Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorKarol Szumiło
   • CommentTimeSep 12th 2013
   • (edited Sep 13th 2013)
   • PermaLink
   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 $m$ 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.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeSep 13th 2013
   • PermaLink
   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
   • PermaLink
   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 $P$ 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.