|
In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
Examples For example, the first isomorphism theorem is a commutative triangle as follows: In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ...
 by me for wiki File links The following pages link to this file: Commutative diagram Categories: GFDL images ...
Since f = h o φ, the left diagram is commutative; and since φ = k o f, so is the right diagram.
 by me for wiki File links The following pages link to this file: Commutative diagram Categories: GFDL images ...
Similarly, the square above is commutative if y o w = z o x.
Verifying commutativity Commutativity makes sense for a polygon of any finite number of sides (including just 1 or 2), and a diagram is commutative if every polygonal subdiagram is commutative.
Diagram chasing Diagram chasing is a method of mathematical proof used especially in homological algebra. Given a commutative diagram, a proof by diagram chasing involves formally using the properties of the diagram, such as injective or surjective maps, or exact sequences. A syllogism is constructed, for which the graphical display of the diagram is just a visual aid. One ends up "chasing" elements around the diagram, until the desired element or result is constructed or verified. In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...
Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
A syllogism (Greek: συλλογισμός — conclusion, inference), more correctly a categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises). ...
Examples of proofs by diagram chasing include those typically given for the five lemma, the snake lemma, the zig-zag lemma, and the nine lemma. In mathematics, especially homological algebra and other applications of Abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. ...
In mathematics, particularly homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology. ...
In mathematics, the nine lemma is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups. ...
External links |
Feeds |
Contact