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• CommentRowNumber1.
• CommentAuthorPeter Heinig
• CommentTimeJul 16th 2017
• (edited Jul 16th 2017)

Is the category Hom of bicategories with homomorphisms as the morphisms, in the sense of

Ross Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no 2 (1980), p. 111-160

already (recognizably) documented on the nLab ? (I did a sem-cursory search in this respect, but did not find it documented (in its own right, I mean, the article of Street appears.)

Should it be?

Should it have an article of its own?

To me it seems it should (my motivation is that I am using and documenting bicategories currently, and are studying Street’s 1980 paper as a sort of background reading to Garner–Shulman, Adv. Math. 289), but its traditional name Hom seems unfortunate, creating yet another meaning of Hom.

My suggestion would be to call it (and its article)

$BiCat$

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeJul 16th 2017

I would have thought Bicat would the the usual name, possibly with a subscript to indicate the type of morphisms. One can have lax maps, pseudo=weak maps, ie homomorphisms, and strict maps. This convention is used for monoidal categories too.

• CommentRowNumber3.
• CommentAuthorPeter Heinig
• CommentTimeJul 16th 2017
• (edited Jul 16th 2017)

Thanks. Added a short entry to database of categories, using Bicat, which turned up a redirect Bicat->2-Cat, which surprised me. This may be worth some editing by someone else, since I thought that 2-Cat, unadorned, is supposed to mean “strict 2-category”, whence the usage “weak 2-category” whenever the numeral-prefix is used.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeJul 16th 2017

The nlab style is to take the weak notion as default, and write strict 2-category explicitly for the strict definition.

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeJul 16th 2017

But I agree that 2-Cat at present looks merely like a placeholder. References (eg to Benabou, to Street, to Gray) would be good.