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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.
Үйлдэл · lcm(4, 6)
Алхам алхмаар
- \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.
Хариулт
Одоо та Даалгавар сонго, эсвэл бич, эсвэл зур. Бүх алхам, зураг, хариулт нь асуулт асуух хүртэл нуугдаж байдаг.
Үнэгүй бүртгэл нь хичээл бүрийн тэмдэглэл, төгссөн хичээл, шийдсэн даалгаврын тэмдэглэл, энэ хуудсан дээр асууж болох багшийг нэгтгэж өгдөг. Математик нь бүртгэлтэй, бүртгэлтэйгүй хэн бүхэнд нээлттэй.
Бүртгүүлэх НэвтрэхЭнд ашигласан тэмдэгтүүд
Бүх тодорхойлолт, зураг, үсгийн утгыг харахын тулд ямар ч тэмдэгтийг дарна уу.
Яаж: 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.
Хүмүүс асуудаг асуултууд
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.