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added to generalized Reedy category a bunch of definitions and propositions from Cisinski’s article, concerning the notion of normal morphisms of presheaves over a generalized Reedy category.
Thanks! I just had a look at this page for the first time since Cisinski’s notion was added. I was a bit confused about which statements and examples were about which definition, so I tried to clarify; but please correct if I got it wrong.
Hi Mike,
yes, sorry, I had added this only today and then was interrupted before I had really cleaned up the expanded entry. I will expand further on this in more detail in the next days.
I think there’s a little problem on this page: Cisinski’s “catégories squelettiques” are introduced as a generalization of Reedy categories, and are actually referred to as “Cisinski generalized Reedy categories”. However, I don’t think that they actually generalize Reedy categories: in a Cisinski category, all negative maps are split epis, but for Reedy categories arising from inverse categories this property only holds in trivial cases (when the inverse category is discrete).
To fix this, I think we have to start by changing terminology. What should we say instead of “Cisinski generalized Reedy category”? Maybe “Cisinski category”? Or “skeletic category”?
Or maybe “Cisinski-skeletal category” in analogy to “Barr-exact”.
I think ’skeletal category’ is good, can always be qualified with ’in the sense of Cisinki’ if necessary.
I like Cisinski-skeletal.
skeletal category has a standard meaning, so any other usage of it should be qualified somehow. But since Cisinski’s categories are so much like Reedy categories, I think it would be good to name them accordingly, rather than perpetuating unnecessary confusion. Maybe “Cisinski-Reedy category”?
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