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Let $C$, $D$ be categories and $F:C\to D$, $G: D\to C$ be functors.

Consider the following properties :

  • $p_1$ : "$F$ is a left adjoint of $G$"
  • $p_2$ : "$F$ has a right adjoint"
  • $q_1$ : "$G$ is a right adjoint of $F$"
  • $q_2$ : "$G$ has a left adjoint"

If you were to choose new names for these properties, what would be your choice ? Furthermore, what would be your notation for these properties?

PS: I'm asking this question because I don't like the usual terminology and I'm looking for a better one.

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    $\begingroup$ So for examxple you could ask "I dont like this terminology because such and such...(...). Do you know of a terminology that avoids this problem?" or something like that. $\endgroup$ Oct 9, 2020 at 2:55
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    $\begingroup$ The chat may be more aproppiate, but in any case you should mention which aspects of the terminology are bothering you. Why it doesn't work for you, do you find it confusing, etc. $\endgroup$ Oct 9, 2020 at 2:58
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    $\begingroup$ @Colas What is it that bothers you so much about this terminology? $\endgroup$ Oct 9, 2020 at 14:21
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    $\begingroup$ It does not convey any intuition. $\endgroup$
    – Colas
    Oct 11, 2020 at 9:02
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    $\begingroup$ I don't like the left/right terminology because (i) it suggests that the distinction between left and right adjoints is only one of convention when in fact they are structurally different, and (ii) I don't know any sensible diagram I can draw where the left adjoint is on the left and the right adjoint is on the right, which makes the terms "left" and "right" unhelpful. I don't know a better terminology though. One could consider something like "inner" and "outer" adjoint, or even "lower" and "upper" - that would convey the asymmetry, but it still doesn't really give any intuition. $\endgroup$
    – N. Virgo
    Oct 14, 2020 at 0:38

1 Answer 1

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You can say "left / right dual" instead of left / right adjoint, and also "left / right dualizable" for the conditions that such adjoints exist. This terminology comes from the theory of dualizable objects in monoidal categories, which turn out to be essentially a specialization of left / right adjoints (once the latter are generalized to arbitrary 2-categories). "Dual" is unfortunately a very overloaded word in mathematics so it's not clear that this is really better.

One of the nice things about "adjoint" is that one of the only other uses of this word in mathematics is for a situation which is very analogous, namely adjoint linear transformations between inner product spaces. The idea is that there's an analogy between the hom-set definition of an adjunction

$$\text{Hom}(F(x), y) \cong \text{Hom}(x, G(y))$$

and the definition of an adjoint pair of linear transformations

$$\langle T(x), y \rangle = \langle x, T^{\dagger}(y) \rangle.$$

This analogy becomes very strong when considering Frobenius reciprocity, where the dimension of a hom space exactly recovers the inner product on characters of a finite or compact group.

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  • $\begingroup$ Thanks Qiaochu. Do you know other places (discord, forum, mailing list, etc.) where I could talk about this question of terminology? $\endgroup$
    – Colas
    Oct 9, 2020 at 4:28
  • $\begingroup$ @Colas: you could try the nForum: nforum.ncatlab.org $\endgroup$ Oct 9, 2020 at 6:09

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