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Stream: learning: questions

Topic: How do I call the problem of completing a limit?

Ignat Insarov (Sep 18 2026 at 12:40):

If I have two parallel arrows, the problem of finding the biggest subobject of the source for which they are equal (maybe we can say «the solution set» for the equation of which the arrows are the two sides) is the equalizer problem. In the same way, a problem of classification (like «find all trajectories in the phase space of a dynamical system») would be a co-equalizer problem, and so on.

But what if I have a chain of two arrows like $X \overset f → Y \overset g → Z$ and I want to find the best arrow $Y \overset {g'} → Z$ such that $f$ is the equalizer of $g$ and $g'$? The solution is only an arrow, without a representing object, so this does not fit the definition of universal construction. For example, maybe I have a set of trajectories in mind and I am looking for the dynamics that would realize them. Then I need to complete a co-equalizer (given by the set of trajectories I have in mind) of my unknown dynamics with the identity on the phase space.

Ignat Insarov (Sep 18 2026 at 12:57):

Well, it also cannot be universal because there are usually multiple solutions.

Ignat Insarov (Sep 18 2026 at 14:03):

We may consider another related problem where $f$, $g$ and $g'$ are to be drawn from three different categories, so that:

$\mathcal A \overset F → \mathcal C \overset G ← \mathcal B \qquad X \overset f → Y \overset {F (g)} {\underset {G (g')} ⇉} Z$

Here, as before, $f$ and $g$ are given but $g'$ is unknown.