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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.
Replace terminology “strength” and “costrength” with “left-strength” and “right-strength”, following the discussion on the nForum. This terminology is more in line with conventional categorical terminology and also widely used in the literature. It is unclear who introduced the terminology “strength” and “costrength”, as this terminology does not appear in Kock’s original papers.
added pointer to:
for the terminology “very strong monad” for the case that the strength is a natural iso.
But I got this just by googling around. Please drop a note if there is other terminology for this.
Please drop a not if there is other terminology for this.
In Gabriel-Morita theory for excisive model categories, a monad satisfying this condition is called “linear”. Since one of the coauthors of the paper is Ratkovic, I suppose they now prefer “linear” over “very strong”.
I have added the actual publication data for these items:
Anders Kock, Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970) 1-10 [doi:10.1007/BF01220868]
Anders Kock, Closed categories generated by commutative monads, Journal of the Australian Mathematical Society 12 4 (Nov 1971) 405-424 [doi:10.1017/S1446788700010272, KockMonoidalMonads.pdf:file]
Anders Kock, Strong functors and monoidal monads, Arch. Math 23 (1972) 113–120 [doi:10.1007/BF01304852, pdf]
Mention costrong monads.
Move some discussion to tensorial costrength.
I’m not an expert on category theory or anything, but shouldn’t Mixed Associativity say “$(X \otimes Y) \odot [Z] \cong X \odot (Y \odot [Z])$” rather than “$(X \otimes Y) \odot [Z] \cong X \otimes (Y \odot [Z])$”?
Plugging $Y=I$ into the latter gives “X \odot [Z] \cong X \otimes [Z]”!
The old equation looks more balanced, i.e. one $\otimes$ and one $\odot” on both sides, which caused the typo?
Carlo
I see that you fixed it, so I gather your question is rethorical.
But yes, this is the “actor isomorphism” of an action of a monoidal category. (Or should be. I didn’t write this, but that’s what it seems to be getting at.)
(That section, “Interaction with the Kleisli category”, is due to revision 41 in March 2022.)
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