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In section 6 of the article, it says that since in some contexts a monad admits a unique strength, a strength can be thought of as a property rather than a structure. I feel this is misleading, since this does not automatically imply that morphisms of monads will preserve the strength. A better term would be property-like structure, rather.
Any thoughts?
It’s true that the statement “a monad has at most one strength” doesn’t itself imply that every morphism of monads preserves the strength. But I wouldn’t be surprised if the same, or slightly stronger, hypotheses actually do imply this stronger result. Have you looked into it?
In particular I would be shocked if a morphism of monads on $Set$ could fail to preserve their strengths. (-:
That’s a good point. No, I haven’t looked into that (yet).
It has been pointed to me that the definition of costrength on this page doesn’t agree with most literature. It also doesn’t agree with the convention that if X goes $A \to B$, coX goes $B \to A$.
To be clear, the current definition of costrength is $T A \otimes B \to T(A\otimes B)$ (so the difference with a strength is in which of the factors of the domain $T$ is applied to, whereas in the literature [1,2,3] I find $T(A \otimes B) \to A \otimes TB$
Who’s right?
[1] First hit for ’costrong comonad’ on Google: https://www.chrisstucchio.com/blog/2014/costrong_comonads_are_boring.html
[2] Def 4.6 in https://arxiv.org/abs/1505.04330
[3] https://library.oapen.org/bitstream/handle/20.500.12657/48221/9783030720193.pdf?sequence=1#page=248
I think “strength” and “costrength” have been used entirely inconsistently in the literature. For example, Comonadic Notions of Computation uses the same terminology as the nLab. However, I agree this usage is very confusing, as it is not consistent with the usage of “co-” in the rest of category theory.
(I’m not sure who introduced this terminology, as “costrength” and “costrong” were not used in the papers of Kock I looked in.)
I think the most appropriate terminology would be “right-strength” for $TA \otimes B \to T(A \otimes B)$ and “left-strength” for $A \otimes TB \to T(A \otimes B)$, and “right-costrength” for $T(A \otimes B) \to TA \otimes B$ and “left-costrength” for $T(A \otimes B) \to A \otimes TB$. I know various other people have the same complaint, so perhaps the nLab page would be an opportunity to provide clearer terminology (though giving a remark to say that the existing literature is inconsistent). This is also consistent with terminology like left closed and right closed for nonsymmetric monoidal categories.
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