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slightly less confusing text

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Rachel Lambda Samuelsson 2022-10-07 23:13:53 +02:00
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src/CwF.lagda.md
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@ -41,9 +41,6 @@ open import Fams
module CwF where module CwF where
``` ```
<!---
TODO: write more about what a CwF actually is and how one should think abut it.
--->
# Categories with families # Categories with families
A Category with families, written as CwF, is a category in which we can interpret the synax of a type theory. The objects of a CwF are contexts, and morphisms are a list of judgements, which capture the notions of substitution and weakening. For a more detailed introduction to CwFs see the introductory paper by Hoffmann. [@hoffmann97] A Category with families, written as CwF, is a category in which we can interpret the synax of a type theory. The objects of a CwF are contexts, and morphisms are a list of judgements, which capture the notions of substitution and weakening. For a more detailed introduction to CwFs see the introductory paper by Hoffmann. [@hoffmann97]
@ -128,19 +125,19 @@ For every context $\Gamma$ and $A ∈ Ty(\Gamma)$ there is a projection map $p_{
q : {Γ : Ob} {A : ∣ Ty Γ ∣} → ∣ Tr (Γ ; A) (A ⟦ p ⟧) ∣ q : {Γ : Ob} {A : ∣ Ty Γ ∣} → ∣ Tr (Γ ; A) (A ⟦ p ⟧) ∣
``` ```
The meaning of $q$ might be a bit confusing, but it's essentialy an inverse of $p$, in the sense made precise below. It's required that the map $p$ weakened by $q$ is the identity.
``` ```
field field
pq-id : {Γ : Ob} {A : ∣ Ty Γ ∣} → ⟨ p , q ⟩ ≡ id {Γ ; A} pq-id : {Γ : Ob} {A : ∣ Ty Γ ∣} → ⟨ p , q ⟩ ≡ id {Γ ; A}
``` ```
There are some additional requirments placed upon $p$ and $q$ to ensure they play nice. Dubbing $p$ a projection is no coincidence, it projects a map out of a weakened map.
``` ```
p-∘ : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣) p-∘ : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣)
→ p ∘ ⟨ γ , a ⟩ ≡ γ → p ∘ ⟨ γ , a ⟩ ≡ γ
``` ```
The condition for q, depending on the condition for p holding, needs to be expressed with a rather annoying path over path. Likewise, $q$ "projects" an object out of a weakened substitution.
``` ```
q-∘-pathp : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣) q-∘-pathp : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣)
→ ∣ Tr Δ (F₁ ⟨ γ , a ⟩ .fst (F₁ p .fst A)) ∣ ≡ ∣ Tr Δ (F₁ γ .fst A) ∣ → ∣ Tr Δ (F₁ ⟨ γ , a ⟩ .fst (F₁ p .fst A)) ∣ ≡ ∣ Tr Δ (F₁ γ .fst A) ∣