cwfs/src/Fams.lagda.md

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 = {!   !}