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1. • CommentRowNumber1.
   • CommentAuthorColin Zwanziger
   • CommentTimeSep 16th 2014
   • (edited Sep 16th 2014)
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   Author: Colin Zwanziger
   Format: MarkdownItexIs there a reason we don't care about these? We could have something like: k-tuply posetal n-categories are the same as k-1-tuply posetal n+1 categories (with distinguished n+1-morphisms in each inhabited hom-set between n-morphisms) with parallel n+1-morphisms equivalent

   Is there a reason we don’t care about these?

   We could have something like: k-tuply posetal n-categories are the same as k-1-tuply posetal n+1 categories (with distinguished n+1-morphisms in each inhabited hom-set between n-morphisms) with parallel n+1-morphisms equivalent

2. • CommentRowNumber2.
   • CommentAuthorColin Zwanziger
   • CommentTimeSep 16th 2014
   • (edited Sep 16th 2014)
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   Author: Colin Zwanziger
   Format: MarkdownItexI suppose it has something to do with the fact that this stabilizes for k>1!

   I suppose it has something to do with the fact that this stabilizes for k>1!

3. • CommentRowNumber3.
   • CommentAuthorColin Zwanziger
   • CommentTimeSep 16th 2014
   • (edited Sep 16th 2014)
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   Author: Colin Zwanziger
   Format: MarkdownItexSo these are just n-posets I guess. The reason I asked the question is we do appear to have the right kind of adjoint pair between Posets ("1-tuply posetal 0-cats") and Categories with a distinguished morphism in each inhabited hom set [distinguished morphisms closed under composition and include identities] ("0-tuply posetal 1 cats") that restricts to the equivalence of posets and posets-viewed-as-categories.

   So these are just n-posets I guess. The reason I asked the question is we do appear to have the right kind of adjoint pair between Posets (“1-tuply posetal 0-cats”) and Categories with a distinguished morphism in each inhabited hom set [distinguished morphisms closed under composition and include identities] (“0-tuply posetal 1 cats”) that restricts to the equivalence of posets and posets-viewed-as-categories.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeSep 16th 2014
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   Author: TobyBartels
   Format: MarkdownItexHigher posets are more fundamental than higher categories. It\'s backwards to think of posetal categories as a modification of categories. Rather, an $n$-category is a $1$-tuply groupoidal $(n+1)$-poset (and an $n$-groupoid is an $(n+1)$-tuply groupoidal $(n+1)$-poset). But the usual numbering scheme goes the other way: an $n$-groupoid is an $(n,0)$-category, an $n$-category is an $(n,n)$-category, and an $(n+1)$-poset is an $(n,n+1)$-category. It\'s sheer bigotry that we use ‘category’ as the base of this naming scheme (rather than the extreme concepts of groupoid and poset), and then use an unnatural numbering scheme (from $0$ to $n+1$) to match that.

   Higher posets are more fundamental than higher categories. It's backwards to think of posetal categories as a modification of categories. Rather, an $n$-category is a $1$-tuply groupoidal $(n+1)$-poset (and an $n$-groupoid is an $(n+1)$-tuply groupoidal $(n+1)$-poset).

   But the usual numbering scheme goes the other way: an $n$-groupoid is an $(n,0)$-category, an $n$-category is an $(n,n)$-category, and an $(n+1)$-poset is an $(n,n+1)$-category. It's sheer bigotry that we use ‘category’ as the base of this naming scheme (rather than the extreme concepts of groupoid and poset), and then use an unnatural numbering scheme (from $0$ to $n+1$) to match that.