Commutative diagrams and equality of composition

In summary, a commutative diagram shows that two morphisms, h and k, are equal because their compositions with other morphisms (f and g, respectively) are also equal. This is demonstrated by the diagram for hf = kg, where the commutativity implies that h(f(a)) = k(g(a)) for all elements a in the domain. Despite potential differences in intermediate steps, the overall compositions h∘f and k∘g are equal.
  • #1
Mr Davis 97
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I am a little bit confused on how commutative diagrams show equality of two morphisms. For example, one can imagine the diagram for hf = kg, where composing f and g is the same morphism as composing h and k:
https://upload.wikimedia.org/wikipedia/commons/9/91/Commutative_square.svg

Why does the commutativity of this diagram imply equality of the composition h and f, and k and g? Wouldn't the commutativity just show that hf and kg have the same domain and codomain but are not necessarily the same map?
 
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  • #2
The commutativity works elementwise: ##h(f(a))=k(g(a))\,\forall\,a\in A##. There might happen different things in between (at ##B## and ##C##), but the compositions are equal ##h\circ f = k \circ g##.
 

Related to Commutative diagrams and equality of composition

1. What are commutative diagrams?

Commutative diagrams are visual representations of mathematical relations between objects. They consist of arrows connecting objects, which represent functions or morphisms, and squares or triangles which represent equality or compatibility between compositions of functions.

2. What does it mean for a commutative diagram to be "commutative"?

A commutative diagram is "commutative" if the order in which the functions or morphisms are composed does not affect the result. In other words, if the diagram can be rearranged without changing the relationships between the objects, it is considered commutative.

3. How are commutative diagrams used in mathematics?

Commutative diagrams are used to visualize and simplify mathematical relationships, particularly in the fields of algebra, category theory, and topology. They allow for complex concepts to be represented in a more intuitive and easy-to-understand way.

4. What is the importance of equality of composition in commutative diagrams?

Equality of composition in commutative diagrams is important because it ensures that the relationships between the objects are preserved, regardless of how the functions or morphisms are composed. This allows for consistency and coherence in mathematical arguments and proofs.

5. How can commutative diagrams be applied in real-world situations?

Commutative diagrams can be applied in various real-world situations, such as in computer science, economics, and engineering. They can be used to model and analyze systems, relationships between variables, and processes. They also have applications in data visualization and network analysis.

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