scratch

§ Category Where Coproducts of Computable Things Is Not Computable

created 2022-05-30 · last edited 2023-04-02
  • Modular lattices are an algebraic variety.
  • Consider the category of modular latties.
  • The free modular lattice on 2 elements and on 3 elements has dediable equality, by virtue of being finite.
  • The free modular lattice on 5 elements does not have decidable equality.
  • The coproduct of free modular lattice on 2 and 3 generators is the free modular lattice on 5 generators, because F(2∪3)=F(2)⊔F(3)F(2 \cup 3) = F(2) \sqcup F(3)F(2∪3)=F(2)⊔F(3) (where 2,32, 32,3 are two and three element sets), because free is left adjoint to forgetful, and the left adjoint FFF preserve colimits!
❦
Newer ৪ Blog ৪ Older