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    Category Theory Illustrated - Functors compsci haskell math abuseofnotation.github.io
    borismarinov avatar authored by borismarinov 14 days ago |
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    Corbin avatar Corbin 14 days ago | link

    Great illustrations. Thanks for your careful approach.

    Categorical isomorphisms are also very rare in practice… That’s because if two categories are isomorphic then there is no reason at all to treat them as different categories - they are one and the same.

    There are several equivalences of categories which a student might optionally consider at this point in the material, revealing structure not often examined:

    • As an instance of Stone duality, the opposite category of Set will describe some sort of Boolean algebra; each set is mapped to the Boolean algebra of its elements.
    • The category of partial orders Pos happens to be equivalent to (0,1)Cat. Also, Set is equivalent to (0,0)Cat. These are merely definitions, but it is immediate that sets are a subcategory of posets, and not the other way around.
    • The category of sets and relations Rel is equivalent to its opposite category. This is a kind of symmetry that is often overlooked in Cartesian-closed courses.
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      borismarinov avatar borismarinov 14 days ago | link

      Thanks for the comment, very few people take the time to read it all and comment here.

      On your suggestions - they are very good, but it seems like I have to cover this material before getting to isomorphisms, else it would be too much of a detour and will spoil the “narration”.

      • Stone duality: I actually don’t know a lot about this. I’ve been meaning to read about the category of topological spaces for a while and maybe write a whole chapter on it, as I did for monoids and orders. So, let me know if you can help me out with some good material.
      • (0, 1)Cat - I think it’s way too early to involve concepts in higher category theory.
      • The category of sets and relations is an interesting one, but I need an appropriate context to talk about it, or a motivating example, related to something that I have covered, or some everyday concept.