Homomorphism v.s. Morphism

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Differences between Homomorphism and Morphism

In mathematics, homomorphisms and morphisms are concepts used to describe structure-preserving maps between algebraic structures, but they are used in slightly different contexts.

Homomorphism

A homomorphism is a specific type of morphism that applies to algebraic structures like groups, rings, or vector spaces. It is a map between two structures of the same type that preserves the operations defined on those structures. For example, if $ f $ is a homomorphism between two groups $ G $ and $ H $, then for all elements $ a $ and $ b $ in $ G $:

$$ f(ab) = f(a)f(b) $$

This means that the operation (like addition or multiplication) in $ G $ is preserved in $ H $ under the map $ f $.

Morphism

A morphism is a more general term used in category theory to describe a map between objects in a category. Categories can include a wide variety of structures, not just algebraic ones. Morphisms must satisfy certain properties defined by the category, but these properties can vary widely. In essence, every homomorphism is a morphism, but not every morphism is a homomorphism.

Key Differences

  1. Scope: Homomorphisms are specific to algebraic structures, while morphisms are used in the broader context of category theory.
  2. Preservation: Homomorphisms specifically preserve algebraic operations, whereas morphisms preserve the structure defined by the category they belong to.
  3. Usage: Homomorphisms are used in fields like group theory, ring theory, and linear algebra. Morphisms are used in category theory, which can encompass many different types of mathematical structures

References