Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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”.