Timeline for Different levels of isomorphism/equivalence/adjunction between bicategories
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| S May 13, 2021 at 0:04 | history | bounty ended | CommunityBot | ||
| S May 13, 2021 at 0:04 | history | notice removed | CommunityBot | ||
| S May 4, 2021 at 22:22 | history | bounty started | Alec Rhea | ||
| S May 4, 2021 at 22:22 | history | notice added | Alec Rhea | Draw attention | |
| Apr 23, 2021 at 22:10 | comment | added | Emily | I also just noticed a mistake: it's not that the above different definitions of "op/lax biadjunctions" need not agree if $F$ and $G$ are merely lax, it's that they can only work if $F$ and $G$ are pseudo: for the first we need whiskerings of functors with pseudonatural transformations, and these can't be defined for lax functors, while for the second (and similarly for the third) the assignment $A,B\mapsto\mathsf{Hom}_{\mathcal{D}}(F(A),B))$ won't define a pseudofunctor $\mathcal{C}^\mathsf{op}\times\mathcal{C}\to\mathsf{Cats}$ if $F$ is lax! | |
| Apr 23, 2021 at 22:10 | comment | added | Emily | @AlecRhea I hope these were at least a bit helpful! :) | |
| Apr 23, 2021 at 22:08 | comment | added | Emily | @MikeShulman I agree; here's a proper question to avoid digressing further here :) | |
| Apr 23, 2021 at 3:04 | comment | added | Alec Rhea | @MikeShulman I always appreciate your input Mike, thank you. (thank you too Théo!) | |
| Apr 23, 2021 at 2:50 | comment | added | Mike Shulman | Hah, actually I should probably have said that "having weakly homotopy equivalent nerves" is a notion of sameness. Functors that are weak homotopy equivalences aren't necessarily symmetric either; you have to pass to zigzags. (-: So "being connected by a zigzag of adjunctions" could be a notion of sameness, but it doesn't seem likely to be a very interesting one. (However, this is rather a digression!) | |
| Apr 23, 2021 at 2:37 | comment | added | Mike Shulman | "Functors whose nerve is a weak homotopy equivalence" are certainly a notion of sameness for categories. But I wouldn't call adjunctions a notion of sameness, even though they lie "in between" equivalences and weak homotopy equivalences, because for instance they're not symmetric: if there is an adjunction $C\rightleftarrows D$ there need not be an adjunction $D\rightleftarrows C$. | |
| Apr 22, 2021 at 20:13 | comment | added | Emily | I'm also a bit split on whether or not to consider adjunctions as a notion of sameness, as there is evidence for both sides: for instance, they correspond in a sense to "homotopy equivalences" of categories in the sense that if $F\colon\mathcal{C}\longrightarrow\mathcal{D}$ admits an adjoint, then the induced map $\mathrm{N}_{\bullet}(F)\colon\mathrm{N}_{\bullet}(\mathcal{C})\longrightarrow\mathrm{N}_{\bullet}(\mathcal{D})$ is a homotopy equivalence of simplicial sets. | |
| Apr 22, 2021 at 20:06 | comment | added | Emily | The reason I'm bringing all these different possible definitions is because it seems that if $F$ and $G$ are merely lax instead of pseudo, then I think these can even fail to be equivalent! This seems to make the "adjunction portion" of the strength poset to be even more difficult to figure out :/ | |
| Apr 22, 2021 at 20:03 | comment | added | Emily | Third: we can declare a pair $(L,R)$ of pseudofunctors to be a "lax biadjunction" if pre/post-composition with them induces an adjunction between categories of lax natural transformations as in $$ \mathsf{LaxNat}(L\circ F,G) \leftrightarrows \mathsf{LaxNat}(F,R\circ G). $$ And fourth (a guess) is via fibrations of bicategories, which I asked about here | |
| Apr 22, 2021 at 19:56 | comment | added | Emily | Second, we can use adjunctions/equivalences/isomorphisms of $\mathsf{Hom}$-categories, where an "op/lax biadjunction" corresponds to requiring a compatible system of adjunctions between the categories $\mathsf{Hom}(F(A),B)$ and $\mathsf{Hom}(A,G(B))$. Then, a "pseudo biadjunction" corresponds to such a compatible system of adjoint equivalences, and a "2-adjunction" would be such a system of isomorphisms. So for our strength poset: being a "2-adjunction" implies being a "pseudo biadjunction" which implies being an "op/lax biadjunction". | |
| Apr 22, 2021 at 19:53 | comment | added | Emily | First, we can use co/units, as done in the nLab page linked above (or as in Gurski's biequivalences of tricategories), though the pasting diagrams involved when working with bicategories and pseudofunctors are slightly different than those for Gray-categories. | |
| Apr 22, 2021 at 19:48 | comment | added | Emily | I've been thinking a lot about this lately, though I haven't quite found an answer yet. I think the weakest possible such notion is that of an "op/lax biadjunction", which I think can be equivalently formulated in three ways (and maybe four): [...] | |
| Apr 22, 2021 at 19:39 | comment | added | Alec Rhea | @MikeShulman That’s fair, but I’m not sure what to edit in its place; any suggestions? (if you’d like to edit please feel free) | |
| Apr 22, 2021 at 14:46 | comment | added | Mike Shulman | I would not call an adjunction a "notion of sameness". | |
| Apr 22, 2021 at 13:14 | history | asked | Alec Rhea | CC BY-SA 4.0 |