Universal arrows and universal elements
Remember that we defined a colimit as an initial object in the comma category , and a limit as a terminal object in the comma category . It turns out that initial and terminal objects in more general comma categories are also useful.
So, say we have a categories and , an object , and a functor , so we can set up the category . Recall that an object of this category consists of an object and an arrow , and a morphism from to is an arrow such that .
Now an initial object in this category is an object of and an arrow so that for any other arrow there is a unique arrow satisfying . We call such an object, when it exists, a “universal arrow from to “. For example, a colimit of a functor is a universal arrow (in ) from to .
Dually, a terminal object in consists of an object and an arrow so that for any other arrow there exists a unique arrow satisfying . We call this a “couniversal arrow from to “. A limit of a functor is a couniversal arrow from to .
Here’s another example of a universal arrow we’ve seen before: let’s say we have an integral domain and the forgetful functor from fields to integral domains. That is, takes a field and “forgets” that it’s a field, remembering only that it’s a ring with no zerodivisors. An object of consists of a field and a homomorphism of integral domains . The universal such arrow is the field of fractions of .
As a more theoretical example, we can consider a functor and the comma category . Since an arrow in is an element of the set , we call a universal arrow from to a “universal element” of . It consists of an object and an element so that for any other element we have a unique arrow with .
As an example, let’s take to be a normal subgroup of a group . For any group we can consider the set of group homomorphisms that send every element of to the identity element of . Verify for yourself that this gives a functor . Any such homomorphism factors through the projection , so the group and the projection constitute a universal element of . We used cosets of to show that such a universal element exists, but everything after that follows from the universal property.
Another example: let be a right module over a ring , and be a left -module. Now for any abelian group we can consider the set of all -middle-linear functions from to — functions which satisfy , , and . This gives a functor from to , and the universal element is the function to the tensor product.
The concept of a universal element is a special case of a universal arrow. Very interestingly, though, when is locally small (as it usually is for us)$ the reverse is true. Indeed, if we’re considering an object and a functor then we could instead consider the set and the functor . Universal arrows for the former setup are equivalent to universal elements in the latter.