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I’m finally getting into Lurie’s work on the cobordism hypothesis, and it sounds like he primarily uses multiplicial sets. I consider these to be a step toward the homotopificiation of matrices, viewed as simplicial profunctors between discrete sets or as simplicial objects in the framed bicategory of matrices (although maybe it is disingenuous to state this way), depending on how one curries. Anyone know what happens to the monoidal structure(s) under these equivalences? What happens when we pass to multiplicial sets in affine spaces over a monad? (Which, I think, admits a bicategorical universal property relating it to the Eilenberg-Moore object of a general monad; more on this under the slightest pressure.) I get the strong sense now that all of this is a pale shadow of something already known and remarked upon coherently by someone somewhere. I would like some directions, please, if that is the case!
I get really great vibes from these categories. They also beg to be unified simultaneously with the remarkable structure of the bicategory of bisimplicial sets, which is most beautifully perceived as the monoidal bicategory of profunctors \Delta \to \Delta, which is (\Delta^{\circ} \times \Delta ) \to Set. I am interested in any of the applications of the simplicial bar construction in this settings. I’d like to contextualize all of this holistically in terms of simplicial algebras over monads, or something to that tune. All reading recommendations are very welcome!
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