3:54 PM
@DenisNardin I think J is making it too complicated
you don't need to bring N into the mix at all
take the BK RelNerve of sCat to get a qCat and take the BK RelNerve of sSet wrt the Joyal equivs
The QEQ of C and N induces an equiv of Qcats by Barwick-Kan
both are models for Cat_infty,
Can you detect if the equivalence has some kind of 'parity' to see if it has a twist (from the ^op) or not
I know that we're implicitly identifying the two
it makes sense if you can somehow attach an "orientation" to both qCats
seems reasonable because they do in fact come from things with natural transformations