Began initial work

.gitignore

@@ -0,0 +1 @@
+_build

README.md

@@ -0,0 +1 @@
+A formalization of CwFs in cubical agda using the 1Lab as a library.

cwfs.agda-lib

@@ -0,0 +1,15 @@
+name: cwfs
+include:
+  src
+
+flags:
+  --cubical
+  --no-load-primitives
+  --postfix-projections
+  --rewriting
+  --guardedness
+  --two-level
+  -W noNoEquivWhenSplitting
+
+depend:
+  cubical-1lab

src/CwF.lagda.md

@@ -0,0 +1,11 @@
+# Category with family
+
+```
+module CwF where
+
+open import Cat.Prelude
+```
+
+```
+open import Fams
+```

src/Fams.lagda.md

@@ -0,0 +1,40 @@
+```
+module Fams where
+
+open import 1Lab.HLevel.Universe
+open import 1Lab.HLevel.Retracts
+open import 1Lab.Type.Sigma
+open import Cat.Prelude
+```
+Families of sets are sets
+```
+Fam-is-set : {o : _} → is-set (Σ[ B ∈ Set o ] (∣ B ∣ → Set o))
+Fam-is-set = Σ-is-hlevel 2 (λ x y → {!   !}) {!   !}
+```
+
+Given a universe level there is a category $\mathcal Fam$ of the families of sets of that level.
+
+```
+module _ where
+  open Precategory
+
+  Fams : (o : _) → Precategory (lsuc o) o
+```
+Objects in $\mathcal Fam$ are pairs $B = (B^0,B^1)$ where $B^0$ is a set and $B^1$ is a family of sets indexed over $B^0$.
+```
+  Fams o .Ob = Σ[ B ∈ Set o ] (∣ B ∣ → Set o)
+```
+A morphism between $B$ and $C$ in $\mathcal Fam$ is a pair of functions $(f^0,f^1)$ where $f^0$ is a function $f^0 : B^0 \to C^0$ and $f^1$ is a family of functions in $B^0$, $b : B^0 \mapsto f^1 : B^1(b) \to C^1(f^0(b))$.
+```
+  Fams o .Hom B C = Σ[ f ∈ (∣ B .fst ∣ → ∣ C .fst ∣) ] 
+                      ((b : ∣ B .fst ∣) → ∣ B .snd b ∣ → ∣ C .snd (f b) ∣)
+```
+Remaning proofs are fairly trivial due to working with sets.
+```
+  Fams o .Hom-set B C f g p q i j = {!   !}
+  Fams o .id = {!   !}
+  Fams o ._∘_ = {!   !}
+  Fams o .idr = {!   !}
+  Fams o .idl = {!   !}
+  Fams o .assoc = {!   !}
+```