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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?

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  • $\begingroup$ 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. $\endgroup$ Dec 10 '14 at 13:08
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    $\begingroup$ Generally a universal property defines an object up to isomorphism, so I think the answer to your second question is 'no'. $\endgroup$
    – Todd Trimble
    Dec 10 '14 at 13:08
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    $\begingroup$ @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. $\endgroup$
    – Todd Trimble
    Dec 10 '14 at 13:10
  • $\begingroup$ @ToddTrimble I was thinking specifically in $\mathbf{Top}$. $\endgroup$ Dec 10 '14 at 13:13
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    $\begingroup$ 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. $\endgroup$ Dec 10 '14 at 13:29
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The full categorical definition of universal morphism was given in Włodzimierz Holsztyński, Universal Mappings and Fixed Points Theorems, Bull Acad Polon Sci, v.XV, No 7, 1967, pp.433-438:

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.

In the topological case one considers the full category of the non-empty topological spaces.

COMMENTS

  • 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.
  • Topic universal maps is a join generalization of the topological dimension theory and of the fixed-point property topic. Some theorems involve--in the same result--the dimension and the fixed point theory (without explicitly mentioning universal maps), and they apply the universal maps in their proof.
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