# The importance of parallel arrows in a commutative square

I noticed that whenever there is a commutative square, the relation it imposes on parallel morphisms is usually very important (e.g. natural transformations, pullbacks). In contrast, there's usually almost nothing to be said about the consecutive morphisms in these diagrams, even though the equation expressed by the square directly involves their composition. Is there a conceptual explanation for this in general?

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Commutativity says that it does not matter which of the two paths you follow. However, this makes no statement about where your intermediate stops are on the two paths. Saying that going from London via Oslo to Moscow fulfills the same travel purpose as going from London via Rome to Moscow (e.g. you can personally transmit greetings from Buckingham Palace to the Kreml) can hardly say a lot about the relation between Rome and Oslo. In fact, Naples or Stockholm might be just as nice. –  Hagen von Eitzen Dec 6 '12 at 22:52
@HagenvonEitzen you don't say... –  Alexei Averchenko Dec 6 '12 at 22:54
exact sequences are a special type of useful consecutive. –  user51427 Dec 6 '12 at 23:07
@sunflower fixed. –  Alexei Averchenko Dec 6 '12 at 23:51
@HagenvonEitzen Excellent metaphor! –  magma Dec 7 '12 at 8:29

This is an interesting observation , but imho this is a case where there is less than meets the eye. Commutative squares by themselves do not really say much more about the constituent arrows - or pairs of arrows - than their commutative relation already says. In particular your examples - natural transformations and pulbacks/pushouts - derive their interest because of other/extra conditions imposed on the squares.

Take natural transformations: the real noteworthy fact is that you can build a commutative square for all pairs of objects in the domain category. In itself a single square is not that special. It is like saying: a single puzzle tile means nothing, it is not even a rectangle, but taken together in a special way, the whole collection of tiles has the remarkable property of forming a rectangular puzzle.

Same with pulbacks/pushouts: the real interesting thing is the the universal property.

Not convinced yet? Ok, let's dig deeper.

Take a very special commutative square: a commutative triangle.

It can be interpreted as either one of the following commutative squares with identity morphisms:

where $\text{Id}_X^{\mathbb{C}}$ stands for "the identity morphism of object X in category $\mathbf{C}$"

Now: in the first square, f is parallel with an identity - which is a very special morphism: iso, mono, epi... What insight can we get from this? None! f can be anything.

Same story with g in the other diagram.

Conclusion: the importance of parallel arrows in commutative squares? In general...none

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The above diagrams were composed with my package WildCats wildcatsformma.wordpress.com . It is a freely available category theory package for Mathematica from Wolfram Research –  magma Dec 7 '12 at 11:25