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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.
Kushanda muenzaniso · {1,2} union {3}
{1,2} union {3}
Step by step
- 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|.
Kuratidza mhinduro
Zvino iwe Choose a problem, or type or draw your own. Every step, a picture, the answer hidden until you ask.
Mubhadharo wakasununguka unowedzera mazita ezvidzidzo, zvinyorwa zvezvidzidzo zvawakati wazvigadzirira, zvidzidzo zvawave uchizvigadzirisa, uye mudzidzisi waungabvunza nezvechidzidzo ichi. Maths inowanikwa kune vese, vasiri vadzidzi kana vadzidzi. Kuwana mazita ezvidzidzo zvinodiwa kunogona kuitwa pawebsite ye Maths.
Kutanga KupindaMazita anoshandiswa pano
Tinya chero chiratidzo kuti uone chirevo chakazara, vhidhiyo, uye kuti chii chinoreva chero chiratidzo.
Maitiro: Categories, functors and natural transformations
- The two sets.
- Keep everything in either set.
- Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.
Mabvunzo vanhu vanobvunza
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.