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commutative square diagram (Definition)
Definition 0.1   A square commutative diagram (as for example in an Abelian category ${\mathcal A}$):

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\ {C}\ar[r]_{h}&{D} } } \end{xy}$
is called commutative iff

$\displaystyle g\circ f = h\circ k ,$
where $A, B, C$, and $D$ are objects of a category $\mathcal{C}$, and $f,g,h$ and $k$ are, in general, arrows or “morphisms” (mappings, functions, homomorphisms, homeomorphisms, and so on) of $\mathcal{C}$.
Remark 0.1   One can intuitively understand commutativity as the equivalence of the two 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 concept of commutative diagram can be thus generalized for any polyhedron with “diagonal mirror symmetry” of morphisms oriented in the same direction of the type described for the square diagram shown above.



"commutative square diagram" is owned by bci1.

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See Also: category

Also defines:  diagram, square diagram, commutativity, commutative diagram
Keywords:  Abelian category, morphisms, homeomorphisms, category, homomorphisms

Cross-references: type, concept, morphism, homeomorphisms, homomorphisms, functions, category, objects, Abelian category, square
There are 30 references to this object.

This is version 6 of commutative square diagram, born on 2009-02-13, modified 2009-05-29.
Object id is 516, canonical name is CommutativeSquareDiagram.
Accessed 1799 times total.

Classification:
Physics Classification: 00. (GENERAL)
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
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