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master
Rachel Lambda Samuelsson 2022-10-06 00:05:15 +02:00
commit f89494419b
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A formalization of CwFs in cubical agda using the 1Lab as a library.

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

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# Category with family
```
module CwF where
open import Cat.Prelude
```
```
open import Fams
```

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```
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 = {! !}
```