Timeline for pseudofunctors and pseudonatural transformations
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jun 6, 2011 at 7:00 | comment | added | Tim Porter | @Alan A homotopy coherent (hc) version of these ideas is obtained using the idea that the 'diagrams' used have to be hc, but a hc square does have the diagonal in it. (See page 366 of the version of the crossed menagerie (available from the n-Lab page of that name).) This use of the diagonal allows both conventions, but still hits a problem as it has to make a choice of `diagonal towards the sides' as against 'sides towards diagonal'. There are papers back in the 1980s(?) by Cordier and myself that take this apart. (It is also discussed in Kamps and Porter's book. (blatant advertisement. :-)) | |
| Jun 5, 2011 at 19:49 | comment | added | Alan Wilder | @Martin: I'm not sure what you mean by the middle paragraph, but assuming you meant a(j)o F(g)=>G(g)o a(j), you actually do have this data on both sides. Clearly it is part of the natural transformation data, and for a p.functor $h:J\times [1]\to C$, this isomorphism is produced by applying the compositors of $h$ to the equal factorizations [ (j'\to j,\text{id}_1)\circ (\text{id}_j, 0\to 1) = (\text{id}_{j'},0\to 1)\circ (j'\to j, \text{id}_0) ] in the middle we have $h(j'\to j,0\to 1)$, which is the datum that has no counterpart on the p.natural transformation side | |
| Jun 5, 2011 at 19:49 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Jun 5, 2011 at 18:11 | comment | added | Alan Wilder | Thanks, Tim, can you go into more detail please? | |
| Jun 5, 2011 at 16:26 | comment | added | Tim Porter | The problem you identify is one of direction of the 2-arrow. I used to work with homotopy coherence and in its simplicial manifestation that gets around the problem by asking for both directions! There are still problems with the directions but they are resolved by the initial structures of the homotopy coherent nerve construction. (I can go into more detail if you want, but this is a comment not an answer.) | |
| Jun 5, 2011 at 14:38 | history | answered | Martin Brandenburg | CC BY-SA 3.0 |