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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.

Nhazi ahụ ejirila · {1,2} union {3}

Set-Theory · Sets

{1,2} union {3}

Nzọụkwụ site n'ụdị

  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|.

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

Ugbu a ị Họrọ nsogbu, mọọbụ tinye mọọbụ depụta gị. Nzọụkwụ ọbụla, inyogo, azịza ahụ echekwara ruo mgbe ịjụla.

Kwụsị ọrụ gị

Akwụkwọ ndebanye n'efu na-egbakwunye ntọala n'ụdị ọmụmụ ihe ọbụla, ndepụta nke ihe ị ga-eme, nsogbu ị ga-ewepụ n'otu ebe, na onye nkụzi ị ga-ajụ banyere ihuakwụkwọ a. Akwụkwọ ndepụta ihe bụ nke ọbụla mepere maka onye ọbụla, ọbụla ma ọ bụ na ọ bụghị.

Akaụntụ Nbanye

Aịkọn ndị ahụ ejirila ebe a

Tinye ọbụla simbụl maka nkọwa zuru ezu, inyogo, na ihe ọbụla ọbụla na-egosi.

Olee otú: Categories, functors and natural transformations

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

Ajụjụ ndị mmadụ jụrụ

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.

Oge Category Theory

Universal properties, limits and adjunctions