Concept
Morphism
Overview
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not be maps, but they can be composed in a way that is similar to function composition.
Morphisms and objects are constituents of a category. Morphisms, also called maps or arrows, relate two objects called the source and the target of the morphism. There is a partial operation, called composition, on the morphisms of a category that is defined if the target of the first morphism equals the source of the second morphism. The composition of morphisms behaves like function composition (associativity of composition when it is defined, and existence of an identity morphism for every object), and the outcome of the composition is a morphism.
Morphisms and categories recur in much of contemporary mathematics. Originally, they were introduced for homological algebra and algebraic topology. They belong to the foundational tools of Grothendieck's scheme theory, a generalization of algebraic geometry that applies also to algebraic number theory.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
There are nevertheless, in mathematics, some examples of so-called Induction, in which the conclusion does bear the appearance of a generalization grounded on some of the particular cases included in it. A mathematician, when he has calculated a sufficient number of the terms of an algebraical or arithmetical series to have ascertained what is called the _law_ of the series, does not hesitate to fill up any number…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Morphism is.
But this does not mean that mathematics can be created in a purely arbitrary fashion. The mark of its origin is upon it in the need of exhibiting some existing situation through which the non-contradictory character of its postulates can be verified. The real advantage of the generalization is that of all generalizations in science, namely, that by looking away from practical applications (as appears in a historical…
John Dewey, Creative Intelligence
Close in meaning to Morphism; the term itself is not named.
As chemical concepts become more and more those of relationships of interaction in constituting other substances, so physical concepts express more and more relations of operation: mathematical, as expressing functions of dependence and order of grouping; biological, relations of differentiation of descent, effected through adjustment of various environments; and so on through the sphere of the sciences. In short…
John Dewey, How We Think
Close in meaning to Morphism; the term itself is not named.
Related ideas
- Integer
- Isomorphism
- Function (mathematics)
- Algebra
- Mathematics
- Group (mathematics)
- Rational number
- Set (mathematics)
- Topological space
- Bijection
- Map (mathematics)
- Category theory
- Continuous function
- Axiom of choice
- Axiom
- Disjoint sets
Place in the tradition
Part of Linear logic.