Free monad sequence versus colimit over injections of ordered sets

Question

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.

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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 $(S,\eta)$.

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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.

Answer 1

Let's talk about free monoid constructions in a monoidal category $(\mathcal{V}, \otimes, I)$. Free monad constructions are a special case, when we take $\mathcal{V}$ to be a monoidal category of endofunctors. There are at least two flavors: the free monoid on an object $X$, and the free monoid on a pointed object $I \to X$.

It's pretty easy to construct the free monoid functor $T: \mathcal{V} \to \mathcal{V}$ in typical monoidal categories -- just take $T: X \mapsto \sum_{n=0}^\infty X^{\otimes n}$, or in the pointed case, $T: I \to X \mapsto \varinjlim_{n\to \infty} X^{\otimes n}$ using the map $I \to X$ to define semicosimplicial connecting maps. The issue is with defining multiplication. We want to say that $TX \otimes TX = \varinjlim_m X^{\otimes m} \otimes \varinjlim_n X^{\otimes n} = \varinjlim_{m,n}X^{\otimes m+n}$, and then define multiplication $\mu_X: T^2 X \to TX$ by the universal property of the colimit, setting $X^{m+n} \to T^2 X \overset{\mu_X}{\to} TX$ equal to $X^{m+n} \to TX$, where the unlabeled maps are the canonical colimit inclusions. If you think about this, you'll see it's the usual formula for multiplication in a free monoid. The diagrams all commute, so we can define this map $\mu_X$ with no problem.

The problem is that the domain of $\mu_X$ is $\varinjlim_{m,n} X^{\otimes m+n}$, and in order to identify this with $T^2 X$, we had to commute a colimit past $\otimes$ -- we needed the functor $TX \otimes(-): \mathcal{V} \to \mathcal{V}$ to commute with the colimit defining $TX$. This is fine if $\otimes$ is (right) closed, in which case $TX \otimes(-)$ commutes with all colimits, so these simple constructions work in the familiar cases of free monoids in categories like $\mathsf{Set},\mathsf{Ab}$, etc.

But right closedness is simply not true in a category of endofunctors $\mathcal{V} = [\mathcal{C},\mathcal{C}]$. The condition becomes slightly different in this setting -- if you start with $F$ and define $M(C) = \varinjlim_n F^n C$, you want to argue that $M^2(C) = \varinjlim_m F^m (\varinjlim_n F^n(C)) = \varinjlim_{m,n} F^{m+n}(C)$, for which it suffices for $F$ to commute with the colimits defining $M$. Unfortunately, this is not often the case, although I believe the semisimplex category is at least sifted, so if $F$ commutes with sifted colimits you might be in luck.

Otherwise I don't know a shortcut to reading Kelly. I think the trick to reading his paper on a first pass must be to keep in mind that you can take the factorization system he uses to be $(M,E)$ = (all maps, isos), and then "preserving $E$-tightness of a cocone" just means preserving a colimit. What I currently understand is that Kelly gives free constructions endofunctor -> pointed endofunctor -> well-pointed endofunctor -> monad. I'm currently most mystified by the middle one.