From this MO question I learned to tentatively think of formally unramified arrows as immersions and of formally smooth arrows as submersions.

I'm trying to semi-formally handwave myself into conviction about this but I'm getting the opposite identifications.

The setting is a commutative square as below posing a unique lifting problem. $$\require{AMScd} \begin{CD} P @>>> U\\ @VVV @VV{f}V\\ N @>>> X \end{CD}$$

Suppose we start in the category of topological spaces and let $P\rightarrow N$ be the inclusion of a point into an open neighborhood, and let all the other arrows also be inclusions into opens of a space $X$. Then our diagram really lives in the lattice of opens of $X$ containing the point $P$. Since we want to deal with infinitesimal neighborhoods, move to the pro-completion of this lattice and replace $N$ by some formal intersection which isn't really there to get a good notion. Now our square lives in the pro-completion with all arrows still "inclusions of opens".

The existence of a diagonal filler signifies that whenever an open $U$ of $X$ contains a point $P$, it also contains every infinitesimal thickening of it. (This fits perfectly with the intuition that open sets contain neighboring points.) Uniqueness follows from the fact all arrows are "on the nose inclusions", or, alternatively, from the fact we're living in the pro-completion of a lattice.

So open inclusions are tentatively right-orthogonal to the class of such infinitesimal thickenings of points. The same goes for open embeddings. Now since $N$ is infinitesimal, this lifting problem should only care about local proeprties of $f$. So let $f$ be a topological immersion, i.e a map which is locally-on-the-domain an embedding. I think being an immersion gives existence by precomposing with elements of a good open cover, but by the analogy suggested above, immersions correspond to unramified arrows whose property gives at most one filler.

What am I doing wrong here?

It seems my mistake must be in saying that open embeddings gives existence, but I don't see why they should give uniqueness (without necessarily existence) instead.


I can't precisely isolate the wrong step, but it might help to consider a concrete motivational example, where $X$ is the $xy$-plane, $N$ is a very small interval in the $x$-axis, and $P$ is the origin. If $f: U \to X$ is the immersion given by inclusion of an interval in the $x$-axis, then you have a unique lift. If $f: U \to X$ is the immersion given by inclusion of an interval in the $y$-axis, then there is no lift, and passing to an open cover won't help.

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