started work on defining a CwF

src/CwF.lagda.md

@@ -1,11 +1,57 @@
-# Category with family
+---
+references:
+- type: article-journal
+  id: abd2008
+  editor:
+  - family: Jacques Carrigue
+  - family: Manuel V. Hermengildo
+  author:
+  - family: Andreas Abel
+  - family: Thierry Coquand
+  - family: Peter Dybjer
+  issued:
+    date-parts:
+    - - 2008
+  title: 'On the Algebraic Foundation of Proof Assistants for Intuitionistic Type Theory'
+  journal: "Lecture Notes in Computer Science"
+  volume: 4989
+  url: http://link.springer.com/10.1007/978-3-540-78969-7_2
+  page: 3-13
+  language: en-GB 
 
+citation-style: ieee
+---
 ```
-module CwF where
-
 open import Cat.Prelude
+open import Fams
+
+module CwF where
 ```
+<!---
+TODO: write more about what a CwF actually is and how one should think abut it.
+--->
+# Categories with families
+
+This file defines a CwF, as well as type formers for pi types, sigma types, identity types and universes. Here CwFs are defined as a GAT (generalised algebraic theory), as is presented in a paper by Abel, Coquand and Dybjer. [@abd2008]
+
+A CwF is a category $\mathcal C$ equipped with a functor $\mathcal F : \mathcal C \to \mathcal Fam$, fufilling the rules of the GAT.
 
 ```
-open import Fams
+record CwF (o h f : Level) : Type (lsuc (o ⊔ h ⊔ f)) where
+  field
+    𝓒 : Precategory o h
+    𝓕 : Functor 𝓒 (Fams f)
+```
+
+Two functions, $Ty$ and $Tm$ are defined, respectively giving the set of syntactic types, and terms in a context.
+```
+  open Precategory
+  open Functor
+
+  Ty : 𝓒 .Ob → Set f
+  Ty Γ = 𝓕 .F₀ Γ .fst 
+
+  Tr : (Γ : 𝓒 .Ob) → ∣ Ty Γ ∣ → Set f
+  Tr Γ σ = 𝓕 .F₀ Γ .snd σ
+
 ```

src/Fams.lagda.md

@@ -3,7 +3,9 @@ open import Cat.Prelude
 
 module Fams where
 ```
-# The category of families (of sets)
+# The category of families of sets
+
+This file defines the category $\mathcal Fam$ of families of sets.
 
 Given a universe level there is a category $\mathcal Fam$ of the families of sets of that level.
 ```
@@ -37,7 +39,7 @@ Composition follows from regular function composition for $(f \circ g)^0$, and f
   (Fams o ∘ (f⁰ , f¹)) (g⁰ , g¹) .snd b x = f¹ (g⁰ b) (g¹ b x)
 ```
 
-This computes nicely.
+Since composition is defined using regular function composition it computes nicely.
 ```
   Fams o .idr f = refl
   Fams o .idl f = refl