#the free cyclic object in the category of rings is the structure of integer polynomials in one indeterminate | Explore Tumblr posts and blogs

bubbloquacious · 2 years ago

Text

You can define the ring of integers as the set of group endomorphisms of the free abelian group on one point (the infinite cyclic group, i.e. the integers under addition) under pointwise addition and composition of morphisms. This works for any category with free objects and morphism objects. The reason you can identify the set of endomorphisms with the original set of elements is exactly the universal property of the free object on one point: a morphism is uniquely determined by its action on the generator. It happens that rings are monoid objects in the category of abelian groups (but not in the category of all groups!), so for which monoidal closed categories (monoidal to define monoid objects, closed to define morphism objects) with free objects does this endomorphism construction give you a monoid object?

#math #the free cyclic object in the category of rings is the structure of integer polynomials in one indeterminate #and the additional operation is composition of polynomials #i.e. replace every occurence of x in the first polynomial with the second polynomial #i don't think there's a name for the kind of algebraic structure that this makes #for vector spaces over a given field you get back that field #for sets and topological spaces (Top does not have all morphism objects but it does have one for the one point space)#you just get the trivial monoid (and the trivial topological monoid!)

16 notes · View notes