Theory of Mathematical Structures

A₁ A₂ Abelian groups adjoint Analogously arbitrary B)-spans B₁ bijection Cartesian product colimits commutes compatible complete lattice complete semilattice concrete congruence coproducts defined Definition denote disjoint element equivalent example exists a unique f₁ f₂ faithful functor fibre-small construct finite forgetful functor free objects full subconstruct functor F given groupoid hence Hint hom-functor hom(A Hom(B homomorphism identity morphisms implies inclusion map indiscrete space initial structure initially complete initially mono-complete Let F linear map map f map ƒ map h metric spaces mono morphism f natural transformation naturally isomorphic numbers one-to-one pair poset preordered set Proof prove Q₂ quotient object Remark semifinal object semifinally complete semilattice set functor small category subobject subset surjective t₁ t₂ Theorem topological space topology unique morphism Vect vector space verify x₁ x₂ y₁ y₂