Editing Commutative diagram (section)

==Description==
A commutative diagram often consists of three parts:
* [[Object (category theory)|objects]] (also known as ''vertices'') 
* [[morphism]]s (also known as ''arrows'' or ''edges'')
* paths or composites
===Arrow symbols===
In algebra texts, the type of morphism can be denoted with different arrow usages: 
* A [[monomorphism]] may be labeled with a <math>\hookrightarrow</math><ref name=":0">{{Cite web|url=https://www.euclideanspace.com/maths/discrete/category/principles/arrow/index.htm|title=Maths - Category Theory - Arrow - Martin Baker|website=www.euclideanspace.com|access-date=2019-11-25}}</ref> or a <math>\rightarrowtail</math>.<ref name="Riehl 2016">{{Cite book|last=Riehl|first=Emily|author-link=Emily Riehl|date=2016-11-17|title=Category Theory in Context|chapter=1|page=11|publisher=Dover Publications|url=https://math.jhu.edu/~eriehl/context.pdf}}</ref>
* An [[epimorphism]] may be labeled with a <math>\twoheadrightarrow</math>.
* An [[isomorphism]] may be labeled with a <math>\overset{\sim}{\rightarrow}</math>.
* The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as <math>\exists</math>. 
** If the morphism is in addition unique, then the dashed arrow may be labeled <math>!</math> or <math>\exists!</math>.
*If the morphism acts between two arrows (such as in the case of [[higher category theory]]), it's called preferably a [[natural transformation]] and may be labelled as <math>\Rightarrow</math> (as seen below in this article).

The meanings of different arrows are not entirely standardized: the arrows used for monomorphisms, epimorphisms, and isomorphisms are also used for [[Injective function|injections]], [[surjection]]s, and [[bijection]]s, as well as the cofibrations, fibrations, and weak equivalences in a [[model category]].

===Verifying commutativity===
Commutativity makes sense for a [[polygon]] of any finite number of sides (including just 1 or 2), and a diagram is commutative if every polygonal subdiagram is commutative.

Note that a diagram may be non-commutative, i.e., the composition of different paths in the diagram may not give the same result.