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1. • CommentRowNumber1.
   • CommentAuthorKarol Szumiło
   • CommentTimeSep 12th 2013
   • (edited Sep 13th 2013)
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   Author: Karol Szumiło
   Format: MarkdownItexI have stumbled upon the following small category. Its objects are all natural numbers and the set morphisms from $m$ to $n$ is $\{ f_0, f_1, \ldots, f_{m \wedge n} \}$ (where $m \wedge n$ is the minimum of $m$ and $n$). The composite of $f_i$ and $f_j$ is $f_{i \wedge j}$. Have you seen this category before? Do you know different descriptions? This category is in a sense the minimal Reedy category whose direct part is $\mathbb{N}$ (consisting of morphisms of the form $f_m : m \to n$) and the inverse part is $\mathbb{N}^\mathrm{op}$ (consisting of $f_n : m \to n$). I'm just curious whether it is good for anything.

   I have stumbled upon the following small category. Its objects are all natural numbers and the set morphisms from to is (where is the minimum of and ). The composite of and is . Have you seen this category before? Do you know different descriptions?

   This category is in a sense the minimal Reedy category whose direct part is (consisting of morphisms of the form ) and the inverse part is (consisting of ). I’m just curious whether it is good for anything.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeSep 13th 2013
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   Author: TobyBartels
   Format: MarkdownItexWell, it\'s equivalent to the category of spans in the category of finite well-ordered sets (in which a morphism is an inclusion of an initial segment). One could easily drop the finiteness condition here (which amounts to changing ‘natural number’ to ‘ordinal number’ in your description). I don\'t know if anybody has studied this either, but you might find something about it in that guise. (Note that the category of well-ordered sets is thin, being essentially the poset of ordinal numbers; so the category of spans is a nice way to derive a non-thin category from it.)

   Well, it's equivalent to the category of spans in the category of finite well-ordered sets (in which a morphism is an inclusion of an initial segment). One could easily drop the finiteness condition here (which amounts to changing ‘natural number’ to ‘ordinal number’ in your description). I don't know if anybody has studied this either, but you might find something about it in that guise. (Note that the category of well-ordered sets is thin, being essentially the poset of ordinal numbers; so the category of spans is a nice way to derive a non-thin category from it.)

3. • CommentRowNumber3.
   • CommentAuthorKarol Szumiło
   • CommentTimeSep 14th 2013
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   Author: Karol Szumiło
   Format: MarkdownItexThat's a good observation. More generally, I think you can take any poset $P$ that is well-founded and has binary meets and then the category of spans in $P$ is a Reedy category with $P$ as the direct part and $P^\mathrm{op}$ as the inverse part. Still, I tried to google some related terms and I didn't find anything relevant. Perhaps this construction is just not very interesting.

   That’s a good observation. More generally, I think you can take any poset that is well-founded and has binary meets and then the category of spans in is a Reedy category with as the direct part and as the inverse part. Still, I tried to google some related terms and I didn’t find anything relevant. Perhaps this construction is just not very interesting.