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Proofs for the exercises for Lawvere and Schanuel's Conceptual Mathematics

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Conceptual Mathematics

Curriculum

  • Week 1
    • Article 2.i: Isomorphisms - isomorphisms
  • Week 2
    • Article 2.ii: Isomorphisms - choice and determination
    • Article 2.iii: Isomorphisms - retracts, sections, and idempotents
  • Week 3
    • Article 2.iv: Isomorphisms - isomorphisms and automorphisms
    • Discussion:
      • Every section has a retraction and every retraction has a section
      • Sections and retractions are two parts of an inverse
      • Every section maps from a smaller object to a larger object (embeds)
      • Every retraction maps from a larger object to a smaller object (exemplifies)
      • If a function f has a section s and a retraction r then r = s; f has an inverse which is both the section and retraction f-1 = r = s
      • Isomorphisms give us a notion of equal cardinality, we can use this to prove that for A -f-> B there are as many isomorphisms f as there are automorphisms on A
    • Day 2
      • Categories can pack a lot more structure than SET can! Eg, PERM requires a big commuting square for all its morphisms, and these are quite restrictive. It took us 30 minutes to find a PERM morphism!
      • A permutation morphism needs to preserve cycles. For every cycle in the domain, there must be a cycle of the same length in the codomain.
      • The embed / exemplify metaphor for sections and retractions are a really good intuition! They hold in PERM too!

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