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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 7th 2012
   • (edited Dec 7th 2012)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI've been playing around with an old construction of mine, and I've realised that what it gives is a[[multicategory]] (or coloured operad). However, in a certain case this degenerates into a simplicial group, and I'd expect that somehow the more general case is "groupoidal". However, this doesn't seem like it is generally possible. However, I can think of the following: Let $x_1,\ldots,x_n \stackrel{f}{\to} y$ be an arrow, and assume there exist arrows $y\stackrel{\tilde{f_i}}{\to} x_i$ such that the composition $F := f\circ (\tilde{f_1},\ldots,\tilde{f_n})\colon y,\ldots,y \to y$ is such that it "acts like an identity". The structure necessary for this to happen seems to me to be quite special (at the very least, it probably has to be cartesian). I have an example, but I'll keep it secret for now, to see if people have ideas about how this might work, or if they've seen something like this before. I'm going to have a read of Tom's _Higher Operads, Higher Categories_ this weekend, and also a gander and [Mike's joint paper](http://golem.ph.utexas.edu/category/2010/01/generalized_multicategories.html), so if there is something obvious in there, don't sniff in derision that I haven't seen it ;-)

   I’ve been playing around with an old construction of mine, and I’ve realised that what it gives is amulticategory (or coloured operad). However, in a certain case this degenerates into a simplicial group, and I’d expect that somehow the more general case is “groupoidal”. However, this doesn’t seem like it is generally possible. However, I can think of the following:

   Let $x_1,\ldots,x_n \stackrel{f}{\to} y$ be an arrow, and assume there exist arrows $y\stackrel{\tilde{f_i}}{\to} x_i$ such that the composition $F := f\circ (\tilde{f_1},\ldots,\tilde{f_n})\colon y,\ldots,y \to y$ is such that it “acts like an identity”. The structure necessary for this to happen seems to me to be quite special (at the very least, it probably has to be cartesian). I have an example, but I’ll keep it secret for now, to see if people have ideas about how this might work, or if they’ve seen something like this before.

   I’m going to have a read of Tom’s Higher Operads, Higher Categories this weekend, and also a gander and Mike’s joint paper, so if there is something obvious in there, don’t sniff in derision that I haven’t seen it ;-)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 7th 2012
   • (edited Dec 7th 2012)
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   Author: Urs
   Format: MarkdownItexA notion of groupoidal _symmetric_ multi-$\infty$-category is given by those [[dendroidal sets]] that are dendroidal Kan complexes, hence that also have the outer horn fillers. They are equivalent to connective spectra: Matija Bašić, Thomas Nikolaus, Dendroidal sets as models for connective spectra ([arXiv:1203.6891](http://arxiv.org/abs/1203.6891))

   A notion of groupoidal symmetric multi-$\infty$-category is given by those dendroidal sets that are dendroidal Kan complexes, hence that also have the outer horn fillers. They are equivalent to connective spectra:

   Matija Bašić, Thomas Nikolaus, Dendroidal sets as models for connective spectra (arXiv:1203.6891)

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 7th 2012
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   Author: DavidRoberts
   Format: MarkdownItexThanks, Urs! That looks promising.

   Thanks, Urs! That looks promising.