2
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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 $\pi_n \circ \kappa_m = f_{mn}$. $\endgroup$ Dec 16, 2019 at 15:17

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.