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Commutative Diagram


A commutative diagram is a collection of maps A_i-->^(phi_i)B_i in which all map compositions starting from the same set A and ending with the same set B give the same result. In symbols this means that, whenever one can form two sequences

 A=A_(i_0)-->^(phi_(i_0))B_(i_0)=A_(i_1)-->^(phi_(i_1))B_(i_1)=A_(i_2)-->^(phi_(i_2))...-->^(phi_(i_(n-1)))B_(i_(n-1))=A_(i_n)-->^(phi_(i_n))B_(i_n)=B,
(1)

and

 A=A_(j_0)-->^(phi_(j_0))B_(j_0)=A_(j_1)-->^(phi_(j_1))B_(j_1)=A_(j_2)-->^(phi_(j_2))...-->^(phi_(i_(m-1)))B_(i_(m-1))=A_(i_m)-->^(phi_(i_m))B_(i_m)=B,
(2)

the following equality holds:

 phi_(i_n) degreesphi_(i_(n-1)) degrees... degreesphi_(i_1) degreesphi_(i_0)=phi_(j_m) degreesphi_(j_(m-1)) degrees... degreesphi_(j_1) degreesphi_(j_0).
(3)
CommutativeTriangle
CommutativeSquare

Commutative diagrams are usually composed by commutative triangles and commutative squares.

CommutativePlane
CommutativeCube

Commutative triangles and squares can also be combined to form plane figures or space arrangements.

CommutativeMultiarrow

A commutative diagram can also contain multiple arrows that indicate different maps between the same two sets.

CommutativeLoopedArrow

A looped arrow indicates a map from a set to itself.

CommutativeInverseMap

The above commutative diagram expresses the fact that g is the inverse map to f, since it is a pictorial translation of the map equalities g degreesf=id_A and f degreesg=id_B.

CommutativeInverseMap2

This can also be represented using two separate diagrams.

Many other mathematical concepts and properties, especially in algebraic topology, homological algebra, and category theory, can be formulated in terms of commutative diagrams.

CommutativeModuleHomomorphism

For example, a module M is projective iff any surjective module homomorphism U-->^sV and any module homomorphism M-->^phiV can be completed to a commutative diagram.

CommutativeInjectiveModule

Similarly, one can characterize the dual notion of injective module: a module M is injective iff any injective module homomorphism and U-->^iV and any module homomorphism U-->^phiM can be completed to a commutative triangle.

According to Baer's criterion, it is sufficient to require this condition for the inclusion maps i of the ideals of R in R.

CommutativeChainHomomorphism

Another example of a notion based on diagrams is the chain homomorphism, which can be visualized as a sequence of commutative squares.

The advantage of drawing commutative diagrams is the possibility to seize any given map configuration at a glance. The picture also facilitates the task of composing maps, which is like following directed paths from set to set. Many homological theorems are proven by studying commutative diagrams: this method is usually referred to as "diagram chasing."


See also

Coequalizer, Diagram Chasing, Diagram Lemma, Eight Lemma, Equalizer, Five Lemma, Four Lemma, Injective Module, Nine Lemma, Projective Module, Pullback Map, Snake Lemma, Split Exact Sequence, Zig-Zag Lemma

This entry contributed by Margherita Barile

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References

Bourbaki, N. "Diagrammes commutatifs." §1.1 in Algèbre. Chap. 10, Algèbre Homologique. Paris, France: Masson, 1-3, 1980.Cartan H. and Eilenberg, S. Homological Algebra. Princeton, NJ: Princeton University Press, 1956.Davis, J. F. and Kirk, P. Lecture Notes in Algebraic Topology. Providence, RI: Amer. Math. Soc., 2001.Eilenberg, S. and Steenrod, N. Foundations of Algebraic Topology. Princeton, NJ: Princeton University Press, 1952.Herrlich, H. and Strecker, G. E. Category Theory: An Introduction. Boston, MA: Allyn and Bacon, 1973.Hilton, P. J. and Stammbach, U. A Course in Homological Algebra, 2nd ed. New York: Springer-Verlag, 1997.Lang, S. Algebra, rev. 3rd ed. New York: Springer-Verlag, 2002.Mac Lane, S. Categories for the Working Mathematician. New York: Springer-Verlag, 1971.Mac Lane, S. Homology. Berlin: Springer-Verlag, 1967.Mitchell, B. Theory of Categories. New York: Academic Press, 1965.Northcott, D. G. An Introduction to Homological Algebra. Cambridge, England: Cambridge University Press, 1966.Rotman, J. J. An Introduction to Algebraic Topology. New York: Springer-Verlag, 1988.Scott Osborne, M. Basic Homological Algebra. New York: Springer-Verlag, 2000.

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Commutative Diagram

Cite this as:

Barile, Margherita. "Commutative Diagram." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CommutativeDiagram.html

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