Kelly describes a constructive procedure for building the algebraically free monad on a pointed endofunctor. Garner gives a concise summary, which I partially review here for convenience.

Let $V$ be a cocomplete category (feel free to make other strong assumptions about $V$ if I have omitted them) and $(S,\eta)$ a pointed endofunctor of $V$. Define $X_0=\mathrm{id}_V$, $X_1=S$, and $\sigma_0:SX_0\to X_1$ as the identity. Then assuming $\sigma_0,\ldots,\sigma_i$ and $X_0,\ldots,X_{i+1}$ are defined, define $X_{i+2}$ and $\sigma_{i+1}:SX_{i+1}\to X_{i+1}$ as the coequalizer of the following diagram:

$\require{AMScd}$ \begin{CD} SX_i @>\sigma_{i}>>X_{i+1}\\ @VS\eta X_iVV @V\eta X_{i+1}VV\\ SSX_i@>S\sigma_i>>SX_{i+1} \end{CD}

Then putting $\{X_i\}$ together via the maps $X_{i}\xrightarrow{\eta X_i} SX_{i}\xrightarrow{\sigma_i} X_{i+1}$, the resulting diagram $$ X_0\to X_1\to \cdots $$ is the (finite part of the) free monad sequence for $(S,\eta)$. If we make what seems to be a fairly strong convergence assumption, then the colimit $X_\omega$ of the sequence is the (algebraically) free monad on $(S,\eta)$.

On the other hand, another thing one could thing to do is to consider $(S,\eta)$ as generating a semicosimplicial functor $S_*$ where $S_n= S^n$ and the coface maps are given by $S^i\eta S^j$. Then one could consider the colimit of the diagram made up by coface maps up to a certain point:

$$ S_0\to S_1\rightrightarrows S_2\to\cdots \to S_n $$ Call this colimit $Y_n$.

Inductively it looks to me like $X_i\cong Y_i$ and so $X_\omega\cong Y_\omega$.

Here are my questions.

1. Is there something subtly (or not so subtly) wrong with this observation?

2. The presentation of $Y_n$ and $Y_\omega$ seems "more symmetric." Assuming the answer to question 1. is "no," is there a good reason to prefer the presentation of $X_n$ and $X_\omega$? Is this because things will go badly as soon as one needs to extend to infinite ordinals in the free monad sequence?

3. If the answer to question 1. is "no," is there a reference that uses the $Y_n$ presentation to build this sequence where I can see in detail whether there are secret assumptions I am suppressing?

Richard Garner, MR 2506256 Understanding the small object argument, Appl. Categ. Structures 17 (2009), no. 3, 247--285.

G. M. Kelly, MR 589937 A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on, Bull. Austral. Math. Soc. 22 (1980), no. 1, 1--83.

Kelly describes a constructive procedure for building the algebraically free monad on a pointed endofunctor. Garner gives a concise summary, which I partially review here for convenience.

Let $V$ be a cocomplete category (feel free to make other strong assumptions about $V$ if I have omitted them) and $(S,\eta)$ a pointed endofunctor of $V$. Define $X_0=\mathrm{id}_V$, $X_1=S$, and $\sigma_0:SX_0\to X_1$ as the identity. Then assuming $\sigma_0,\ldots,\sigma_i$ and $X_0,\ldots,X_{i+1}$ are defined, define $X_{i+2}$ and $\sigma_{i+1}:SX_{i+1}\to X_{i+2}$ as the coequalizer of the following diagram:

$\require{AMScd}$ \begin{CD} SX_i @>\sigma_{i}>>X_{i+1}\\ @VS\eta X_iVV @V\eta X_{i+1}VV\\ SSX_i@>S\sigma_i>>SX_{i+1} \end{CD}

Then putting $\{X_i\}$ together via the maps $X_{i}\xrightarrow{\eta X_i} SX_{i}\xrightarrow{\sigma_i} X_{i+1}$, the resulting diagram $$ X_0\to X_1\to \cdots $$ is the (finite part of the) free monad sequence for $(S,\eta)$. If we make what seems to be a fairly strong convergence assumption, then the colimit $X_\omega$ of the sequence is the (algebraically) free monad on $(S,\eta)$.

On the other hand, another thing one could thing to do is to consider $(S,\eta)$ as generating a semicosimplicial functor $S_*$ where $S_n= S^n$ and the coface maps are given by $S^i\eta S^j$. Then one could consider the colimit of the diagram made up by coface maps up to a certain point:

$$ S_0\to S_1\rightrightarrows S_2\to\cdots \to S_n $$ Call this colimit $Y_n$.

Inductively it looks to me like $X_i\cong Y_i$ and so $X_\omega\cong Y_\omega$.

Here are my questions.

1. Is there something subtly (or not so subtly) wrong with this observation?

2. The presentation of $Y_n$ and $Y_\omega$ seems "more symmetric." Assuming the answer to question 1. is "no," is there a good reason to prefer the presentation of $X_n$ and $X_\omega$? Is this because things will go badly as soon as one needs to extend to infinite ordinals in the free monad sequence?

3. If the answer to question 1. is "no," is there a reference that uses the $Y_n$ presentation to build this sequence where I can see in detail whether there are secret assumptions I am suppressing?

Richard Garner, MR 2506256 Understanding the small object argument, Appl. Categ. Structures 17 (2009), no. 3, 247--285.

G. M. Kelly, MR 589937 A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on, Bull. Austral. Math. Soc. 22 (1980), no. 1, 1--83.