Timeline for Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ?
Current License: CC BY-SA 2.5
12 events
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| Feb 23, 2011 at 7:13 | answer | added | Tim Porter | timeline score: 2 | |
| Feb 17, 2011 at 21:42 | comment | added | Alan Wilder | @steve I mean everything has both a left and right adjoint. @david I mean: pick an ordering on the simplicial coordinates, and use it to induce a direction on the arrows. E.g. In $[1]\times[1]$, the 2-morphism starts with the 1-morphism passing through 01 and ends at 10. | |
| Feb 17, 2011 at 4:49 | comment | added | Steve Lack | What do you mean by "has adjoints for 1-morphisms"? Every morphism has a left adjoint? a right adjoint? both? either? or that composition with a given morphism induces an adjunction between hom-categories? | |
| Feb 17, 2011 at 0:00 | comment | added | David Roberts♦ | @Alan, you have lots of choices for your non-invertible 2-morphisms in $[i]\times [j]$. I presume they all 'point the same way'? | |
| Feb 16, 2011 at 23:59 | comment | added | David Roberts♦ | @Harry - if you forget horizontal composition, then a bicategory is a category over $Obj \times Obj$, so you get an object of $sSet/(Obj \times Obj)$. This gives you a truncated simplicial object in $sSet$, which applying one of your favourite functors $sSet \to ssSet$ gives you something that is interesting. The paper I was thinking of does only treat 2-categories, so I can't say this is exactly the right thing. | |
| Feb 16, 2011 at 23:37 | comment | added | Alan Wilder | I just read about Duskin nerve on nLab. I'm not sure I want that because I would like to end up in bisimplicial sets. If there's a nice answer for Duskin nerve though, I would love to hear it! | |
| Feb 16, 2011 at 23:30 | comment | added | Alan Wilder | What I was thinking of was [ N\mathcal{C}_{i,j} = \textrm{Fun}([i]\times [j], \mathcal{C}), ] where in the product $[i]\times [j]$, each square gets 'filled in' with a noninvertible 2-morphism. I hope this is (weak?) equivalent to the other constructions there are. Sorry I'm not more familiar with this stuff. References would helpful! | |
| Feb 16, 2011 at 23:17 | comment | added | Harry Gindi | @David: I thought you could only apply functors homwise in the case where the enrichment is strict (strict 2-categories are categories enriched in the cartesian monoidal category $Cat$.) | |
| Feb 16, 2011 at 22:34 | comment | added | David Roberts♦ | Well my deleted comment was silly. @Harry - he's taking the hom-wise nerve to get a (weakened) simplicial category and the the other nerve to get a bisimplicial set. Personally I would take the Duskin nerve, which is the '2-simplices are 2-commuting triangles, etc.' version. | |
| Feb 16, 2011 at 22:26 | comment | added | Harry Gindi | Dear Alan, how do you define the nerve of a bicategory. | |
| Feb 16, 2011 at 22:03 | history | edited | Alan Wilder | CC BY-SA 2.5 |
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| Feb 16, 2011 at 21:57 | history | asked | Alan Wilder | CC BY-SA 2.5 |