Universal property of smash products (#865)

I added some functions for computing with the universal property of
pushouts of pointed types and a function that computes the
identifications created by the map from the product to the smash
product.
I also started working on one of the sides of the universal property,
but the last hole ended up as a very complicated dependent
identification that I'm not sure how to handle. If anyone wants to look
at it, you are welcome to do so :)

---------

Co-authored-by: Fredrik Bakke <[email protected]>

src/foundation-core/equality-dependent-pair-types.lagda.md

@@ -105,6 +105,14 @@ module _
 
   η-pair : (t : Σ A B) → (pair (pr1 t) (pr2 t)) ＝ t
   η-pair t = eq-pair-Σ refl refl
+
+  eq-base-eq-pair : {s t : Σ A B} → (s ＝ t) → (pr1 s ＝ pr1 t)
+  eq-base-eq-pair p = pr1 (pair-eq-Σ p)
+
+  dependent-eq-family-eq-pair :
+    {s t : Σ A B} → (p : s ＝ t) →
+    dependent-identification B (eq-base-eq-pair p) (pr2 s) (pr2 t)
+  dependent-eq-family-eq-pair p = pr2 (pair-eq-Σ p)
 ```
 
 ### Lifting equality to the total space

src/foundation/homotopies.lagda.md

@@ -20,6 +20,7 @@ open import foundation.identity-types
 open import foundation.path-algebra
 open import foundation.universe-levels
 
+open import foundation-core.dependent-identifications
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-function-types
@@ -208,12 +209,19 @@ module _
 
   compute-dependent-identification-eq-value :
     {x y : A} (p : x ＝ y) (q : eq-value f g x) (r : eq-value f g y) →
-    (((apd f p) ∙ r) ＝ ((ap (tr B p) q) ∙ (apd g p))) ≃
-    (tr (eq-value f g) p q ＝ r)
+    coherence-square-identifications (ap (tr B p) q) (apd f p) (apd g p) r ≃
+    dependent-identification (eq-value f g) p q r
   pr1 (compute-dependent-identification-eq-value p q r) =
     map-compute-dependent-identification-eq-value f g p q r
   pr2 (compute-dependent-identification-eq-value p q r) =
     is-equiv-map-compute-dependent-identification-eq-value p q r
+
+  inv-map-compute-dependent-identification-eq-value :
+    {x y : A} (p : x ＝ y) (q : eq-value f g x) (r : eq-value f g y) →
+    dependent-identification (eq-value f g) p q r →
+    coherence-square-identifications (ap (tr B p) q) (apd f p) (apd g p) r
+  inv-map-compute-dependent-identification-eq-value p q r =
+    map-inv-equiv (compute-dependent-identification-eq-value p q r)
 ```
 
 ### Computing dependent-identifications in the type family `eq-value` of ordinary functions
@@ -224,23 +232,30 @@ module _
   where
 
   is-equiv-map-compute-dependent-identification-eq-value-function :
-    {x y : A} (p : x ＝ y) (q : eq-value f g x) (q' : eq-value f g y) →
-    is-equiv (map-compute-dependent-identification-eq-value-function f g p q q')
-  is-equiv-map-compute-dependent-identification-eq-value-function refl q q' =
+    {x y : A} (p : x ＝ y) (q : eq-value f g x) (r : eq-value f g y) →
+    is-equiv (map-compute-dependent-identification-eq-value-function f g p q r)
+  is-equiv-map-compute-dependent-identification-eq-value-function refl q r =
     is-equiv-comp
       ( inv)
-      ( concat' q' right-unit)
-      ( is-equiv-concat' q' right-unit)
-      ( is-equiv-inv q' q)
+      ( concat' r right-unit)
+      ( is-equiv-concat' r right-unit)
+      ( is-equiv-inv r q)
 
   compute-dependent-identification-eq-value-function :
-    {a0 a1 : A} (p : a0 ＝ a1) (q : f a0 ＝ g a0) (q' : f a1 ＝ g a1) →
-    ( coherence-square-identifications q (ap f p) (ap g p) q') ≃
-    ( tr (eq-value f g) p q ＝ q')
-  pr1 (compute-dependent-identification-eq-value-function p q q') =
-    map-compute-dependent-identification-eq-value-function f g p q q'
-  pr2 (compute-dependent-identification-eq-value-function p q q') =
-    is-equiv-map-compute-dependent-identification-eq-value-function p q q'
+    {x y : A} (p : x ＝ y) (q : f x ＝ g x) (r : f y ＝ g y) →
+    coherence-square-identifications q (ap f p) (ap g p) r ≃
+    dependent-identification (eq-value f g) p q r
+  pr1 (compute-dependent-identification-eq-value-function p q r) =
+    map-compute-dependent-identification-eq-value-function f g p q r
+  pr2 (compute-dependent-identification-eq-value-function p q r) =
+    is-equiv-map-compute-dependent-identification-eq-value-function p q r
+
+  inv-map-compute-dependent-identification-eq-value-function :
+    {x y : A} (p : x ＝ y) (q : f x ＝ g x) (r : f y ＝ g y) →
+    dependent-identification (eq-value f g) p q r →
+    coherence-square-identifications q (ap f p) (ap g p) r
+  inv-map-compute-dependent-identification-eq-value-function p q r =
+    map-inv-equiv (compute-dependent-identification-eq-value-function p q r)
 ```
 
 ### Relation between between `compute-dependent-identification-eq-value-function` and `nat-htpy`

src/structured-types/commuting-squares-of-pointed-maps.lagda.md

@@ -7,6 +7,8 @@ module structured-types.commuting-squares-of-pointed-maps where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.commuting-squares-of-maps
+open import foundation.function-types
 open import foundation.universe-levels
 
 open import structured-types.pointed-homotopies
@@ -35,12 +37,24 @@ between the pointedness preservation proofs.
 ## Definition
 
 ```agda
-coherence-square-pointed-maps :
+module _
   {l1 l2 l3 l4 : Level}
   {A : Pointed-Type l1} {B : Pointed-Type l2}
   {C : Pointed-Type l3} {X : Pointed-Type l4}
-  (top : C →∗ B) (left : C →∗ A) (right : B →∗ X) (bottom : A →∗ X) →
-  UU (l3 ⊔ l4)
-coherence-square-pointed-maps top left right bottom =
-  (bottom ∘∗ left) ~∗ (right ∘∗ top)
+  (top : C →∗ B) (left : C →∗ A) (right : B →∗ X) (bottom : A →∗ X)
+  where
+
+  coherence-square-pointed-maps : UU (l3 ⊔ l4)
+  coherence-square-pointed-maps =
+    bottom ∘∗ left ~∗ right ∘∗ top
+
+  coherence-square-maps-coherence-square-pointed-maps :
+    coherence-square-pointed-maps →
+    coherence-square-maps
+      ( map-pointed-map top)
+      ( map-pointed-map left)
+      ( map-pointed-map right)
+      ( map-pointed-map bottom)
+  coherence-square-maps-coherence-square-pointed-maps =
+    htpy-pointed-htpy (bottom ∘∗ left) (right ∘∗ top)
 ```

src/structured-types/pointed-homotopies.lagda.md

@@ -79,6 +79,8 @@ _~∗_ :
   pointed-Π A B → pointed-Π A B → UU (l1 ⊔ l2)
 _~∗_ {A = A} {B} = htpy-pointed-Π
 
+infix 6 _~∗_
+
 htpy-pointed-htpy :
   {l1 l2 : Level} {A : Pointed-Type l1} {B : Pointed-Fam l2 A} →
   (f g : pointed-Π A B) → f ~∗ g →

src/structured-types/pointed-unit-type.lagda.md

@@ -12,6 +12,7 @@ open import foundation.identity-types
 open import foundation.unit-type
 open import foundation.universe-levels
 
+open import structured-types.pointed-homotopies
 open import structured-types.pointed-maps
 open import structured-types.pointed-types
 ```
@@ -33,12 +34,26 @@ pr2 unit-Pointed-Type = star
 ## Properties
 
 ```agda
-terminal-pointed-map : {l : Level} (X : Pointed-Type l) → X →∗ unit-Pointed-Type
-pr1 (terminal-pointed-map X) _ = star
-pr2 (terminal-pointed-map X) = refl
-
-inclusion-point-Pointed-Type :
-  {l : Level} (X : Pointed-Type l) → unit-Pointed-Type →∗ X
-pr1 (inclusion-point-Pointed-Type X) = point (point-Pointed-Type X)
-pr2 (inclusion-point-Pointed-Type X) = refl
+module _
+  {l : Level} (X : Pointed-Type l)
+  where
+
+  terminal-pointed-map : X →∗ unit-Pointed-Type
+  pr1 terminal-pointed-map _ = star
+  pr2 terminal-pointed-map = refl
+
+  map-terminal-pointed-map : type-Pointed-Type X → unit
+  map-terminal-pointed-map =
+    map-pointed-map {A = X} {B = unit-Pointed-Type}
+      terminal-pointed-map
+
+  inclusion-point-Pointed-Type :
+    unit-Pointed-Type →∗ X
+  pr1 inclusion-point-Pointed-Type = point (point-Pointed-Type X)
+  pr2 inclusion-point-Pointed-Type = refl
+
+  is-initial-unit-Pointed-Type :
+    ( f : unit-Pointed-Type →∗ X) → f ~∗ inclusion-point-Pointed-Type
+  pr1 (is-initial-unit-Pointed-Type f) _ = preserves-point-pointed-map f
+  pr2 (is-initial-unit-Pointed-Type f) = inv right-unit
 ```

src/synthetic-homotopy-theory/pushouts-of-pointed-types.lagda.md

@@ -115,3 +115,42 @@ module _
     ( preserves-point-pointed-map
       ( horizontal-pointed-map-cocone-Pointed-Type f g c))
 ```
+
+### Computation with the cogap map for pointed types
+
+```agda
+module _
+  { l1 l2 l3 l4 : Level}
+  {S : Pointed-Type l1} {A : Pointed-Type l2} {B : Pointed-Type l3}
+  ( f : S →∗ A) (g : S →∗ B)
+  { X : Pointed-Type l4} (c : type-cocone-Pointed-Type f g X)
+  where
+
+  compute-inl-cogap-Pointed-Type :
+    ( x : type-Pointed-Type A) →
+    ( map-cogap-Pointed-Type
+      ( f)
+      ( g)
+      ( c)
+      ( map-pointed-map (inl-pushout-Pointed-Type f g) x)) ＝
+    ( horizontal-map-cocone-Pointed-Type f g c x)
+  compute-inl-cogap-Pointed-Type =
+    compute-inl-cogap
+      ( map-pointed-map f)
+      ( map-pointed-map g)
+      ( cocone-type-cocone-Pointed-Type f g c)
+
+  compute-inr-cogap-Pointed-Type :
+    ( y : type-Pointed-Type B) →
+    ( map-cogap-Pointed-Type
+      ( f)
+      ( g)
+      ( c)
+      ( map-pointed-map (inr-pushout-Pointed-Type f g) y)) ＝
+    ( vertical-map-cocone-Pointed-Type f g c y)
+  compute-inr-cogap-Pointed-Type =
+    compute-inr-cogap
+      ( map-pointed-map f)
+      ( map-pointed-map g)
+      ( cocone-type-cocone-Pointed-Type f g c)
+```