From 82b59b9e4aa96a769c566c09ae73e8189e5eb9a9 Mon Sep 17 00:00:00 2001
From: Tom de Jong <43781735+tomdjong@users.noreply.github.com>
Date: Thu, 9 Nov 2023 12:13:02 +0000
Subject: [PATCH] Composition and left factor for epis/acyclic maps (#909)

---
 src/foundation/epimorphisms.lagda.md          | 19 +++++++++++++
 .../acyclic-maps.lagda.md                     | 28 +++++++++++++++++++
 2 files changed, 47 insertions(+)

diff --git a/src/foundation/epimorphisms.lagda.md b/src/foundation/epimorphisms.lagda.md
index 5505581352..d3a14b661e 100644
--- a/src/foundation/epimorphisms.lagda.md
+++ b/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}
+  (g : B → C) (f : A → B)
+  where
+
+  is-epimorphism-comp :
+    is-epimorphism g → is-epimorphism f → is-epimorphism (g ∘ f)
+  is-epimorphism-comp eg ef X =
+    is-emb-comp (precomp f X) (precomp g X) (ef X) (eg X)
+
+  is-epimorphism-left-factor :
+    is-epimorphism (g ∘ f) → is-epimorphism f → is-epimorphism g
+  is-epimorphism-left-factor ec ef 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)
diff --git a/src/synthetic-homotopy-theory/acyclic-maps.lagda.md b/src/synthetic-homotopy-theory/acyclic-maps.lagda.md
index a9f4aa9d5b..bceb52e31a 100644
--- a/src/synthetic-homotopy-theory/acyclic-maps.lagda.md
+++ b/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}
+  (g : B → C) (f : A → B)
+  where
+
+  is-acyclic-map-comp :
+    is-acyclic-map g → is-acyclic-map f → is-acyclic-map (g ∘ f)
+  is-acyclic-map-comp ag af =
+    is-acyclic-map-is-epimorphism (g ∘ f)
+      ( is-epimorphism-comp g f
+        ( is-epimorphism-is-acyclic-map g ag)
+        ( is-epimorphism-is-acyclic-map f af))
+
+  is-acyclic-map-left-factor :
+    is-acyclic-map (g ∘ f) → is-acyclic-map f → is-acyclic-map g
+  is-acyclic-map-left-factor ac af =
+    is-acyclic-map-is-epimorphism g
+      ( is-epimorphism-left-factor g f
+        ( is-epimorphism-is-acyclic-map (g ∘ f) ac)
+        ( is-epimorphism-is-acyclic-map f af))
+```
+
 ## See also
 
 - [Dependent epimorphisms](foundation.dependent-epimorphisms.md)