Last Updated: September 02, 2019
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· beezee

# Relating commutativity of binary operators and diagrams

Does that imply relating binary operators and diagrams more generally?

If a diagram commutes we can say g.f = h.

If I am chasing a diagram in a category and arrive at a limit of a sub-diagram, I have to choose a path. This can be done using a semigroup. If the pathways are commutative, then I can use a trivial semigroup, and the operation commutes.

``a = b -> fst (a, b) = fst (b, a)``

In some sense, saying that a diagram commutes is a weaker statement than saying two paths are equal, but strictly in the context of choosing a path using a free semigroup, commutativity and equality are one and the same.

Is equality an initial object in some category of binary relations? Is it a free construction in a category of commutative binary relations?

### Some category theory

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#### 1 Response

also are free constructions iff initial objects?

over 1 year ago ·

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