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1. • CommentRowNumber1.
   • CommentAuthorEricson2314
   • CommentTimeDec 18th 2020
   • (edited Dec 18th 2020)
   • PermaLink
   Author: Ericson2314
   Format: MarkdownItexIs 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: $$ \frac{ \Gamma, x : A \vdash A : e : B \dashv B : x, \Delta }{ \Gamma \vdash A \leftrightarrow B : \lambda x. e : A \leftrightarrow B \dashv \Delta } $$ abstraction elimination: $$ \frac{ \Gamma \vdash A \leftrightarrow B : f : A \leftrightarrow B \dashv \Delta \qquad \Gamma \vdash B: e : A \dashv \Delta }{ \Gamma \vdash A : f e : B \dashv \Delta } $$ for a closed grouoid, which "seems truthy".

   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:

   $$\frac{ \Gamma, x : A \vdash A : e : B \dashv B : x, \Delta }{ \Gamma \vdash A \leftrightarrow B : \lambda x. e : A \leftrightarrow B \dashv \Delta }$$

   abstraction elimination:

   $$\frac{ \Gamma \vdash A \leftrightarrow B : f : A \leftrightarrow B \dashv \Delta \qquad \Gamma \vdash B: e : A \dashv \Delta }{ \Gamma \vdash A : f e : B \dashv \Delta }$$

   for a closed grouoid, which “seems truthy”.