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Universal properties, limits and adjunctions
Products, coproducts, limits and the adjunctions that unify constructions.
A product is defined by what maps into it, not by what it is made of; the same shape of definition gives limits, colimits and free objects. Adjoint functors pair a "free" construction with a "forgetful" one. Picture it: the lcm of 4 and 6 is the least common multiple in the divisibility order, a categorical join. Think it: Yoneda: an object is determined by the arrows into it.
उदाहरण · lcm(4, 6)
चरणद्वारा चरण
- \gcd(4, 6)
Use Euclid's algorithm: replace the larger number by its remainder on division by the smaller, until the remainder is 0.
- 6 = 1 \times 4 + 2
Divide 6 by 4: quotient 1, remainder 2.
- 4 = 2 \times 2 + 0
Divide 4 by 2: quotient 2, remainder 0.
- \gcd(4, 6) = 2
Euclid gives the gcd.
- \operatorname{lcm}(4, 6) = \frac{4 \times 6}{\gcd(4, 6)} = \frac{24}{2} = 12
lcm × gcd = a × b, so divide the product by the gcd.
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: Universal properties, limits and adjunctions
- Use Euclid's algorithm: replace the larger number by its remainder on division by the smaller, until the remainder is 0.
- Divide 6 by 4: quotient 1, remainder 2.
- Divide 4 by 2: quotient 2, remainder 0.
- Euclid gives the gcd.
- lcm × gcd = a × b, so divide the product by the gcd.
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.