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