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Categories for the Working Mathematician says

Definition. A monad $T= \langle T, \eta, \mu\rangle $ in a category $X$ consists of a functor $T: X \to X$ and two natural transformations

$$\eta : I_X \Rightarrow T, \mu : T^2 \Rightarrow T $$

which make the following diagrams commute

enter image description here

We shall thus call $\eta$ the unit and $\mu$ the multiplication of the monad $T$; the first commutative diagram of (2) is then the associative law for the monad, while the second and third diagrams express the left and right unit laws, respectively.

Why is $\eta$ called the unity, and $\mu$ multiplication?

How is the first diagram the associative law? It says $μ∘Tμ=μ∘μT$, which equates $Tμ$ and $μT$ up to $μ$, so seems to me the commutative law instead.

Why are the second and third diagrams the second the left and right unit laws? They say that $μ∘ηT=μ∘Tη=1_T$, They seem to me that $ηT$ and $Tη$ are the same up to $\mu$ and they are both inverse of $μ$.

Thanks.

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Given a category $\mathcal C$, the category $[\mathcal C,\mathcal C]$ of endofunctor is naturally a (non symmetric) strict monoidal category: the tensor product is the composition, the unit object if the identity functor and the associators and unitors are actual identities.

A monad on $\mathcal C$ is exactly a monoid in this monoidal category $[\mathcal C,\mathcal C]$.

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  • $\begingroup$ Thanks. In the category [C,C] of endofunctor, "the unit object is the identity functor and the associators and unitors are actual identities." But I am asking in the monad T=⟨T,η,μ⟩ in a category X (i.e. your C), why is η called the unity, and μ multiplication? η isn't the identity functor, and μ isn't the composition between functors. $\endgroup$ – Tim Jul 24 at 10:56
  • $\begingroup$ Some other questions. (2) How is the first diagram the associative law? It says $μ∘Tμ=μ∘μT$, which equates $Tμ$ and $μT$ up to $μ$, so seems to me the commutative law instead. (3) Why are the second and third diagrams the second the left and right unit laws? They say that $μ∘ηT=μ∘Tη=1_T$, They seem to me that $ηT$ and $Tη$ are both inverse of $\mu$. $\endgroup$ – Tim Jul 24 at 12:01
  • $\begingroup$ A monoid object $C$ in a monoidal category $(\mathcal C, \otimes, I)$ has unit $I\to C$ and composition $C\otimes C\to C$, satisfying the same diagrams. Consider first what it means in $(Set,\times,\{*\})$, and then in $([\mathcal C,\mathcal C],\,\circ,\,id_{\mathcal C}) $. $\endgroup$ – Berci Jul 24 at 15:07

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