This series covers the basics of Enriched Category Theory. You can also find these videos organized as a playlist on YouTube.
I give the definition for a (symmetric) monoidal category, followed by a series of examples. These include the category of sets with the cartesian product, the category of modules over a commutative ring, and the category of endofunctors on a category C.
I define closed symmetric monoidal categories and give examples. These are symmetric monoidal categories that have internal hom objects, defined to be right adjoint to the monoidal product.
In this video, I motivate the definition of an enriched category by showing how the on-elements definition of a regular category can be generalized using set-maps.
I give the definition of an enriched category.
I illustrate the concept of an enriched category with several examples. These include one-object Ab-categories (which turn out to be rings), Bool-categories (which turn out to be preorders), and Cost-categories (which generalize metric spaces).
I define enriched functors and show how this definition reduces to that of a usual functor when we enrich over the category of sets.
I cover several examples of enriched functors. These include Cat-functors, Ab-functors, Bool-functors, and Cost-functors. We repeatedly see how the concept of enrichment yields familiar notions.
I define enriched natural transformations and show that an enriched transformation between two Ab-functors can be interpreted as an R-module homomorphism.