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There are a few very useful websites when it comes to either finding a specific object with certain properties (and maybe lacking other properties) or finding out which properties a certain object has. For example, there are $\pi$-Base and Topospaces for topological spaces (the former having an amazingly powerful search language). I suppose on a more classical level there is also the Atlas of Finite Groups for (as one would hope) finite groups.

Is there a similar reference (website, article, book, etc.) for categories? That is if I want to know if a certain category is complete, or admits a monoidal structure, or if I want an example of a locally-small-but-not-small category, is there somewhere where I can look such things up. I've done a fair amount of looking for such a thing but have not found one so far. I know that the nLab often has information, but it often goes far beyond what I need and omits some of the more basic things.

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    $\begingroup$ There isn't. There should be. $\endgroup$ Feb 3, 2016 at 21:55
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    $\begingroup$ I'm coming up to exam season, so I won't have much time, but I've started a project for this: github.com/thosgood/CatZoo . Any help or comments would be more than appreciated, if anybody else wants to get involved. $\endgroup$
    – Tim
    Feb 8, 2016 at 17:29
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    $\begingroup$ The nLab ncatlab.org/nlab is continuously being built. If everyone goes and adds his favorite basic property of a category there, eventually it will contain them. $\endgroup$ Mar 17, 2017 at 7:17
  • $\begingroup$ It doesn't answer your question, but there are also computation frameworks for category theory, like: algebraicjulia.github.io/Catlab.jl/stable I think those are important because at some point it would be great to have a "search for an object of this kind" possibility. And that could be realised if categories are properly represented in some framework rather than a book/wiki. $\endgroup$ Feb 20, 2021 at 10:05
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    $\begingroup$ I'm one of the maintainers for topology.pi-base.org. If any category theorist is interested in building categories.pi-base.org please feel free to reach out to discuss! $\endgroup$ Apr 1, 2023 at 20:47

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