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Stream: theory: category theory

Topic: Limits and unnatural transformations

Nathaniel Virgo (Feb 04 2025 at 10:49):

Given functors $F,G: I \to C$, an unnatural transformation $\varepsilon:F\leadsto G$ is a family of maps $\varepsilon_i:F(i)\to G(i)$ for each object $i\in I$, which doesn't have to obey a naturality condition or any other conditions.

Suppose that $F$ and $G$ both have limits, and that we have an unnatural transformation $\varepsilon:F\leadsto G$, together with a cone $K$ over $F$, given by maps $k_i:K\to F(i)$. Suppose also that the maps $k_i{;}\varepsilon_i$ form a cone over $G$ (which is not guaranteed, since $\varepsilon$ isn't natural).

Then we have the unique maps $!_{K,G}:K\to \lim F$ and $!_{K,F}:K\to\lim G$. If $\varepsilon$ were a natural transformation we would also have a map $\lim\varepsilon:\lim F\to \lim G$, but we don't have that in this case.

However, my instinct is that even though we don't have that, $!_{K,G}$ should still factor through $!_{K,F}$. That is, I conjecture that there exists a morphism $\hat\varepsilon:\lim F\to \lim G$ in $C$ (not necessarily unique) such that

$$K\xrightarrow{!_{K,G}} \lim G = K\xrightarrow{!_{K,F}} \lim F\xrightarrow{\hat \varepsilon}\lim G.$$

The morphism $\hat\varepsilon$ will in general depend on the cone $K$ as well as the unnatural transformation $\varepsilon$.

The reason for thinking this is that if we can think of $\lim F$ as "the object of cones over $F$", then the cone $K$ shouldn't "contain any information" that isn't preserved by its map into $\lim F$.

Although this seems kind of plausible to me I don't see a way to prove it, so my question is whether it's true, or if it's not true in general, what other assumptions might need to be in place to make it so.

Morgan Rogers (he/him) (Feb 04 2025 at 15:32):

If the $k_i$ happen to be epimorphisms, the condition that the $k_i ; \epsilon_i$ form a cone over $G$ imply naturality of $\epsilon$. Beware that your intuition might be built on examples where this happens to be the case.

Kevin Carlson (Feb 04 2025 at 17:35):

9B645116-49E0-44BD-A905-A6FDEE8E9AB1.jpg

Here’s the minimal counterexample I can think of: the universal category containing two cospans, cones over those cospans, and an unnatural transformation between them, adjoined with an initial object. The two cones summitting the filled squares are indeed pullbacks since the only other cones over the cospans are those under the initial object $k.$