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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.
उदाहरण · {1,2} union {3}
{1,2} union {3}
चरणद्वारा चरण
- 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|.
जवाफ प्रकट गर्नुहोस्
अब तपाईँ समस्या रोज्नुहोस्, वा टाइप गर्नुहोस् वा तपाईँको आफ्नै रेखाचित्र गर्नुहोस् । प्रत्येक चरण, एउटा तस्वीर, तपाईँले सोध्नु अघि उत्तर लुकेको छ ।
एक निःशुल्क खाता प्रत्येक पाठ मा नोट थप्छ, तपाईं के समाप्त गरेको छ को एक रेकर्ड, एक ठाउँमा आफ्नो समाधान समस्या, र एक शिक्षक तपाईं यो पृष्ठ बारेमा सोध्न सक्नुहुन्छ. गणित आफैलाई सबैलाई खुला छ, मा साइन इन वा छैन.
साइन अप गर्नुहोस् लगइनयहाँ प्रयोग गरिएको प्रतीक
पूर्ण परिभाषा, एउटा तस्वीर, र यो प्रत्येक अक्षर अर्थ के लागि कुनै पनि प्रतीक ट्याप गर्नुहोस्।
कसरी: Categories, functors and natural transformations
- The two sets.
- Keep everything in either set.
- Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.
प्रश्नहरू मानिसहरूले सोध्छन्
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.