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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.
   • CommentAuthorAli Caglayan
   • CommentTimeJan 19th 2019
   • PermaLink
   Author: Ali Caglayan
   Format: MarkdownItexcleaned up page and added a small ideas section too <a href="https://ncatlab.org/homotopytypetheory/revision/diff/circle/5">diff</a>, <a href="https://ncatlab.org/homotopytypetheory/revision/circle/5">v5</a>, <a href="https://ncatlab.org/homotopytypetheory/show/circle">current</a>

   cleaned up page and added a small ideas section too

   diff, v5, current

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 19th 2019
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   Author: Todd_Trimble
   Format: MarkdownItexI didn't understand 'apd'. Is it a typo?

   I didn’t understand ’apd’. Is it a typo?

3. • CommentRowNumber3.
   • CommentAuthorAli Caglayan
   • CommentTimeJan 19th 2019
   • (edited Jan 19th 2019)
   • PermaLink
   Author: Ali Caglayan
   Format: MarkdownItexapd is dependent ap $apd_f(p)$ is the application of $f : \prod_{(a:A)} B(a)$ to $p : x =_A y$ So instead of applying a function to both sides, we apply a dependent function (the result isn't a path type anymore but a dependent path). See page ~92 in the HoTT book. In this case this was written before I modified the page. Edit: For most things you will only ever use circle recursion which tells you how to build a function out of the circle. Circle induction tells you how to build a dependent function out the circle.

   apd is dependent ap

   $apd_f(p)$ is the application of $f : \prod_{(a:A)} B(a)$ to $p : x =_A y$

   So instead of applying a function to both sides, we apply a dependent function (the result isn’t a path type anymore but a dependent path). See page ~92 in the HoTT book.

   In this case this was written before I modified the page.

   Edit:

   For most things you will only ever use circle recursion which tells you how to build a function out of the circle. Circle induction tells you how to build a dependent function out the circle.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 19th 2019
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   Author: Todd_Trimble
   Format: MarkdownItexThanks!

   Thanks!

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJan 20th 2019
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   Author: Mike Shulman
   Format: MarkdownItex> For most things you will only ever use circle recursion which tells you how to build a function out of the circle. I'm not sure what you mean by that. I can't think offhand of much that you can prove about the circle using recursion but not induction.

   For most things you will only ever use circle recursion which tells you how to build a function out of the circle.

   I’m not sure what you mean by that. I can’t think offhand of much that you can prove about the circle using recursion but not induction.

6. • CommentRowNumber6.
   • CommentAuthorAli Caglayan
   • CommentTimeJan 20th 2019
   • PermaLink
   Author: Ali Caglayan
   Format: MarkdownItexThere isn’t much you can prove but when I was first learning about it in HoTT I found it a lot easier and intuitive to use circle recursion. I wouldn’t have gotten very far with induction without some more work. So perhaps I should of said: it’s easier to use circle recursion than induction. I’ll take back what I said about most things.

   There isn’t much you can prove but when I was first learning about it in HoTT I found it a lot easier and intuitive to use circle recursion. I wouldn’t have gotten very far with induction without some more work. So perhaps I should of said: it’s easier to use circle recursion than induction. I’ll take back what I said about most things.