Monads

The maybe monad on $\mathbf{Set}$ has $TX = X \sqcup \{\bot\}$, $\eta_X(x) = x$, and $\mu_X$ collapsing the two copies of $\bot$. What is a $T$-algebra $(A, a)$?

How many algebra structures $a : \mathcal{P}A \to A$ for the power-set monad are there on the three-element set $A = \{p, q, r\}$?

In the Kleisli category of the list monad, let $f(n) = [n, n + 1]$ and $g(n) = [10n]$ (both from $\mathbb{Z}$ to $\mathrm{List}(\mathbb{Z})$). What is $(g \circ_T f)(1) = \mu(Tg(f(1)))$?

Let $F \dashv G$ with $F : \mathcal{C} \to \mathcal{D}$, unit $\eta$ and counit $\varepsilon$. The induced monad is $T = GF$ on $\mathcal{C}$. What is its multiplication $\mu_X : GFGFX \to GFX$?

In proving that $\eta_A$ is a right identity in the Kleisli category, we write, for $f : A \to TB$,

$f \circ_T \eta_A = \mu_B \circ Tf \circ \eta_A = \mu_B \circ \eta_{TB} \circ f = f.$

Which two facts justify the second and third equalities, in order?

For the list monad $T$ on $\mathbf{Set}$, $\mathbf{Set}^T \simeq \mathbf{Mon}$. Which argument shows that the Kleisli category $\mathbf{Set}_T$ is not equivalent to $\mathbf{Mon}$?

Which forgetful functors to $\mathbf{Set}$ are monadic? (Choose all that apply.)

In the Lean file, a list algebra on X has act : List X → X. Which field states the associativity law $a \circ Ta = a \circ \mu_A$?

Write kleisli(f, g) returning the Kleisli composite (first f, then g) for the list monad: f and g are Python functions returning lists, and the composite applies g to every output of f and concatenates. Also write unit(x) returning [x]. The tests check the Kleisli category laws on examples.