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Book: CategoryTheoryGlossary


See: Category theory

Investigation: Organize the concepts in category theory to reveal underlying themes



This is a glossary of properties and concepts in category theory in mathematics. It is based especially on the following sources:

The notations and the conventions used throughout the article are:


Types of object

Variations of objects

Types of morphism

Types of category

  1. A class of objects,
  2. For each pair of objects X, Y, a set {$\operatorname{Hom}(X, Y)$}, whose elements are called morphisms from X to Y,
  3. For each triple of objects X, Y, Z, a map (called composition)
  4. :{$\circ: \operatorname{Hom}(Y, Z) \times \operatorname{Hom}(X, Y) \to \operatorname{Hom}(X, Z), \, (g, f) \mapsto g \circ f$},
  5. For each object X, an identity morphism {$\operatorname{id}_X \in \operatorname{Hom}(X, X)$} subject to the conditions: for any morphisms {$f: X \to Y$}, {$g: Y \to Z$} and {$h: Z \to W$}, {$(h \circ g) \circ f = h \circ (g \circ f)$} and {$\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f$}.
  1. A class of objects,
  2. For each pair of objects X, Y in D, an object {$\operatorname{Map}_D(X, Y)$} in C, called the mapping object from X to Y,
  3. For each triple of objects X, Y, Z in D, a morphism in C,
  4. :{$\circ: \operatorname{Map}_D(Y, Z) \otimes \operatorname{Map}_D(X, Y) \to \operatorname{Map}_D(X, Z)$},
  5. :called the composition,
  6. For each object X in D, a morphism {$1_X: 1 \to \operatorname{Map}_D(X, X)$} in C, called the unit morphism of X subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.

Variation of categories

Generalizations of a category

Types of functor

Generalizations of a functor

Natural transformations

Types of diagram

Spaces

Theorems

Techniques

Dualities

Areas of math

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