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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · {1,2} union {3}

Set-Theory · Sets

{1,2} union {3}

Àwọn ìṣàmúlò-ètò

A = \left\{1, 2\right\},\quad B = \left\{3\right\}
The two sets.
A \cup B = \left\{1, 2, 3\right\}
Keep everything in either set.
|A| = 2,\ |B| = 1,\ |A \cup B| = 3
Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.

Fi àwọn àgbèwọlé hàn

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

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Bii o ṣe le: Categories, functors and natural transformations

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

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Category Theory

Universal properties, limits and adjunctions