### Mapping (Co)cylinder Factorizations via the Small Object Argument

#### Posted by Emily Riehl

I’ve been thinking about the small object argument recently and noticed something curious. Even though I can’t think of any applications, I still find it interesting. Fortunately, the $n$-Category Café seems like the perfect place to record curiosities.

Here I don’t mean Quillen’s small object argument but rather an improved construction due to Richard Garner. And I don’t quite mean the functorial factorization he describes in his paper but rather an enriched version that you can read more about in chapter 13 of these notes.

The reason I like the calculation I’m about to describe is that it illustrates that what I might call the *enriched algebraic small object argument* can be remarkably computable. Like Quillen’s original construction, this form of the small object argument produces a factorization into a pair of maps whose lifting properties can be easily characterized. But the algebraic small object argument always converges and sometimes you get lucky and the factorization it spits out, rather than being some transfinite beast, instead turns out to be the elementary functorial factorization you already had in mind.

### Chain complexes, in analogy with spaces

Consider the category of unbounded chain complex over your favorite commutative ring $R$. There is an analogy between spaces and chain complexes that might begin as follows. Let $I$ denote the chain complex with a generator [I] in degree one and two generators [0] and [1] in degree zero, with $d$[I] = [0]-[1]. A chain map $X \otimes I \to Y$ restricting to $f$ and $g$ along the “endpoints” of the interval is then exactly a chain homotopy between $f$ and $g$.

With respect to this interval, you can define *cofibrations* and *fibrations* to be respectively the maps that satisfy the homotopy extension property and the homotopy lifting property. There is a model structure on the category of unbounded chain complexes of $R$-modules with these cofibrations and fibrations and with the chain homotopy equivalences as weak equivalences. This is variously called the *absolute*, *relative*, or $h$-*model structure*. You can find more details in chapter 18 of More Concise Algebraic Topology by Peter May and Kate Ponto.

This model structure is not cofibrantly generated for general rings (e.g., for $R=\mathbb{Z}$) as is proven in a paper of Dan Christensen and Mark Hovey. But there are functorial factorizations that are easy to describe constructed using the precise algebraic analogs of the mapping cylinder and dual mapping path space or mapping cocylinder. In the notation of May-Ponto, given $f: X \to Y$ define $Mf$ and $Nf$ to be the pushout and pullback

$\begin{matrix} X & \overset{f}{\longrightarrow} & Y & & & Nf & \longrightarrow & Y^I \\ i_0 \downarrow & & \downarrow & & & \downarrow & & \downarrow p_0 \\ X \otimes I & \longrightarrow & Mf & & & X & \overset{f}{\longrightarrow} & Y \end{matrix}$

The map $f$ then factors as

$\begin{matrix} X & \overset{i_1}{\longrightarrow} & Mf & \longrightarrow & Y & & & X & \longrightarrow & Nf & \overset{p_1}{\longrightarrow} & Y \end{matrix}$

What I claim is that this model structure is actually cofibrantly generated, not in the usual sense of course, but in a suitable *enriched* sense. Indeed the generating cofibrations and trivial cofibrations can be taken to be the exact same generating sets that generate the more-standard Quillen-type model structure in the unenriched sense! This result will appear in a forthcoming paper with Peter May and Tobias Barthel.

Furthermore — and this is what I want to explain here — when these generating sets are used in the enriched algebraic small object argument, the result is exactly the two mapping (co)cylinder factorizations!

### On enrichment

I don’t want to say too much about enriched weak factorization systems here — you can read more in chapter 13 — but the point is that the usual lifting property is replaced by a strengthened enriched lifting property. Here the enrichment is over the category of $R$-modules. (Note that the category of $R$-modules is not usually thought of as a model category. The sort of enriched model structure I’m describing here is not the standard notion of enriched model category.)

I’ll illustrate via an example. Let $J$ denote the set of maps $\{ 0 \to D^n\}$ for $n \in \mathbb{Z}$, where $D^n$ is the chain complex with $R$ in degrees $n$ and $n-1$ and the identity differential. As the notation suggests, it should be thought of as analogous to the $n$-disk. A map satisfies the right lifting property with respect to the set $J$ if and only if it is degreewise surjective (i.e., iff it is a fibration in the Quillen-type model structure). A map satisfies the $R$-module enriched right lifting property with respect to $J$ if and only if it is a degreewise split map of graded $R$-modules, where the splittings need not commute with the differentials. These turn out to be exactly the fibrations in the sense defined above.

### The enriched algebraic small object argument

Now it’s easy enough to describe the enriched small object argument: In the colimit defining the functorial factorization, replace any coproducts indexed by sets of commutative squares with a tensor with the analogous $R$-module of commutative squares.

Applied to $J$, the zeroth step of the enriched small object argument forms a coproduct indexed by $J$ of the generating trivial cofibrations $0 \to D^n$, now tensored with the $R$-module of commutative squares

$\begin{matrix} 0 & \longrightarrow & X \\ \downarrow & & \downarrow \\ D^n & \longrightarrow & Y \end{matrix}$

This $R$-module is just $Y_n$; maps $D^n \to Y$ are in bijection with elements of $Y_n$ and this bijection is a module homomorphism. The tensor of $0 \to D^n$ with $Y_n$ is the map $0 \to D^n[Y_n]$, where the codomain is the chain complex with the module $Y_n$ in degrees $n$ and $n-1$.

So the “step-one” functorial factorization is defined via the pushout

$\begin{matrix} 0 & \longrightarrow & X \\ \downarrow & & \downarrow & \searrow \\ \oplus_n D^n[Y_n] & \longrightarrow & X \oplus \oplus_n D^n[Y_n] & \longrightarrow & Y \end{matrix}$

The enriched form of Quillen’s small object argument would proceed by just iterating this construction, but the enriched *algebraic* small object argument avoids attaching redundant cells, where a cell is redundant if the attaching map factors through a previous stage. Here all cells attached after step one are redundant because the domains of the generating trivial cofibrations are all 0. So this enriched algebraic small object converges at step one — and indeed you can check that

$\begin{matrix} (\ast) & & & X & \longrightarrow & X \oplus \oplus_n D^n[Y_n] & \longrightarrow & Y \end{matrix}$

is exactly the mapping cocylinder factorization.

### Mapping cylinders via the enriched algebraic small object argument

Let $I$ denote the set of maps $\{ S^{n-1} \to D^n\}$ for $n \in \mathbb{Z}$, where $S^n$ is the chain complex with $R$ in degree $n$ and zeros elsewhere; this is the analog of the $n$-sphere. A map $S^n \to X$ corresponds to an $n$-cycle in $X$. It turns out, though it’s less obvious, that a map satisfies the $R$-module enriched right lifting property with respect to $I$ if and only if it is a fibration and chain homotopy equivalence.

Our aim is to use $I$ and the enriched algebraic small object argument to factor $f\colon X \to Y$. The $R$-module of commutative squares

$\begin{matrix} S^{n-1} & \longrightarrow & X \\ \downarrow & &\downarrow \\ D^n & \longrightarrow & Y \end{matrix}$

is the pullback

$\begin{matrix} P_n & \overset{q}{\longrightarrow} & Y_n \\ d\downarrow & & \downarrow d \\ Z_{n-1}X & \overset{f}{\longrightarrow} & Z_{n-1} Y \end{matrix}$

So the step-one functorial factorization is defined by the pushout

$\begin{matrix} \oplus_n S^{n-1}[P_n] & \longrightarrow & X \\ \downarrow & & \downarrow & \searrow \\ \oplus_n D^n[P_n] & \longrightarrow & X \oplus \oplus_n P_n[n] & \longrightarrow & Y \end{matrix}$

Here, the map $f$ is factored through the chain complex with $X_n \oplus P_n$ in degree $n$. The differential on $P_n$ is the map $d$ in the pullback defining this module. The map $X_n \to X_n \oplus P_n$ is the inclusion and the map $X_n \oplus P_n \to Y_n$ is $f+q$.

For step two, we must first compute the $R$-module of commutative squares

$\begin{matrix} S^{n-1} & \longrightarrow & X \oplus \oplus_k P_k[k] \\ \downarrow & &\downarrow \\ D^n & \longrightarrow & Y \end{matrix}$

Here the bottom map is given by $y \in Y_n$, and the top map is given by a pair $x' \in X_{n-1}$ and $p = (x,y') \in P_{n-1}$. The pair $(x',p)$ must be an $(n-1)$-cycle, which is the case just when $d x' = x$, so the data of attaching map turns out to be the pair $x' \in X_{n-1}$ and $y' \in Y_{n-1}$ subject to the condition $d y' + d f x' = 0$. Commutativity of the square corresponds to the condition that $d y = y' + f x'$, which of course implies that $d y' + d f x' = 0$. So in summary, the $R$-module of commutative squares is given by the pullback

$\begin{matrix} E_n & \longrightarrow & Y_n \\ \downarrow & & \downarrow d \\ X_{n-1} \oplus Y_{n-1} & \overset{f+1}{\longrightarrow} & Y_{n-1} \end{matrix}$

The step-two factorization in the enriched Quillen small object argument is then given by the pushout

$\begin{matrix} \oplus_n S^{n-1}[E_n] & \longrightarrow & X \oplus \oplus_k P_k[k] \\ \downarrow & & \downarrow & \searrow \\ \oplus_n D^n[E_n] & \longrightarrow & X \oplus \oplus_k P_k[k] \oplus E_k[k] & \longrightarrow & Y \end{matrix}$

But this isn’t yet step two of the enriched *algebraic* small object argument: we must avoid attaching redundant cells. Doing so replaces the chain complex $X \oplus \oplus_k P_k[k] \oplus E_k[k]$ with a quotient formed by a coequalizer that identifies those cells attached in step two with the corresponding cells attached in step one. In degree $n$, this coequalizer has the form:

$\begin{matrix} X_n \oplus P_n & \rightrightarrows & X_n \oplus P_n \oplus E_n \end{matrix}$

where one map is the inclusion and the other sends the pair $x \oplus (y,x') \in X_n \oplus P_n$ to $x \oplus 0 \oplus (y, x' \oplus 0)$. This identifies $P_n$ as a submodule of $E_n$, so the coequalizer is $X_n \oplus E_n$. Thus, the step-two factorization of the enriched algebraic object argument is

$\begin{matrix} (\ast\ast) & & & X & \longrightarrow & X \oplus \oplus_n E_n[n] & \longrightarrow & Y \end{matrix}$ where the second factor sends $x \oplus (y \oplus (x',y'))$ to $f x + y$ and the differential sends $x \oplus (y \oplus (x',y'))$ to $(d x + x') \oplus ( y' \oplus (-d x',0))$.

Using the differential we’ve just described on $X \oplus \oplus_k E_k[k]$, one can see that the $n$-cycles are in the image of the map $X \oplus \oplus_k P_k[k] \to X \oplus \oplus_k E_k[k]$ from the step-one factorization to the step-two factorization. This says that any attaching map from an element of $I$ must factor through a previous stage. So the enriched algebraic small object argument converges at step two.

And indeed, the factorization $(\ast\ast)$ is exactly the mapping cylinder factorization! By definition $(Mf)_n = X_n \oplus X_{n-1} \oplus Y_n$. I’ll leave it to you to check that the map

$\begin{matrix} X_n \oplus E_n & \longrightarrow & X_n \oplus X_{n-1} \oplus Y_n \\ x \oplus (y \oplus (x',y')) & \mapsto & x \oplus (-1)^n x' \oplus y \end{matrix}$

is an isomorphism of chain complexes commuting with the maps from $X$ and to $Y$.

## Re: Mapping (co)cylinder factorizations via the small object argument

That looks really nice though I haven’t gone through it in as much detail as I would like to. I want to ask: how does this story relate to its topological counterpart?

There is an analogous model structure on topological spaces, called the Strøm or Hurewicz model structure with Hurewicz fibrations and (closed/strong) Hurewicz cofibrations. As far as I understand in an earlier paper of yours you used the algebraic small object argument to fix Cole’s generalization of this model structure to quite arbitrary topological categories.

What I would like to understand better is that (unlike in the case of chain complexes you discussed above) the classical mapping cylinder factorization is a factorization into a Hurewicz cofibration followed by a homotopy equivalence, but this homotopy equivalence is not a Hurewicz fibration. Instead, it is a Dold fibration (i.e. has the weak covering homotopy property in the sense of Dold). (The dual statement holds for mapping path object factorizations.)

In your earlier paper, did you use the enriched version of the algebraic small object argument to construct factorizations for the Cole model structures? If not, what would be the difference between factorizations produced by the non-enriched version and the enriched version?

Can you pinpoint the reason why for chain complexes the mapping cylinder (or path object) factorizations already factor into Hurewicz cofibrations and cofibrations while this is not the case for topological spaces?

Do you think that algebraic factorization systems could shed any light on how Dold type (co)fibrations fit into the story?