Note: I figured this out, but haven't had the time to write an answer for it: see the comment.

For reference: the discussed material appears in http://www.cs.ru.nl/B.Jacobs/CLG/JacobsCoalgebraIntro.pdf page 111.

On page 285 of "Introduction to Coalgebra" by Bart Jacobs, Proposition 5.3.3 is stated as follows:

Let $\mathbb{C}$ be a dcpo-enriched category. Assume an $\omega$-chain $$X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1} \longrightarrow X_2 \overset{f_2}{\longrightarrow} \cdots$$ with colimit $A \in \mathbb C$. If the maps $f_i$ are embeddings, then the colimit $A$ is also a limit in $\mathbb C$, namely the limit of the $\omega$-chain of associated projections $f_i^p : X_{i+1} \to X_i$.

In order to prove this theorem, Jacobs first gives a partial proof of the following lemma:

The coprojection maps $\kappa_n : X_n \to A$ associated with the colimit $A$ are embeddings, and their projections $\pi_n = \kappa_n^{p} : A \to X_n$ form a cone, i.e. satisfy $f^p_n \circ \pi_{n+1} = \pi_n$.

Here is the partial proof:

For each $n \in \mathbb N$ we first show that the object $X_n$ forms a cone: for $m \in \mathbb N$ there is a map $f_{mn} : X_m \to X_n$, namely: $$f_{mn} \overset{def}{=} \left \{ \begin{array}{ll} f_{n-1} \circ \cdots \circ f_m : X_m \to X_{m+1} \to \cdots \to X_n & \text{if}~~m \leq n \\ f_n^p \circ \cdots \circ f_{m-1}^p : X_m \to X_{m-1} \to \cdots \to X_n & \text{if}~~m > n \end{array} \right \}$$ These maps $f_{mn}$ commute with the maps $f_i : X_i \to X_{i+1}$ in the chain: $f_{(m+1)n} \circ f_m = f_{mn}$, and thus form a cone. Since $A$ is a colimit , there is a unique map $\pi_n : A \to X_n$ with $\pi_n \circ \kappa_m = f_{m n}$. In particular, we get $\pi_n \circ \kappa_n = f_{n n} = \mathit{id}$. We postpone the proof that $\kappa_n \circ \pi_n \leq \mathit{id}_A$ for a moment.

It seems to me that the above proof doesn't go far enough: it does not prove that the proposed projections $\pi_n$ actually form a cone, i.e. that $f^p_n \circ \pi_{n+1} = \pi_n$. This fact is used immediately in the proof of the next lemma, so it cannot be postponed! Can anyone explain to me how to prove $f^p_n \circ \pi_{n+1} = \pi_n$? If our embeddings were epic then this would be straightforward, but I don't think that they are.

  • $\begingroup$ I figured it out: we just have to prove that $(f^p_n \circ \pi_{n+1}) \circ \kappa_m = f_{mn}$, because $\pi_n$ was defined as the unique morphism such that $\pi_n \circ \kappa_m = f_{mn}$. $\endgroup$ – Kevin Clancy Dec 16 '19 at 15:17

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