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Apr
21
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Mar
7
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Dec
7
comment What categorical property of monoidal categories picks out the ones with duals?
...if I understand Mike's suggestion, he wants to translate the property of being rigid along a (monoidal) equipment into such a 2-star-autonomous category. Therefore, in such framework, you won't work with "data available to the bicategory", but with data available to: 1) your monoidal bicategory W, 2) another 2-star-autonomous category D, 3) monoidal 2-functor F : W -> D that equips W with proarrows D. This actually requires tons of additional structures --- much more than the information about discrete/groupoidal objects.
Dec
7
comment What categorical property of monoidal categories picks out the ones with duals?
@TheoJohnson-Freyd, I don't understand you, but it seems that you are happy with another answer, so this is, perhaps, no longer relevant. However... you say that you want to use "only data available to the bicategory". On the other hand, monoidal operations from monoidal bicategory are not "available" in a plain bicategory (monoidalness is not a property of a bicategory). So, perhaps, you mean "data available in a monoidal bicategory". However... if I recall the paper mentioned by Mike (although I may be wrong), the authors work in a 2-star-autonomous bicategory, and (cont...)
Dec
6
comment What categorical property of monoidal categories picks out the ones with duals?
@TheoJohnson-Freyd, I don't know what you mean by "categorical" here. However, if you are not comfortable with the notion of "underlying set of objects", then you may take "the underlying set of object up to equivalence". In fact, you may substitute $|\mathcal C|$ with the underlying groupoid of $\mathcal C$ as well...
Dec
6
answered What categorical property of monoidal categories picks out the ones with duals?
Sep
29
comment Relationship between two universal properties of the category of elements?
2) In my opinion, the biggest problem with Street's definition of pointwise Kan extension is actually not in the condition of stability under comma objects, but in the fact that the usual definition of comma object usually does not work properly in general 2-categories (it does work in so-called representable 2-categories).
Sep
29
comment Relationship between two universal properties of the category of elements?
Hi Tim, two quick comments. 1) There is a well-known definition of pointwise Kan extension in 2-categories equipped with proarrows, which dates back to early 80s and is due to R. J. Wood (in fact it mimics the idea from Yoneda structures introduced by Street and Walters in 70s). This definition can be restated in the language of double-categories. (cont...)
Sep
27
revised Relationship between two universal properties of the category of elements?
fixed grammar
Sep
27
answered Relationship between two universal properties of the category of elements?
Sep
27
revised The category of elements, enrichment, and weighted limits
typos
Sep
25
revised The category of elements, enrichment, and weighted limits
Some grammar corrections
Sep
24
answered The category of elements, enrichment, and weighted limits
Sep
10
comment Enriched Cauchy completions and underlying categories
Hi Richard, I am not sure if I understand your question --- of course, the notion of Cauchy completion does not depend on the internal structure of objects of a category.
Sep
8
answered Enriched Cauchy completions and underlying categories
Jun
25
comment Van Kampen colimits
Thank you for putting this into answer.
Jun
25
accepted Van Kampen colimits
Jun
24
comment Van Kampen colimits
@JoonasIlmavirta, I am afraid some people are overenthusiastic about the moderation toolbox. Yes, I wanted to remove the question from both "unanswered" and "open" question lists. In fact, I would have deleted the question itself, if it were not for Marc Hoyois, who offered me his time. I think it is fair to keep the track of the conversation. On the other hand, it is clear that my question based on some terminological confusion and therefore, strictly speaking, it cannot be answered --- because there can be no answer to a no-question.
Jun
23
comment Van Kampen colimits
If you post your observations as an answer than I will be happy to accept it.
Jun
23
comment Van Kampen colimits
@ZhenLin, MarcHoyois --- that's right. Somehow I thought that if you strictify a pseudofunctor, then the concept of limits strictifies automatically. But it seems that it strictifies only in one direction. Hm... I just have unlearned something...