# "Universal maps" as a universal property

In the question Fixed points and universal maps for posets, we find the following definition: if $P, Q$ are partially ordered sets, an order-preserving map $u:P\to Q$ is said to be universal if for every order-preserving map $f:P\to Q$ there is $p\in P$ such that $f(p) = u(p)$.

This definition is not poset-specific; it makes sense in a lot of other categories such as $\mathbf{Top}$.

Can we decribe the essence of a universal map in the language of category theory? Is there a universal property that captures universal maps?

• Hmm, so in the example, this just says that the equalizer of $u$ with any parallel arrow is non-empty (i.e. not initial). Not sure if that can be rephrased as some sort of universal property. Dec 10 '14 at 13:08
• Generally a universal property defines an object up to isomorphism, so I think the answer to your second question is 'no'. Dec 10 '14 at 13:08
• @TobiasKildetoft Not quite. It says that the equalizer $E$ of $u$ with any other arrow admits a map $1 \to E$ from the terminal object, which is very different from being non-initial. Dec 10 '14 at 13:10
• @ToddTrimble I was thinking specifically in $\mathbf{Top}$. Dec 10 '14 at 13:13
• I was about to ask if there were any interesting examples of such maps in $\mathbf{Top}$ when I realized that many fixed point theorems are simply statements that the identity is universal in this sense. Dec 10 '14 at 13:29

DEFINITION.   A morphism $\ u: Y\rightarrow X\$ in category $\ K\$ is universal $\ \Leftarrow:\Rightarrow\$ for any morphism $\ f: Y\rightarrow X\$ there is an object $\ Z\$ in $\ K\$ and a morphism $\ p:Z\rightarrow Y\$ such that $\ g\circ p=u\circ p.\$ Object X is stable (or has the fixed morphism property; analogous to the fixed-point property) if the identity $\ i_X:X\rightarrow X\$ is a universal morphism.
• There are topological theorems but only one non-obvious general categorical theorem (from late 1960s): Let the finite categorical product $\ \prod_{k=1}^n f_n \$ of morphisms $\ f:X_{k-1}\rightarrow X_K\$ be well-defined and universal. Then $\ f_n\circ\ldots\circ f_1 : X_0\rightarrow X_n\$ is universal too.  (This gives examples of universal maps of finite polyhedra which have non-universal product, already in dimension 2). I have a feeling thought that I can generalize the main theorem of the paper mentioned above to address all categories.
• Categories with exactly one object are virtually monoids. Thus in the case of monoids we can talk about universal elements rather than universal morphisms. When $\ 1\$ is universal then we can say that the monoid itself has the fixed-morphism property. The whole theory here is peculiar since there is only one object. If $\ 1\$ is not universal then no element is.