Timeline for Are all binoidal categories (in the literature) actually strict?
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| Apr 2, 2011 at 21:56 | comment | added | Todd Trimble | on the nLab page, I believe your interpretation is not what Power and Robinson intended. This was the point of my answer: to make clear what the domain and codomain are. As a sanity check, notice that the domain and codomain (as I have it) is consistent with their definition of strict premonoidal category as a monoid in $Cat'$. I urge you to think about this from that point of view -- the formalities of this alternate monoidal structure on $Cat$ are not difficult. Or -- if you are unwilling to take my word for it, why not contact John Power or Edmund Robinson directly, and point them here? | |
| Apr 2, 2011 at 21:52 | comment | added | Todd Trimble | Dear Adam: I take it that you agree that Power and Robinson unpacked definition 3.2 correctly to the nuts-and-bolts definition contained in the paragraph that followed. It follows that the definition I used is equivalent to the standard one. The heart of the problem as I see it -- and this is what Finn was gently urging you to consider in his comment -- is whether you are correctly interpreting the domain and codomain of the intended associativity transformation in the notion of premonoidal category (as you touch upon in your reply to Finn). Based on your reply and what you wrote (cont.) | |
| Apr 2, 2011 at 19:58 | comment | added | Adam | There's an analogy here to the definition of monoidal categories. I suppose that category theorists more advanced than I intuitively think of a (strict) monoidal category as a monoid internal to Cat. But the "first order" definition which doesn't refer to large categories is useful in practice for a number of reasons. It seems that binoidal categories similarly have two definitions; my question relates to the correctness/adequacy of the "first order" version of the definition | |
| Apr 2, 2011 at 19:58 | comment | added | Adam | @Todd, that is the only place where that definition (in terms of the auxiliary Cat-prime) appears. The paragraph immediately following Definition 3.2 is the definition used in every other paper, since it doesn't require the (lengthy) formal definition of Cat-prime, the proof that it is a category, the proof that it is a monoidal category, and so forth -- all of the things needed to make Definition 3.2 make sense. | |
| Apr 2, 2011 at 18:38 | comment | added | Todd Trimble | (Unfortunately, I cannot view the papers you linked to from where I am; they are behind a pay-wall.) | |
| Apr 2, 2011 at 18:33 | comment | added | Todd Trimble | @Adam: the definition I used is precisely definition 3.2 in the paper by Power and Robinson. | |
| Apr 2, 2011 at 18:00 | comment | added | Adam | @Todd, the definition you give for a binoidal category -- as a property of the non-cartesian symmetric monoidal structure of Cat rather than a family of endofunctors -- is quite different from the one which is standard in the literature (dx.doi.org/10.1007/BFb0014560 or dx.doi.org/10.1017/S0956796809007308 for example). Are you saying that the one currently in use cannot be repaired without completely rephrasing? | |
| Apr 2, 2011 at 16:45 | history | edited | Todd Trimble | CC BY-SA 2.5 |
added 75 characters in body
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| Apr 2, 2011 at 16:29 | history | answered | Todd Trimble | CC BY-SA 2.5 |