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<p>After/in vacation I am still not back to full speed. There are a couple of remarks of yours that I would still like to get back to, when I get the time. Here is a quick remark on the above:</p>
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an example of this is limit. in my point of view, this happens because we are still not really adopting nPOV, that is, in most entries an higher category is "something like a category, with some additional structure". but this is not really nPOV; rather it is 1POV on higher structures. and coherently, in 1POV every higher concept has to be described as a generalization of a categorical concept. the nPOV on category theory, should instead rather be something like "a category is a very special kind of infinity-category";
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<p>I see what you mean. We had occasionally some discussion on whether it is wise to start an entry right away with the most abstract nonsense perspective on the subject.</p>
<p>I have come to think that good style of an entry is something that roughly follows this pattern:</p>
<p>The Idea section should start with mentioning the most basic motivations/examples and then gradually raise the height of the perspective, indicating how one may pass to nPOVs of higher n.</p>
<p>With that done well, then the definition section may sensibly go the other way round, and start with stating the most abstract nonsense ooPOV. This should then be unwrapped gradually the other way round, to show how the lower n POVs are included in this.</p>
<p>Finally the Examples section reasnoably proceeds in the opposite direction once again and first states the low-brow examples from a low n POV and then gradually comes to discussion of the examples for the full ooPOV.</p>
<p>Not many entries currently come close to this ideal structure, of course. But i think eventually we can usefully bring more of them into a form of this sort. You should feel challenged to reorganize for instance <a href="https://ncatlab.org/nlab/show/limit">limit</a> in such a style. (If you agree with this style, of course.)</p>
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Since you have a personal web, you can create your beta version at limit (domenicofiorenza) until you consider it finished and fixed.
sounds good. If and when you include a link to the "beta version" on your personal web to the entry on the main web, it's for all practical purposes as visible as that entry.
I think I didn't understand what the original post here meant. I think it would be a terrible idea for limit to give only an idea and then start out with a definition for quasi-categories. (In fact, I'd rather that it not discuss quasi-categories at all, put that rather at limit in a quasicategory.) The nPOV doesn't mean that we have to start with the most complicated notions first. Think about all the poor graduate students who are coming to the nLab to learn something about higher category theory, but who don't yet know what a simplicial set is!
I also disagree with this:
in most entries an higher category is "something like a category, with some additional structure". but this is not really nPOV; rather it is 1POV on higher structures
I think that is a perfectly good part of the nPOV. A higher category is like a category, but with more structure; that's just a fact. In most cases, the only way we can get anywhere in higher category theory is by starting with an analogy to 1-categories.
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