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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.
Eżempju maħdum · {1,2} union {3}
{1,2} union {3}
Pass b'pass
- 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|.
Jiżvelaw it-tweġiba
Issa inti Agħżel problema, jew ittajpja jew iġbed tiegħek stess.Kull pass, stampa, it-tweġiba moħbija sakemm inti tistaqsi.
Kont b'xejn iżid noti fuq kull lezzjoni, rekord ta' dak li lest, problemi solvuti tiegħek f'post wieħed, u għalliem inti tista' tistaqsi dwar din il-paġna.Il-matematika nnifisha hija miftuħa għal kulħadd, log fil jew le.
Irreġistra issa LoginSimboli użati hawnhekk
Taptap kwalunkwe simbolu għad-definizzjoni sħiħa, stampa, u dak li kull ittra fiha tfisser.
Kif għandek tuża: Categories, functors and natural transformations
- The two sets.
- Keep everything in either set.
- Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.
Mistoqsijiet li n-nies jistaqsu
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.