Composition and left factor for epis/acyclic maps

src/foundation/epimorphisms.lagda.md

@@ -182,6 +182,25 @@ If the map `f : A → B` is epi, then its codiagonal is an equivalence.
   is-epimorphism-is-pushout = is-epimorphism-is-equiv-codiagonal-map
 ```
 
+### The class of epimorphisms is closed under composition and has the right cancellation property
+
+```agda
+module _
+  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3}
+  (f : A → B) (g : B → C)
+  where
+
+  is-epimorphism-comp :
+    is-epimorphism f → is-epimorphism g → is-epimorphism (g ∘ f)
+  is-epimorphism-comp ef eg X =
+    is-emb-comp (precomp f X) (precomp g X) (ef X) (eg X)
+
+  is-epimorphism-left-factor :
+    is-epimorphism f → is-epimorphism (g ∘ f) → is-epimorphism g
+  is-epimorphism-left-factor ef ec X =
+    is-emb-right-factor (precomp f X) (precomp g X) (ef X) (ec X)
+```
+
 ## See also
 
 - [Acyclic maps](synthetic-homotopy-theory.acyclic-maps.md)

src/synthetic-homotopy-theory/acyclic-maps.lagda.md

@@ -209,6 +209,34 @@ module _
       ( is-acyclic-map-is-epimorphism f e)
 ```
 
+### The class of acyclic maps is closed under composition and has the right cancellation property
+
+Since the acyclic maps are precisely the epimorphisms this follows from the
+corresponding facts about [epimorphisms](foundation.epimorphisms.md).
+
+```agda
+module _
+  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3}
+  (f : A → B) (g : B → C)
+  where
+
+  is-acyclic-map-comp :
+    is-acyclic-map f → is-acyclic-map g → is-acyclic-map (g ∘ f)
+  is-acyclic-map-comp af ag =
+    is-acyclic-map-is-epimorphism (g ∘ f)
+      ( is-epimorphism-comp f g
+        ( is-epimorphism-is-acyclic-map f af)
+        ( is-epimorphism-is-acyclic-map g ag))
+
+  is-acyclic-map-left-factor :
+    is-acyclic-map f → is-acyclic-map (g ∘ f) → is-acyclic-map g
+  is-acyclic-map-left-factor af ac =
+    is-acyclic-map-is-epimorphism g
+      ( is-epimorphism-left-factor f g
+        ( is-epimorphism-is-acyclic-map f af)
+        ( is-epimorphism-is-acyclic-map (g ∘ f) ac))
+```
+
 ## See also
 
 - [Dependent epimorphisms](foundation.dependent-epimorphisms.md)