Diagrams consisting of triangles and squares - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:22:16Z http://mathoverflow.net/feeds/question/59243 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59243/diagrams-consisting-of-triangles-and-squares Diagrams consisting of triangles and squares Dmitry K 2011-03-22T22:35:02Z 2011-03-22T23:18:14Z <p>S. Lang gives a statement on page x of his 'Algebra':</p> <blockquote> <p>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.</p> </blockquote> <p>If we want to prove this statement the problem arises of defining precisely what 'consisting of squares and triangles' means.</p> <p>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)).</p> <p>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?</p> http://mathoverflow.net/questions/59243/diagrams-consisting-of-triangles-and-squares/59249#59249 Answer by Martin Brandenburg for Diagrams consisting of triangles and squares Martin Brandenburg 2011-03-22T23:06:41Z 2011-03-22T23:06:41Z <p>Commutative diagrams visualize calculations of the form</p> <p>\$f_1 * ... * f_n = g_1 * ... * g_m\$</p> <p>of morphisms in the given category. A decomposition of the diagram corresponds to a chain of equations leading to the above equation. A triangle corresponds to an equation of the type \$f_1 f_2 = g_1\$ and a square to an equation of the type \$f_1 f_2 = g_1 g_2\$.</p> http://mathoverflow.net/questions/59243/diagrams-consisting-of-triangles-and-squares/59251#59251 Answer by Gjergji Zaimi for Diagrams consisting of triangles and squares Gjergji Zaimi 2011-03-22T23:18:14Z 2011-03-22T23:18:14Z <p>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.</p> <p>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.</p>