S. Lang gives a statement on page x of his 'Algebra':



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> Most of our diagrams are composed of triangles and squares as above, and to verify that a diagram consisting of triangles and squares is commutative it suffices to verify that each triangle and square in it is commutative.
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If we want to prove this statement the problem arises of defining precisely what 'consisting of squares and triangles' means.


The most obvious definitions of such diagrams (every vertex (arrow) belongs to a triangle\square) turn out to be unsatisfactory (the Lang's statement is wrong then, think of a pentagonal diagram with a commutative triangle on each side (all suitably oriented)).


How could the intuitive notion of a 'diagram consisting of squares and triangles' be strictly formulated such that the Lang's statement is always true?



Diagrams consisting of triangles and squares
Think of a diagram as a planar directed graph with an equivalence relation on directed paths which respects concatenations of paths. A commutative diagram is one where this equivalence relation identifies all paths with common sources and sinks.


The remark above says that to check that a diagram is commutative it is enough to check that each face is a commutative diagram. This is a simple combinatorial fact that you can show by induction on the number of faces.


