Literature – Idempotents in Intensional Type Theory

references.bib

@@ -531,18 +531,24 @@ @online{Shu14UniversalProperties
 }
 
 @article{Shu17,
-  title     = {Idempotents in Intensional Type Theory},
-  author    = {Shulman, Michael},
-  date      = {2017-04-27},
-  year      = {2017},
-  month     = {04},
-  journal   = {Logical Methods in Computer Science},
-  volume    = {Volume 12, Issue 3},
-  publisher = {Episciences.org},
-  issn      = {1860-5974},
-  doi       = {10.2168/LMCS-12(3:9)2016},
-  url       = {https://lmcs.episciences.org/2027},
-  abstract  = {We study idempotents in intensional Martin-L\"of type theory, and in particular the question of when and whether they split. We show that in the presence of propositional truncation and Voevodsky's univalence axiom, there exist idempotents that do not split; thus in plain MLTT not all idempotents can be proven to split. On the other hand, assuming only function extensionality, an idempotent can be split if and only if its witness of idempotency satisfies one extra coherence condition. Both proofs are inspired by parallel results of Lurie in higher category theory, showing that ideas from higher category theory and homotopy theory can have applications even in ordinary MLTT. Finally, we show that although the witness of idempotency can be recovered from a splitting, the one extra coherence condition cannot in general; and we construct "the type of fully coherent idempotents", by splitting an idempotent on the type of partially coherent ones. Our results have been formally verified in the proof assistant Coq.}
+  title        = {Idempotents in Intensional Type Theory},
+  author       = {Shulman, Michael},
+  date         = {2017-04-27},
+  year         = {2017},
+  month        = {04},
+  eprint       = {1507.03634},
+  eprinttype   = {arxiv},
+  eprintclass  = {math},
+  primaryClass = {math.LO},
+  journal      = {Logical Methods in Computer Science},
+  volume       = {12},
+  issue        = {3},
+  pages        = {1--24},
+  publisher    = {Episciences.org},
+  issn         = {1860-5974},
+  doi          = {10.2168/LMCS-12(3:9)2016},
+  url          = {https://lmcs.episciences.org/2027},
+  abstract     = {We study idempotents in intensional Martin-L\"of type theory, and in particular the question of when and whether they split. We show that in the presence of propositional truncation and Voevodsky's univalence axiom, there exist idempotents that do not split; thus in plain MLTT not all idempotents can be proven to split. On the other hand, assuming only function extensionality, an idempotent can be split if and only if its witness of idempotency satisfies one extra coherence condition. Both proofs are inspired by parallel results of Lurie in higher category theory, showing that ideas from higher category theory and homotopy theory can have applications even in ordinary MLTT. Finally, we show that although the witness of idempotency can be recovered from a splitting, the one extra coherence condition cannot in general; and we construct "the type of fully coherent idempotents", by splitting an idempotent on the type of partially coherent ones. Our results have been formally verified in the proof assistant Coq.}
 }
 
 @article{Shu18,

src/foundation-core/sets.lagda.md

@@ -25,7 +25,9 @@ open import foundation-core.truncation-levels
 
 ## Idea
 
-A type is a set if its identity types are propositions.
+A type is a {{#concept "set" Agda=is-set}} if its
+[identity types](foundation-core.identity-types.md) are
+[propositions](foundation-core.propositions.md).
 
 ## Definition
 
@@ -86,6 +88,27 @@ module _
       ( contraction (is-proof-irrelevant-is-prop (H x x) refl) p)
 ```
 
+### A type is a set if and only if it satisfies uniqueness of identity proofs
+
+A type `A` is said to satisfy
+{{#concept "uniqueness of identity proofs" Agda=has-uip}} if for all elements
+`x y : A` all equality proofs `x ＝ y` are equal.
+
+```agda
+has-uip : {l : Level} → UU l → UU l
+has-uip A = (x y : A) → all-elements-equal (x ＝ y)
+
+module _
+  {l : Level} {A : UU l}
+  where
+
+  is-set-has-uip : is-set A → has-uip A
+  is-set-has-uip is-set-A x y = eq-is-prop' (is-set-A x y)
+
+  has-uip-is-set : has-uip A → is-set A
+  has-uip-is-set uip-A x y = is-prop-all-elements-equal (uip-A x y)
+```
+
 ### If a reflexive binary relation maps into the identity type of `A`, then `A` is a set
 
 ```agda

src/literature.lagda.md

@@ -1,13 +1,18 @@
 # Formalization of results from the literature
 
+This page is a work in early progress. To see what's happening behind the
+scenes, you can have a look at the associated GitHub issue
+[#1055](https://github.com/UniMath/agda-unimath/issues/1055).
+
 ## References
 
-{{#bibliography}} {{#reference SvDR20}}
+{{#bibliography}} {{#reference SvDR20}} {{#reference Shu17}}
 
 ## Files in the namespace
 
 ```agda
 module literature where
 
+open import literature.idempotents-in-intensional-type-theory public
 open import literature.sequential-colimits-in-homotopy-type-theory public
 ```