We call an object $C\in\mathscr{C}$ is $T$-closed if $TC=C$.
Suppose that $(T,\eta,\mu)$ is a monad on a poset category $\mathscr{C}$. Prove that the Eilenberg-Moore category $\mathscr{C}^T$ is isomorphic to the full subcategory of $\mathscr{C}$ spanned by the $T$-closed objects in $\mathscr{C}$.
(I think) I saw a more complicated version of this question on Category Theory in Context, which requires knowledge about the Kleisli category (which I don't have yet). I am basically lost at what to start with because I don't actually know what to do with the whole "full subcategory of $\mathscr{C}$ spanned by the $T$-closed objects in $\mathscr{C}$" and how to find a relation back to $\mathscr{C}^T$. Thank you in advance!
Best Answer
Hint 1: we already have an arrow $C \to TC$ since $T$ is a monad. Since $\mathcal{C}$ is a poset category, saying that $TC = C$ is just the same as saying that we have an arrow $TC \to C$.
Hint 2: since $\mathcal{C}$ is a poset category, all diagrams commute automatically, simplifying the definition of the Eilenberg-Moore category considerably.
The below was written before I noticed you said poset category. Feel free to ignore this.
I don't think this is true as stated. The double powerset monad on $\mathcal{Set}$ with $TX = P(P(X))$ has no $T$-closed objects. Even if we loosen $T$-closed to mean $TX \cong X$ (which may be what you meant to start with), the double powerset of $X$ has much larger cardinality than $X$. So the full subcategory of $\mathcal{Set}$ spanned by the $T$-closed objects is the empty subcategory.
On the other hand, the algebras for the double powerset monad are the complete atomic boolean algebras. The category of complete atomic boolean algebras is not empty, so it can't be isomorphic to the empty subcategory of $\mathcal{Set}$.
More generally, the Eilenberg-Moore category isn't necessarily equivalent to any full subcategory of $\mathcal{C}$. For example, the category of algebras for the free group monad are the groups, but the category of groups can't be equivalent to a full subcategory of $\mathcal{Set}$.
An easy way to see that is that the automorphism groups of objects are the same in any full subcategory (due to fullness). The automorphism groups of objects of $\mathcal{Set}$ are the symmetric groups. But the category of groups contains $\mathbb{Z}_5$, whose automorphism group is $\mathbb{Z}_4$, which isn't a symmetric group.