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**](http://dx.doi.org/10.1007/s10485-008-9137-4), *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**](http://dx.doi.org/10.1017/S0004972700006353), *Bull. Austral. Math. Soc.* **22** (1980), no. 1, 1--83.



Free monad sequence versus colimit over injections of ordered sets
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.


