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!