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Evan Patterson (Apr 19 2020 at 00:19):In 2-category theory it is common to have weak structures, called lax when the morphisms go in one direction (the "default" direction) and oplax or colax when they go in the other. In his "2-categories companion," Lack says:
Typically one speaks of pseudo widgets for the up-to-isomorphism notion, lax widgets for the up-to-not-necessarily-invertible comparison notion, and when the direction of the comparison is reversed, either oplax widget or colax widget, depending on the specific case.
What I am wondering is whether there is any rhyme or reason to what is called "oplax" and what is called "colax." For example, the nLab uses "oplax" for monoidal functors and "colax" for slice 2-categories. Should I make anything of this?
sarahzrf (Apr 19 2020 at 01:05):i think it's just different people writing different pages
sarahzrf (Apr 19 2020 at 01:06):i might be wrong though
John Baez (Apr 19 2020 at 02:13):The Australians use "op", and this is part of a systematic thing for them: just as a category has an "op", a 2-category has an "op" where you turn around the morphisms and a "co" where you turn around the 2-morphisms. But it's hard to stay consistent.
Once upon a time there some people talked about Grothendieck fibrations and Grothendieck "cofibrations", I think - or maybe I'm just imagining it. But when Quillen invented model categories, with fibrations and cofibrations, it became better to talk about Grothendieck "opfibrations".
Evan Patterson (Apr 19 2020 at 02:29):OK, thanks. The examples I gave from the nLab seem to be consistent with that convention.