Commutative diagram

In mathematics , a commutative diagram describes that different chains of figures produce the same result.

An illustration from to can be represented by an arrow. ${\ displaystyle f}$ ${\ displaystyle A}$ ${\ displaystyle B}$

The concatenation with another figure from to can be expressed by connecting the arrows. Such a chain of arrows is called a diagram . ${\ displaystyle g}$ ${\ displaystyle B}$ ${\ displaystyle C}$

If you want to give this chain a name, you can draw another arrow from to . ${\ displaystyle h}$ ${\ displaystyle A}$ ${\ displaystyle C}$

It would be conceivable that any mapping is from to . If it matches the concatenation , the diagram is said to be commuting . ${\ displaystyle h}$ ${\ displaystyle A}$ ${\ displaystyle C}$ ${\ displaystyle g \ circ f}$

In general, for a diagram to commute, the linkages of the associated images must match for all paths from to . ${\ displaystyle X}$ ${\ displaystyle Y}$

In short: a diagram commutes "if it doesn't matter which way you choose".

Examples

This diagram commutes exactly if and applies. These are exactly the conditions that make the mapping too inverse . ${\ displaystyle f ^ {- 1} \ circ f = \ mathrm {id}}$ ${\ displaystyle f \ circ f ^ {- 1} = \ mathrm {id}}$ ${\ displaystyle f ^ {- 1}}$ ${\ displaystyle f}$

${\ displaystyle \ mu}$ in this diagram denotes the multiplication, that is . The diagram commutes exactly when it is true, i.e. it expresses the associative law of the multiplication of real numbers. ${\ displaystyle \ mu (x, y) = xy}$ ${\ displaystyle x (yz) = (xy) z}$

Chart hunting

The diagram of hunting ( engl. Diagram chasing ) is a proof method, particularly in the homological algebra is used. Formal properties of mappings ( e.g. injectivity , surjectivity or exactness ) are used on the basis of a given commutative diagram . Here, elements of the objects are “chased” through the diagram in various ways and the results obtained are compared. The diagram serves only as an aid to visualize a formally valid proof even without this.

Examples of chart hunts are the usual evidences of the five lemma , the snake lemma , the zigzag lemma, or the nine lemma .

Note that a proof by diagram hunt is only directly valid in categories whose objects are sets (with additional structure) and whose morphisms are certain mappings between these sets, which are linked as usual by executing them one after the other, etc. For more general categories one can either use the embedding theorem of Mitchell , who allows every (small) Abelian category to be understood as such a concrete category of modules , or instead of elements to use equivalence classes of morphisms with the corresponding goal; the calculation rules are the same as for elements.

If you use diagram hunting to construct images, these are generally "natural": If you have two copies of the diagram, but with different objects and homomorphisms, and a homomorphism between these diagrams (i.e. homomorphisms from all objects of the one diagram to the corresponding object of the second diagram in such a way that all the resulting meshes are commutative), then the two constructed mappings with these homomorphisms will also commute.

Web links

Commons : Commutative diagrams - collection of images, videos and audio files

literature

Saunders Mac Lane : Homology , Springer-Verlag (2008), ISBN 3-5405-8662-8 , Chapter 1, §3: "Diagrams" (English)