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You also wrote Serre fibration.
I changed the discs in the definition to simplices, which are topologically equivalent; I hope that this doesn't mess up any exposition that you're trying to do. The simplicies seem to me to fit in better with what is in the rest of the Lab … although now that I think about it, maybe cubes would be nice, so that we're discussing the inclusions of cubes into cubes of one higher dimension.
I don't think anyone is ever going to reach agreement on simplices vs cubes vs discs. The nice thing about topological spaces is that all such shapes are homeomorphic there, so I think who ever is writing a given page should be free to use whatever they feel is most appropriate. And I think it's good for people who are used to thinking of one shape to be exposed to the same ideas expressed using different shapes.
So why not including more variants of the equivalent definition...rather than merely replacing one by another.
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the nLab is a little too biased towards simplices
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<p>Hm, and I thought we invested quite a bit of energy into discussing all the different shapes. Not many resources will go into the comparative shapeology that we have at <a href="https://ncatlab.org/nlab/show/geometric+shapes+for+higher+structures">geometric shapes for higher structures</a>.</p>
<p>But one reason why combinatorial simplices play such a prominent role is that <em>only for them is there a well developed homotopy theory</em> . This does not mean that such a theory does not exist in principle for the other shapes (I am sure it does) but it does mean that it is not known well, or at all.</p>
<p>More concretely: there are lots of tools for working with simplicially enriched model categories. As soon as similar tools exist for cubically enriched model categories, we'll be able to transfer much of the simplicial technology over to the cubical world.</p>
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The freedom between the choice of shapes comes with Grothendieck's theory based on "test categories"...but this has not been much followed except for seminal works of Cisinski and support from Jardine and Maltsiniotis. Jardine wrote a nice survey on categorical homotopy theory which can be found at arXiv, which outlines the Cisinski's work in a bit less technical manner than the original work.
I added lots of references and few links to Andre Joyal and similar improvements to torsor.
I added some more comments to model category and Quillen equivalence, including organizing the "Examples" section of "model category" a bit more.
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I added some more comments to model category [...] including organizing the "Examples" section of "model category" a bit more.
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<p>Thanks, nice!</p>
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I added lots of references and few links to <a href="https://ncatlab.org/nlab/show/Andre+Joyal">Andre Joyal</a>
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<p>Thanks! I added various hyperlinks. Do we have anything on species?</p>
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Do we have anything on species?
See structure type.
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Do we have anything on species?
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See <a href="https://ncatlab.org/nlab/show/structure+type">structure type</a>.
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<p>Thanks. I guess that requires more discussion, but just so people looking for "species" find anything (it is requested by a handful of entries), I created <a href="https://ncatlab.org/nlab/show/species">species</a>. At the moment it just points to <a href="https://ncatlab.org/nlab/show/structure+type">structure type</a>.</p>
Is there a policy on the nLab on how to denote the homotopy category of a model category? At Quillen equivalence I changed all to to make it consistent within that page (previously there were both and on the same page).
No, there are no Lab-wide conventions, for there is hardly a way to enforce them – or even to agree on them. (We tend to have – sometimes unfortunately lengthy – disagreements about tiny aspects of single entries already.)
So every page should try to be as self-contained as necessary for it to be readable. Thanls for taking care of this in the present case!
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