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1. • CommentRowNumber1.
   • CommentAuthornonemenon
   • CommentTimeSep 12th 2020
   • PermaLink
   Author: nonemenon
   Format: TextI much prefer working in the "full" category of profunctors to the category of ordinary functors. I have read accounts that treat the relationship between these categories in analogy to the relationship between functions and linear maps. I still haven't sorted all of the moving parts of this analogy. Can you share any resources and references that you know which elucidate the relationships between Set, Rel(Set), Span(Set), Vect, Mat (matrix category), Cat, and Span(Cat)=Prof (per Joyal I think). I think the spirit of my investigations is similar to the driving interest in Chu spaces, but I am not very familiar with these yet. I know that Eilenberg-Watts promises some nice structure for bimodule categories which probably controls the connective adjunctions that sew all of these categories together. My ultimate goal is to do homotopy theory with simplicially-enriched profunctors between simplicial sets, where I feel things like twisted cohomology theories will admit very easy and natural generalizations (since slice oo-topoi can be treated nicely using the collage constrution). Thanks for your time!

   I much prefer working in the "full" category of profunctors to the category of ordinary functors. I have read accounts that treat the relationship between these categories in analogy to the relationship between functions and linear maps. I still haven't sorted all of the moving parts of this analogy. Can you share any resources and references that you know which elucidate the relationships between Set, Rel(Set), Span(Set), Vect, Mat (matrix category), Cat, and Span(Cat)=Prof (per Joyal I think). I think the spirit of my investigations is similar to the driving interest in Chu spaces, but I am not very familiar with these yet. I know that Eilenberg-Watts promises some nice structure for bimodule categories which probably controls the connective adjunctions that sew all of these categories together. My ultimate goal is to do homotopy theory with simplicially-enriched profunctors between simplicial sets, where I feel things like twisted cohomology theories will admit very easy and natural generalizations (since slice oo-topoi can be treated nicely using the collage constrution). Thanks for your time!
2. • CommentRowNumber2.
   • CommentAuthornonemenon
   • CommentTimeSep 14th 2020
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   Author: nonemenon
   Format: MarkdownItexI'm finally getting into Lurie's work on the cobordism hypothesis, and it sounds like he primarily uses [multiplicial sets](https://ncatlab.org/nlab/show/multisimplicial+set). 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](https://www.ncatlab.org/nlab/show/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!

   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!