PlanetPhysics/Commutative Square Diagram

A [[../PiecewiseLinear/|square]] commutative diagram (as for example in an [[../AbelianCategory2/|abelian category]] Failed to parse (unknown function "\A"): {\displaystyle {\A}} ): Failed to parse (unknown function "\begin{xy}"): {\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\\ {C}\ar[r]_{h}&{D} } }\end{xy}} is called commutative iff $g\circ f=h\circ k,$ where $A,B,C$, and $D$ are [[../TrivialGroupoid/|objects]] of a [[../Cod/|category]] $𝒞$, and $f,g,h$ and $k$ are, in general, arrows or "morphisms" (mappings, [[../Bijective/|functions]], [[../TrivialGroupoid/|homomorphisms]], [[../TrivialGroupoid/|homeomorphisms]], and so on) of $𝒞$.

One can intuitively understand commutativity as the equivalence of the two [[../TrivialGroupoid/|morphism]] paths involved, or as an internal, mirror-like symmetry property of the square diagram with respect to the top-right to bottom-left diagonal. The diagonal morphism, $d:A\to D$ (not shown) is thus equal to both $g\circ f$ and $h\circ k$. The [[../PreciseIdea/|concept]] of commutative diagram can be thus generalized for any polyhedron with "diagonal mirror symmetry" of morphisms oriented in the same direction of the [[../Bijective/|type]] described for the square diagram shown above.