Today in #Math544 we finally state the Seifert-van Kampen Theorem! We’ve introduced just enough category theory to make the statement extremely natural: the functor π₁ takes certain pushouts in based spaces to pushouts in groups!

#math #topology #categories

Suppose C is a concrete category whose forgetful functor U:C→Set is a right adjoint. The left adjoint F:Set→C is called the free functor and FS is called the free object on S. For instance, if C = Ab is the category of Abelian groups, then FS is the free Abelian group on S.

Pop quiz: Consider the case C=Top. What is the free space on S?

#Math544 #topology #categories

I'm (finally!) introducing #categories and #functors in my #Math544 topology course this term. What are your favorite defamiliarizing-but not-too-crazy examples? I'm thinking in the vein of Mat (natural numbers + matrices) and *//G for cats and, say, diagrams as functors.