Friday, Oct 13, 2017 - 3:30pm - Allen 411
Categorical Isomorphisms, Initial and Terminal objects, Illustrations and Applications
Dr. Paul Fabel, Mathematics, Msstate
Title: Categorical Isomorphisms, Initial and Terminal objects, Illustrations and Applications
Abstract: Recall a category is a `universe' comprised of objects (dots) and morphisms (you can think of a morphism as an arrow with a unique initial object and a unique target object), obeying very simple rules:
1) Composition of morphisms is defined when two arrows meet tip to tail.
2) Composition of morphisms is associative.
3) Each object partners with an identity morphism, comprising the left and right identities of the all the morphisms.
You may have noticed in group theory that there is a unique homomorphism from an arbitrary group G to a trivial group, and also a unique homomorphism from a trivial group to G. Can you think of another group with these mentioned properties? What can you say about two `different' trivial groups? Why in practice can we say `the trivial group' rather than `a trivial group', and rarely cause confusion? The previous questions motivate the categorical notions of `terminal object' `initial object', `isomorphism' and `unique up to unique isomorphism.'