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.

Isibonelo esisebenza · {1,2} union {3}

Set-Theory · Sets

{1,2} union {3}

Isigaba

  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|.

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

Manje Khetha izinkinga, noma ubhale noma udwebe zakho. Isigaba ngasinye, isithombe, impendulo ifihlakele kuze kube ubuza.

Qhubeka usebenza

I-akhawunti emahhala ifaka iziphawulo kulesifundo ngasinye, irekhodi lokuthi yini oqedile, izinkinga zakho ezixazululiwe endaweni eyodwa, nodokotela owaziyo ongabuza mayelana nalekhasi. I-matematics ivulekile kubo bonke, noma ngabe babhalise noma cha.

Bhala Ngena

Amaphawu asetshenziswa lapha

Chofoza noma yiluphi uphawu lwesibonakaliso ukuze uthole ukuchaza okuphelele, isithombe, kanye nokuthi iyiphi inhlamvu ekhona ekhuluma ngaso.

Indlela: Categories, functors and natural transformations

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

Imibuzo abantu bebuza

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.

Okuningi Category Theory

Universal properties, limits and adjunctions