The Unapologetic Mathematician

Mathematics for the interested outsider

Universal arrows and universal elements

Remember that we defined a colimit as an initial object in the comma category (F\downarrow\Delta), and a limit as a terminal object in the comma category (\Delta\downarrow F). It turns out that initial and terminal objects in more general comma categories are also useful.

So, say we have a categories \mathcal{C} and \mathcal{D}, an object C\in\mathcal{C}, and a functor F:\mathcal{D}\rightarrow\mathcal{C}, so we can set up the category (C\downarrow F). Recall that an object of this category consists of an object D\in\mathcal{D} and an arrow f:C\rightarrow F(D), and a morphism from f_1:C\rightarrow F(D_1) to f_2:C\rightarrow F(D_2) is an arrow d:D_1\rightarrow D_2 such that f_2=F(d)\circ f_1.

Now an initial object in this category is an object U of \mathcal{D} and an arrow u:C\rightarrow F(U) so that for any other arrow f:C\rightarrow F(D) there is a unique arrow d:U\rightarrow D satisfying f=F(d)\circ u. We call such an object, when it exists, a “universal arrow from C to F“. For example, a colimit of a functor F is a universal arrow (in \mathcal{C}^\mathcal{J}) from F to \Delta.

Dually, a terminal object in (F \downarrow C) consists of an object U\in\mathcal{D} and an arrow u:F(U)\rightarrow C so that for any other arrow f:F(D)\rightarrow C there exists a unique arrow d:D\rightarrow U satisfying f=u\circ F(d). We call this a “couniversal arrow from F to C“. A limit of a functor F is a couniversal arrow from \Delta to F.

Here’s another example of a universal arrow we’ve seen before: let’s say we have an integral domain D and the forgetful functor U:\mathbf{Field}\rightarrow\mathbf{Dom} from fields to integral domains. That is, U takes a field and “forgets” that it’s a field, remembering only that it’s a ring with no zerodivisors. An object of (D\rightarrow U) consists of a field F and a homomorphism of integral domains D\rightarrow U(F). The universal such arrow is the field of fractions of D.

As a more theoretical example, we can consider a functor F:\mathcal{C}\rightarrow\mathbf{Set} and the comma category (*\downarrow F). Since an arrow x:*\rightarrow X in \mathbf{Set} is an element of the set X, we call a universal arrow from * to F a “universal element” of F. It consists of an object U\in\mathcal{C} and an element u\in F(U) so that for any other element c\in F(C) we have a unique arrow f:U\rightarrow C with \left[F(f)\right](u)=c.

As an example, let’s take N to be a normal subgroup of a group G. For any group H we can consider the set of group homomorphisms f:G\rightarrow H that send every element of N to the identity element of H. Verify for yourself that this gives a functor F:\mathbf{Grp}\rightarrow\mathbf{Set}. Any such homomorphism factors through the projection \pi_{(G,N)}:G\rightarrow G/N, so the group G/N and the projection \pi_{(G,N)} constitute a universal element of F. We used cosets of N to show that such a universal element exists, but everything after that follows from the universal property.

Another example: let M_1 be a right module over a ring R, and M_2 be a left R-module. Now for any abelian group A we can consider the set of all R-middle-linear functions from M_1\times M_2 to A — functions f which satisfy f(m_1+m_1',m_2)=f(m_1,m_2)+f(m_1',m_2), f(m_1,m_2+m_2')=f(m_1,m_2)+f(m_1,m_2'), and f(m_1r,m_2)=f(m_1,rm_2). This gives a functor from \mathbf{Ab} to \mathbf{Set}, and the universal element is the function M_1\times M_2\rightarrow M_1\otimes_RM_2 to the tensor product.

The concept of a universal element is a special case of a universal arrow. Very interestingly, though, when \mathcal{C} is locally small (as it usually is for us)$ the reverse is true. Indeed, if we’re considering an object C\in\mathcal{C} and a functor F:\mathcal{D}\rightarrow\mathcal{C} then we could instead consider the set * and the functor \hom_\mathcal{C}(C,F(\underline{\hphantom{X}})):\mathcal{D}\rightarrow\mathbf{Set}. Universal arrows for the former setup are equivalent to universal elements in the latter.

June 25, 2007 Posted by | Category theory | 2 Comments