Any time a mathematical concept comes in a “left-handed” and a “right-handed” flavor, I can almost guarantee that I’m going to have trouble remembering which is which. I’ll either have to write it somewhere I’ll see it every day until, after a few months, I pick it up via osmosis, or I’ll have to find some way of relating it to the small number of chiral concepts that I actually understand.
Here’s one such way. We’ll learn to tell the difference between a slice category and a coslice category.
First, recall the definitions. Given a category , and an object in this category, the slice category is the category with
In other words, objects of are objects of equipped with maps to , and a map in is a map in which preserves this additional structure. The coslice category is, of course, the categorical dual.
Now, before we move on, we need a slight generalization of this concept. For some collection of objects of , define the generalized slice category (nonstandard terminology!) to be
So, for instance, objects of are objects of equipped with maps to both and , and morphisms in this category are structure-respecting morphisms from . Generalized coslice categories, then, will be the same thing.
Now we can introduce a pair of facts which will prevent us from ever confusing slice and coslice categories again:
- Products are terminal objects in generalized slice categories;
- Coproducts are terminal objects in generalized coslice categories.
Bonus explanation: having trouble remembering the difference between products and coproducts? Just think of them in the category of sets. A product of sets is the usual cartesian product (together with projection maps) — hence the name — while a coproduct of sets is their disjoint union (together with inclusion maps), which is not something you’d generally call a product.