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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.
Kugwira ntchito chitsanzo · lcm(4, 6)
Gawo ndi Gawo
- \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.
Kusonyeza yankho
Tsopano inu Sankhani vuto, kapena kulemba kapena kujambula wanu. Pakalipano, yankho limeneli lidzapezeka patsamba lino.
M’malo mwake, maakaunti aulere amawonjezera malemba pazophunzira zonse, mnda wa zomwe mwamaliza, mavuto anu omaliza m’malo limodzi, ndi m’bale amene mungafunse za nkhaniyo. Maphunziro a matekinoloje ndi otsegulira kwa aliyense, olembetsa kapena osalembetsa.
Kulembetsa KulowaZithunzi
Pitani pa dzina lililonse lachifaniziro kuti mudziwe tanthauzo lake, chithunzi chake, ndi zimene limatanthauza.
Momwe mungagwiritsire ntchito: 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.
Mafunso omwe anthu amafunsa
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.