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Stream: learning: questions

Topic: pseudolimits from products, powers, and pseudo-equalisers

Nathanael Arkor (Aug 10 2024 at 11:23):

In Street's Fibrations in bicategories, he states (1.25) that weighted pseudolimits can be constructed from products, powers, and pseudo-equalisers. However, in Flexible limits for 2-categories, the authors state (p. 9) that they do not know what examples of pseudolimits to add to products and powers to get all pseudolimits.

There appears to be a contradiction here, because products, powers, and pseudo-equalisers are pseudolimits, and Street claims they generate all pseudolimits. What's going on here?

One possible answer is that what Street calls a "pseudo-equalizer" is not really a pseudo-equaliser, but is actually an iso-inserter (see Example 6.16 of Flexible limits for 2-categories), which is not an example of a pseudolimit. If this is the answer, is there an example of a pseudolimit that cannot be built from products, powers, and pseudo-equalisers?

Clémence Chanavat (Aug 10 2024 at 19:16):

Street's Correction to “Fibrations in bicategories” is about the fact that his (1.25) was problematic as is (as it is also pointed out at the end of page 13 of the Flexible limit paper)

Clémence Chanavat (Aug 10 2024 at 19:23):

and the corrected (1.25) at the end: image.png

Ivan Di Liberti (Aug 10 2024 at 20:05):

I would be surprised if this discussion is not settled in Elementary observation on 2-categorical limits. Am I wrong?

Nathanael Arkor (Aug 11 2024 at 00:08):

Clémence Chanavat said:

Street's Correction to “Fibrations in bicategories” is about the fact that his (1.25) was problematic as is (as it is also pointed out at the end of page 13 of the Flexible limit paper)

Thank you!

Nathanael Arkor (Aug 11 2024 at 00:09):

Ivan Di Liberti said:

I would be surprised if this discussion is not settled in Elementary observation on 2-categorical limits. Am I wrong?

Kelly has essentially the same result as Street's in the correction: namely, a construction that requires iso-inserters.

Nathanael Arkor (Aug 11 2024 at 00:10):

So my understanding is that it is an open question whether the class of pseudo limits is saturated.