# Proof of the limit-colimit coincidince

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.

• 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 $\pi_n \circ \kappa_m = f_{mn}$. – Kevin Clancy Dec 16 '19 at 15:17