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    • CommentRowNumber1.
    • CommentAuthorEricson2314
    • CommentTimeDec 18th 2020
    • (edited Dec 18th 2020)

    Is there any existing literature on what the internal logical of a groupoid might look like?

    Please excuse the syntax bashing of this complete amateur, but i came up with:

    abstraction introduction:

    Γ,x:AA:e:BB:x,ΔΓAB:λx.e:ABΔ \frac{ \Gamma, x : A \vdash A : e : B \dashv B : x, \Delta }{ \Gamma \vdash A \leftrightarrow B : \lambda x. e : A \leftrightarrow B \dashv \Delta }

    abstraction elimination:

    ΓAB:f:ABΔΓB:e:AΔΓA:fe:BΔ \frac{ \Gamma \vdash A \leftrightarrow B : f : A \leftrightarrow B \dashv \Delta \qquad \Gamma \vdash B: e : A \dashv \Delta }{ \Gamma \vdash A : f e : B \dashv \Delta }

    for a closed grouoid, which “seems truthy”.

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