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Created elegant Reedy category.
Why ’elegant’?
Beats me; ask Julie and Charles. My guess would be that they just wanted a word with connotations of “nice” or “well-behaved” that hadn’t been overused already.
I’ve added some additional characterizations and examples to elegant Reedy category.
I’ve corrected the stated theorem at elegant Reedy category to only claim that the Reedy and injective model structures coincide for simplicial presheaves. The more general version (for any model category whose cofibrations are the monos) was copied from v1 of Bergner-Rezk, but v2 removed that claim.
I’ve added a proof of the main theorem about elegant Reedy categories. Writing it more categorically shows the generality in which it is true: it seems to work for presheaves valued in any Cisinski model category.
Isn’t condition 1 of the third equivalent formulation of elegant Reedy category (definition 1.1) redundant? In Bergner-Rezk, they define elegant by just condition 2 there, and prove that it is equivalent to the formulation in terms of absolute pushouts, which of course then implies condition 1 as well.
You’re right. It looks like the definition on the nLab page was taken from their arXiv v1 and no one updated it after their final version was posted. Please feel free to fix it!
Alright, I removed the extra condition and added a corollary to show that it follows from the others. I think it would also be nice to add a proof of the equivalence of the three formulations, since I don’t think it’s all that obvious. I’ll try to do that when I have some time.
Thanks!
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