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define the category of families

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Rachel Lambda Samuelsson 2022-10-06 19:43:55 +02:00
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src/Fams.lagda.md
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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
module Fams where
```
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 → {! !}) {! !}
```
# The category of families (of sets)
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 $B^0$, $b : B^0 \mapsto f^1 : B^1(b) \to C^1(f^0(b))$.
A morphism between objects $B$ 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) ∣)
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.
The identity is given by a pair of identity functions.
```
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 = {! !}
Fams o .id .fst x = x
Fams o .id .snd b x = x
```
Composition follows from regular function composition for $(f \circ g)^0$, and function composition with a translation of $b : B^0$ for $(f \circ g)^1$
```
(Fams o ∘ (f⁰ , f¹)) (g⁰ , g¹) .fst x = f⁰ (g⁰ x)
(Fams o ∘ (f⁰ , f¹)) (g⁰ , g¹) .snd b x = f¹ (g⁰ b) (g¹ b x)
```
This computes nicely.
```
Fams o .idr f = refl
Fams o .idl f = refl
Fams o .assoc f g h = refl
```
Since homs are functions between families of sets, they form a set.
```
Fams o .Hom-set B C = Σ-is-hlevel 2 p⁰ p¹
where
p⁰ : is-set (∣ B .fst ∣ → ∣ C .fst ∣)
p⁰ f g p q i j x = C .fst .is-tr (f x) (g x) (happly p x)
(happly q x) i j
p¹ : (f⁰ : ∣ B .fst ∣ → ∣ C .fst ∣)
→ is-set ((b : ∣ B .fst ∣) → ∣ B .snd b ∣ → ∣ C .snd (f⁰ b) ∣)
p¹ h f g p q i j b x = C .snd (h b) .is-tr (f b x)
(g b x)
(λ k → p k b x)
(λ k → q k b x) i j
```