maths.free › Category Theory › Categories, functors and natural transformations

Categories, functors and natural transformations

Objects and arrows, composition, and maps between categories.

A category is objects plus arrows that compose associatively with identities: sets and functions, groups and homomorphisms, spaces and continuous maps. A functor maps one category to another preserving composition. Picture it: a commutative diagram — a square of arrows where both routes agree. Think it: "naturality" makes precise what it means for a construction to involve no arbitrary choices.

مثال عملي: {1,2} union {3}

Set-Theory · Sets

{1,2} union {3}

خطوة خطوة

  1. A = \left\{1, 2\right\},\quad B = \left\{3\right\}

    The two sets.

  2. A \cup B = \left\{1, 2, 3\right\}

    Keep everything in either set.

  3. |A| = 2,\ |B| = 1,\ |A \cup B| = 3

    Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.

كشفت الإجابة
A \cup B = \left\{1, 2, 3\right\}

Symbols used here

A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
F: \mathcal{C} \to \mathcal{D},\ \eta: F \Rightarrow G
functor, natural transformation
A structure-preserving map between categories; a map between functors.
\operatorname{Hom}(A, B),\ f \circ g
arrows from A to B, composition
The set of morphisms; do g then f.

How to: Categories, functors and natural transformations

  1. The two sets.
  2. Keep everything in either set.
  3. Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.

Questions people ask

What is a functor?

A map between categories that sends objects to objects and arrows to arrows while respecting composition. Taking the fundamental group of a space is a functor from spaces to groups.

جرّب نفسك

أكثر في Category Theory

Universal properties, limits and adjunctions