defn: absoluteness of split coequalisers

Author: TOTBWF

src/Cat/Diagram/Coequaliser/Split/Properties.lagda.md

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+```agda
+open import Cat.Prelude
+
+import Cat.Diagram.Coequaliser.Split as SplitCoeq
+import Cat.Reasoning
+import Cat.Functor.Reasoning
+
+module Cat.Diagram.Coequaliser.Split.Properties where
+```
+
+# Properties of split coequalizers
+
+This module proves some general properties of [split coequalisers].
+
+[split coequalisers]: Cat.Diagram.Coequaliser.Split.html
+
+## Absoluteness
+
+The property of being a split coequaliser is a purely diagrammatic one, which has
+the lovely property of being preserved by _all_ functors. We call such colimits
+absolute.
+
+```agda
+module _ {o o′ ℓ ℓ′}
+         {C : Precategory o ℓ} {D : Precategory o′ ℓ′}
+         (F : Functor C D) where
+```
+<!--
+```agda
+  private
+    module C = Cat.Reasoning C
+    module D = Cat.Reasoning D
+    open Cat.Functor.Reasoning F
+    open SplitCoeq
+    variable
+      A B E : C.Ob 
+      f g e s t : C.Hom A B
+```
+-->
+
+The proof follows the fact that functors preserve diagrams, and reduces to a bit
+of symbol shuffling.
+
+```agda
+  is-split-coequaliser-absolute
+    : is-split-coequaliser C f g e s t
+    → is-split-coequaliser D (F₁ f) (F₁ g) (F₁ e) (F₁ s) (F₁ t)
+  is-split-coequaliser-absolute
+    {f = f} {g = g} {e = e} {s = s} {t = t} split-coeq = F-split-coeq
+    where
+      open is-split-coequaliser split-coeq
+
+      F-split-coeq : is-split-coequaliser D _ _ _ _ _
+      F-split-coeq .coequal = weave coequal
+      F-split-coeq .rep-section = annihilate rep-section
+      F-split-coeq .witness-section = annihilate witness-section
+      F-split-coeq .commute = weave commute
+```