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• CommentRowNumber1.
• CommentAuthorSam Staton
• CommentTimeAug 14th 2019

Explain the connection with enriched monads

• CommentRowNumber2.
• CommentAuthorSam Staton
• CommentTimeAug 15th 2019

“alternative definition” -> “concrete definition”, since it is not an alternative concept, it is the same

• CommentRowNumber3.
• CommentAuthorSam Staton
• CommentTimeAug 15th 2019

parameterized formulation

• CommentRowNumber4.
• CommentAuthorSam Staton
• CommentTimeAug 15th 2019

• CommentRowNumber5.
• CommentAuthorPaoloPerrone
• CommentTimeJan 17th 2020

Added prettier diagrams, and generalized to monoidal categories. More changes to come soon.

• CommentRowNumber6.
• CommentAuthorPaoloPerrone
• CommentTimeJan 23rd 2020

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?

• CommentRowNumber7.
• CommentAuthorPaoloPerrone
• CommentTimeJan 24th 2020

Initiated makeover. (No content will be deleted.)

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJan 24th 2020

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. (-:

• CommentRowNumber9.
• CommentAuthorPaoloPerrone
• CommentTimeJan 27th 2020

That’s a good point. No, I haven’t looked into that (yet).

• CommentRowNumber10.
• CommentAuthorPaoloPerrone
• CommentTimeJan 27th 2020

• CommentRowNumber11.
• CommentAuthorPaoloPerrone
• CommentTimeJan 27th 2020

• CommentRowNumber12.
• CommentAuthorPaoloPerrone
• CommentTimeJan 28th 2020

Started part on closed monoidal categories

• CommentRowNumber13.
• CommentAuthorPaoloPerrone
• CommentTimeFeb 2nd 2020

Added equivalence of costrength and pointwise structure, with examples and references.

• CommentRowNumber14.
• CommentAuthorMike Shulman
• CommentTimeFeb 17th 2021

• CommentRowNumber15.
• CommentAuthoranuyts
• CommentTimeMar 5th 2021

Add section on interaction with Kleisli category (in previous edit), try to get it to render properly.

• CommentRowNumber16.
• CommentAuthoranuyts
• CommentTimeMar 5th 2021

Fixing lists. Correcting diagram.

• CommentRowNumber17.
• CommentAuthoranuyts
• CommentTimeMar 5th 2021
So I added a section to make sense of what happens at https://ncatlab.org/nlab/show/call-by-push-value#as_an_adjoint_logic
• CommentRowNumber18.
• CommentAuthorTim_Porter
• CommentTimeJun 1st 2021

Fixed some awkward formatting

• CommentRowNumber19.
• CommentAuthormattecapu
• CommentTimeSep 22nd 2021
• (edited Sep 22nd 2021)

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?

[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

• CommentRowNumber20.
• CommentAuthorvarkor
• CommentTimeSep 22nd 2021
• (edited Sep 22nd 2021)

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.

• CommentRowNumber21.
• CommentAuthormattecapu
• CommentTimeSep 23rd 2021
• (edited Sep 23rd 2021)
I like right/left strength, Nathanael. If we agree on this I can overhaul the page as soon as I'm on a train and clear up the confusion, with due warnings to the readers.
• CommentRowNumber22.
• CommentAuthorvarkor
• CommentTimeMay 25th 2022

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.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeNov 11th 2022
• (edited Nov 11th 2022)

• Kruna Segrt Ratkovic, Def. 3.1.6 in: Morita theory in enriched context (2013) [arXiv:1302.2774]

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.

• CommentRowNumber24.
• CommentAuthorvarkor
• CommentTimeNov 11th 2022

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”.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeNov 11th 2022

Thanks! That makes sense.

I have added the pointer here.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeNov 11th 2022

for what it’s worth, this thesis also uses the terminology “very strong” (Def. 8.10):

• Daniel Schmitter, On Operations with Binders and Operations with Equations (2020) [web, pdf]

Not that it matters very much, either way. But I have added that pointer to the entry now, too.

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeNov 22nd 2022

I have added the actual publication data for these items: