I'm trying to solve Emily Riehl's "category theory in context", question 1.6.vi (1.6.6). The exercise says:
A coalgebra for an endofunctor $T: C \rightarrow C$ is an object $c \in C$ equipped with a map $\gamma: c \rightarrow Tc$. A morphism $f: (c, \gamma) \rightarrow (c', \gamma)$ of coalgebra is a map $f: c \rightarrow c'$ such that the diagram commutes:
$$ \begin{matrix} c & \xrightarrow{f} & c' \\ \gamma \downarrow & & \downarrow \gamma' \\ Tc & \xrightarrow{Tf} & Tc' \end{matrix} $$
Show that if $(c, \gamma: c \rightarrow Tc)$ is a terminal coalgebra (that is, a terminal object in the category of coalgebras), then $\gamma$ is an isomorphism.
What I've tried
Suppose we have a terminal coalgebra $(c, \gamma: c \rightarrow Tc)$. Consider the coalgebra $(Tc, T\gamma: Tc \rightarrow TTc)$. Since $(c, \gamma)$ is terminal, there must be a unique map of coalgebras $f: Tc \rightarrow c$. This means the following diagram commutes:
$$ \begin{matrix} Tc & \xrightarrow{f} & c \\ T\gamma \downarrow & & \downarrow \gamma \\ TTc & \xrightarrow{Tf} & Tc \end{matrix} $$
So we have the equation $Tf \circ T\gamma = \gamma \circ f$. By functoriality, this implies $T(f \circ \gamma) = \gamma \circ f$. At this point I'm stuck, since I don't seem to have any more data to use.
The assumption I need to proceed: $c$ is terminal in $C$
If I now assume the extra data that $c$ itself is a terminal object in $C$ (not just that $(c, \gamma)$ is a terminal object in $Coalg(C)$), then I can proceed with the proof.
I have a morphism $f \circ \gamma: c \rightarrow c$ which must be equal to $id_c$ by terminality of $c$. Thus, $f \circ \gamma = id_c$.
Substituting $f \circ \gamma = id_c$ in the equation equation $T(f \circ \gamma) = \gamma \circ f$, I get $T(id_c) = \gamma \circ f$, or $id_{Tc} = \gamma \circ f$.
In total, we have $f \circ \gamma = id_c$, $\gamma \circ f = id_{Tc}$. So $\gamma : c \rightarrow Tc$ is an isomorphism, and thus we are done.
How do I prove the assumption that $c$ is terminal in $C$?
I just don't see what data I have to be able to show that $c$ is terminal in $C$. Suppose there is some other $d \in C$. I cannot build a coalgebra $\delta: d \rightarrow Td$ out of thin air to use the fact that $(c, \gamma)$ is terminal as a coalgebra. So I am unsure how to prove the assumption I need to finish the proof.
Questions
- Is the assumption that $c$ is terminal a valid one? If so, how do I prove it?
- If the assumption is invalid, how must I proceed with the proof?
Form this commutative square
\begin{matrix} c & \xrightarrow{\gamma} & Tc & \xrightarrow{f} & c \\ \gamma \downarrow & & \downarrow T\gamma & & \downarrow \gamma \\ Tc & \xrightarrow{T \gamma} & TTc & \xrightarrow{Tf} & Tc \end{matrix}
Note that the identity map from $(c,\gamma) \to (c,\gamma)$ also exists, so by uniqueness of universal maps these are equal thus $f \gamma = id$.
and (as you demonstrated) this implies $T(f \gamma) = T(id)$ which can rewrite to $\gamma f = id$