but the former type of duality is a much more trivial one $\endgroup$
    – FShrike
    Nov 15, 2022 at 23:22
  • $\begingroup$ @FShrike Well that's exactly my question, can you give an example of a non-categorical dual, that is impossible to make sense of in the categorical sense? $\endgroup$
    – Derek
    Nov 16, 2022 at 0:51
  • $\begingroup$ The right dual of an object in a monoidal category $(\mathsf C,\otimes)$ is a left dual in the monoidal category $(\mathsf C,\otimes^{r})$ where $x\otimes^r y = y\otimes x$. This isn't a categorical dual - the base category is $\mathsf C$, not $\mathsf C^{\mathrm{op}}$. $\endgroup$ Nov 16, 2022 at 15:53
  • $\begingroup$ @Mariano So are you saying my question is equivalent to asking "When is a normal vector a normal subgroup"? $\endgroup$
    – Derek
    Nov 16, 2022 at 17:18


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