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    • CommentRowNumber1.
    • CommentAuthorLeopold
    • CommentTimeAug 24th 2021
    I was thinking about making a diagram of examples of categories, organized by which category is a subcategory of another (up to isomorphism). Then I realized that this might lend itself nicely to a collaborative effort, and also might be a nice addition to the "database of categories" entry. I'm drawing the diagram with TikZ's automated graph drawing with LuaLaTeX, saving any fine-tuning for the end (so right now it's somewhat messy). If you like this idea or would like to contribute, I suggest that the tex file just be reposted with changes. If this is the wrong place for this discussion, please let me know too, since I'm knew here. Thanks!

    Current tex file (surely with some mistakes): (output: )

    % !TeX engine = lualatex
    \usetikzlibrary{cd, arrows, graphs, graphdrawing}
    \begin{tikzpicture}[>={Stealth}, rounded corners]
    \graph [layered layout, nodes={font=\bfseries}]{
    Vect -> Mod -> Ab -> {Grp, CMon -> Mon -> Cat};
    Grp -> {Loop, ISGrp, Mon};
    SGrp -> SGrpd;
    Grp -> Grpd -> {SGrpd, pSet, Cat -> SGrpd -> Set};
    {UMag, Loop} -> pSet -> Set;
    {{ISGrp, Mon} -> SGrp, {Loop, ISGrp} -> QGrp, {Loop, Mon} -> UMag} -> Mag -> Set;
    Field -> CRing -> Ring -> Grp;
    CRing -> Ab;
    Poset -> Set;
    Hilb -> IPS -> Norm -> TVect -> Top -> Set;
    Hilb -> Ban -> CMet -> Met -> "Top\textsubscript{Haus}" -> Top;
    Ban -> Norm -> Met;
    TVect -> Vect;
    Set -> Rel;
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 25th 2021

    Beware that, without some qualification, organizing examples of categories is a task like tabulating sand grains on the beach. That attempt of a “database of categories” – by an optimistic soul from the early days of the nLab – failed to take off due to this problem.

    What may work is organizing the typical categories used in a small, well-prescribed topic of mathematics. A good example for this is the diagram of categories of topological vector spaces here, which is actually useful.

    Similarly, some of the examples you have in #1 might be organized to “typical categories used in linear algebra” etc., and a decent diagram of these, containing maybe a dozen examples, might work well on the page for linear algebra.

    But a general page meaning to organize general categories without any restriction to mathematical topics is hopeless – and needless.