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September 16, 2013

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 nn-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 RR. There is an analogy between spaces and chain complexes that might begin as follows. Let II denote the chain complex with a generator [I] in degree one and two generators [0] and [1] in degree zero, with dd[I] = [0]-[1]. A chain map XIYX \otimes I \to Y restricting to ff and gg along the “endpoints” of the interval is then exactly a chain homotopy between ff and gg.

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 RR-modules with these cofibrations and fibrations and with the chain homotopy equivalences as weak equivalences. This is variously called the absolute, relative, or hh-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=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:XYf: X \to Y define MfMf and NfNf to be the pushout and pullback

X f Y Nf Y I i 0 p 0 XI Mf X f Y \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 ff then factors as

X i 1 Mf Y X Nf p 1 Y \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 RR-modules. (Note that the category of RR-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 JJ denote the set of maps {0D n}\{ 0 \to D^n\} for nn \in \mathbb{Z}, where D nD^n is the chain complex with RR in degrees nn and n1n-1 and the identity differential. As the notation suggests, it should be thought of as analogous to the nn-disk. A map satisfies the right lifting property with respect to the set JJ 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 RR-module enriched right lifting property with respect to JJ if and only if it is a degreewise split map of graded RR-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 RR-module of commutative squares.

Applied to JJ, the zeroth step of the enriched small object argument forms a coproduct indexed by JJ of the generating trivial cofibrations 0D n0 \to D^n, now tensored with the RR-module of commutative squares

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

This RR-module is just Y nY_n; maps D nYD^n \to Y are in bijection with elements of Y nY_n and this bijection is a module homomorphism. The tensor of 0D n0 \to D^n with Y nY_n is the map 0D n[Y n]0 \to D^n[Y_n], where the codomain is the chain complex with the module Y nY_n in degrees nn and n1n-1.

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

0 X nD n[Y n] X nD n[Y n] Y\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

(*) X X nD n[Y n] Y\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 II denote the set of maps {S n1D n}\{ S^{n-1} \to D^n\} for nn \in \mathbb{Z}, where S nS^n is the chain complex with RR in degree nn and zeros elsewhere; this is the analog of the nn-sphere. A map S nXS^n \to X corresponds to an nn-cycle in XX. It turns out, though it’s less obvious, that a map satisfies the RR-module enriched right lifting property with respect to II if and only if it is a fibration and chain homotopy equivalence.

Our aim is to use II and the enriched algebraic small object argument to factor f:XYf\colon X \to Y. The RR-module of commutative squares

S n1 X D n Y\begin{matrix} S^{n-1} & \longrightarrow & X \\ \downarrow & &\downarrow \\ D^n & \longrightarrow & Y \end{matrix}

is the pullback

P n q Y n d d Z n1X f Z n1Y\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

nS n1[P n] X nD n[P n] X nP n[n] Y\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 ff is factored through the chain complex with X nP nX_n \oplus P_n in degree nn. The differential on P nP_n is the map dd in the pullback defining this module. The map X nX nP nX_n \to X_n \oplus P_n is the inclusion and the map X nP nY nX_n \oplus P_n \to Y_n is f+qf+q.

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

S n1 X kP k[k] D n Y\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 yY ny \in Y_n, and the top map is given by a pair xX n1x' \in X_{n-1} and p=(x,y)P n1p = (x,y') \in P_{n-1}. The pair (x,p)(x',p) must be an (n1)(n-1)-cycle, which is the case just when dx=xd x' = x, so the data of attaching map turns out to be the pair xX n1x' \in X_{n-1} and yY n1y' \in Y_{n-1} subject to the condition dy+dfx=0d y' + d f x' = 0. Commutativity of the square corresponds to the condition that dy=y+fxd y = y' + f x', which of course implies that dy+dfx=0d y' + d f x' = 0. So in summary, the RR-module of commutative squares is given by the pullback

E n Y n d X n1Y n1 f+1 Y n1\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

nS n1[E n] X kP k[k] nD n[E n] X kP k[k]E k[k] Y\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 kP k[k]E k[k]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 nn, this coequalizer has the form:

X nP n X nP nE n \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(y,x)X nP nx \oplus (y,x') \in X_n \oplus P_n to x0(y,x0)x \oplus 0 \oplus (y, x' \oplus 0). This identifies P nP_n as a submodule of E nE_n, so the coequalizer is X nE nX_n \oplus E_n. Thus, the step-two factorization of the enriched algebraic object argument is

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

Using the differential we’ve just described on X kE k[k]X \oplus \oplus_k E_k[k], one can see that the nn-cycles are in the image of the map X kP k[k]X kE k[k]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 II 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 nX n1Y n(Mf)_n = X_n \oplus X_{n-1} \oplus Y_n. I’ll leave it to you to check that the map

X nE n X nX n1Y n x(y(x,y)) x(1) nxy\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 XX and to YY.

Posted at September 16, 2013 3:06 AM UTC

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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?

Posted by: Karol Szumiło on September 16, 2013 8:29 AM | Permalink | Reply to this

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

Hi Karol,

With regard to your question …

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?

…I did some work in my thesis which gives a precise reason: the cylinder and co-cylinder in chain complexes satisfy a certain ‘strictness of identities’ hypothesis.

I should also say that with regard to factorisation, the significance of this ‘strictness of identities’ hypothesis was independently observed by Richard Garner and Benno van den Berg in this paper.

I identify in the thesis some structures on a cylinder/co-cylinder which, if this strictness of identities hypothesis is satisfied, ensure that the notions of cofibration, fibration, and homotopy equivalence with respect to this cylinder/co-cylinder define a model structure. The strictness of identities hypothesis is crucial for proving not only that the factorisation axioms hold, but that the lifting axioms hold.

I only give one example in the thesis, that of categories/groupoids: the original motivation for the work is towards model structures on algebraic models of homotopy types, which should hopefully be available soon.

I mention a few more examples in the introduction to the thesis, including that of chain complexes. The topological case is also discussed in the introduction: as in Emily and Tobias’s paper (but by different methods!) it is possible to use the Moore co-cylinder to get around the fact that the usual co-cylinder does not satisfy strictness of identities.

Perhaps I can also mention that it is possible, I believe, to use these Hurewicz-type model structures to constructively obtain model structures on categories such as cubical sets, for which the weak equivalences do not in general come from a cylinder or co-cylinder. I hope that work on this will also be available soon.

The thesis is available here.

Posted by: Richard Williamson on September 16, 2013 2:41 PM | Permalink | Reply to this

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

I should acknowledge that it was Richard who first noticed the mistake in the Cole factorization, which is what induced Tobi and I to start to think about this sort of situation.

Richard, this sounds very neat. Does your thesis shed any light on the alternate definition of a Dold fibration given on the nnLab: as a map with the delayed homotopy lifting property?

Posted by: Emily Riehl on September 16, 2013 4:06 PM | Permalink | Reply to this

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

Hi Emily,

Does your thesis shed any light on the alternate definition of a Dold fibration given on the nLab: as a map with the delayed homotopy lifting property?

Interesting question, I had not thought about it before! My first guess would be that the methods of my thesis would not be very suited to saying anything about the ‘delayed homotopy lifting property’. The fact that the latter is equivalent to the other definition mentioned on the nLab page strikes me as something that is likely to be very specific to topological spaces, or a category very close to topological spaces.

It reminds me particularly of the work of Strøm here, here, and here. I somehow see the Moore co-cylinder as a way to carry out the idea behind Strøm’s work in a more ‘structural’ rather than ‘specifically topological’ way. If I recall correctly, this idea is due to Tobias and Bill Richter, who came up with at roughly the same time in the spring of 2012. I like to think of your very nice work with Tobias as taking this point of view a lot further.

As an aside, I feel that there are lot of insights in Strøm’s work from a ‘structural’ point of view, even though it may appear at a first glance very ‘specific to topological spaces’, which have perhaps not been as much appreciated as they could have been, at least amongst our generation.

For example, Theorem 9 in the second paper above is the first place that I know of where the significance of the ‘covering homotopy extension property’ for establishing the lifting axioms for a model category is recognised.

Crucial in this context is the lemma on the third page of the first paper, which strikes me as a fundamental fact about cofibrations that everybody should know! One of the things that I do in the thesis is to give a proof, different to Strøm’s, of this lemma in the abstract cylindrical setting in which I work.

To get back to your question, it seems to me at first glance plausible that one can use abstract cylindrical methods to prove that we have a Dold fibration in the sense of the first definition on the nLab page, by ‘keeping track of certain homotopies’ everywhere. My work is very focused upon proving that we have a Hurewicz fibration when strictness of identities is satisfied: whilst it might definitely be possible to ‘relax’ my methods to show that we have a Dold fibration for topological spaces, it might be easier to take a different approach, using for instance work on double mapping cylinders due to Marco Grandis. This is described for example in section 4.6 of this book. The book is written in the language of directed homotopy theory, but everything in section 4.6, and in many places elsewhere, works just as well in an ordinary cylindrical setting.

On a different note, thank you for the interesting blog post, which I enjoyed reading! It’s great to see that your work with Tobias is being taken further, and I look forward to the new paper!

Posted by: Richard Williamson on September 17, 2013 7:51 AM | Permalink | Reply to this

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

Thanks for an answer. Indeed, it seems that your “strictness of identities” answers my first question. Moreover, I think there is a connection between this condition and Dold fibrations. In fact, “strictness of identities” has a distinctly “delayed” feel to me. In notation of your Definition III.31, isn’t precomposing with q lsq_l s the same as “making a homotopy delayed”? Then “strictness of identities” would say that every homotopy actually agrees with its “delayification”, thus every lifting problem is delayed thus every Dold fibration is a Hurewicz fibration. If I’m right about this, then perhaps “subdivision structures on cylinders” are the right tool for talking about Dold type (co)fibrations in an abstract setting.

I am intrigued by your comment about cubical sets, could you elaborate a bit? I don’t know too much about cubical sets, but as far as I understand the main difficulty in constructing a model structure is the same as with simplicial sets, namely, identifying acyclic (co)fibrations. This part simplifies greatly in Hurewicz type structures since all objects are cofibrant and fibrant. In your approach is there some specific reason why this would work for cubical sets but not for simplicial sets? Or does it work for both?

Posted by: Karol Szumiło on September 17, 2013 8:33 AM | Permalink | Reply to this

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

In fact, “strictness of identities” has a distinctly “delayed” feel to me. In notation of your Definition III.31, isn’t precomposing with q lsq_{l}s the same as “making a homotopy delayed”? Then “strictness of identities” would say that every homotopy actually agrees with its “delayification”, thus every lifting problem is delayed thus every Dold fibration is a Hurewicz fibration.

That’s a very nice observation! Thanks!

I am intrigued by your comment about cubical sets, could you elaborate a bit?

Yes, I’ll do so below. My apologies for taking so long to get back to you, I did not get a chance before now. Thanks for your interest!

In your approach is there some specific reason why this would work for cubical sets but not for simplicial sets? Or does it work for both?

It should work for both. It’s just that I usually prefer to work with cubical sets, as I find the homotopy theory of cubical sets (with respect to its monoidal structure, which behaves ‘topologically’) nicer from a constructive and conceptual point of view. In particular, I find it much easier to construct explicit homotopies.

as far as I understand the main difficulty in constructing a model structure is the same as with simplicial sets, namely, identifying acyclic (co)fibrations. This part simplifies greatly in Hurewicz type structures since all objects are cofibrant and fibrant.

The approach I have in mind proceeds ‘in the opposite direction’ from the usual one. The idea is to begin by putting a Hurewicz type model structure on a ‘structured version’ of what will afterwards be the category of fibrant (say) objects in the model structure one is trying to construct. For cubical sets, one could, for example, take the category of algebraic Kan complexes: I believe that one can put a Hurewicz type model structure on this category by the methods of my thesis.

One then explicitly constructs what will afterwards be a fibrant replacement functor. For cubical sets with connections, one should be able to, for example, work with a cubical analogue of Kan’s Ex Ex^\infty functor.

If this fibrant replacement functor has certain nice properties, one can formally obtain a model structure on the category one is interested in: cubical sets in this case.

The method is very general. A similar idea allows one to construct, for example, a Bousfield localisation of this model structure on cubical sets (or the known model structure due to Cisinski, or the standard model structure on simplicial sets,…) to a model category of nn-types, without using the usual machinery.

Posted by: Richard Williamson on September 24, 2013 8:21 AM | Permalink | Reply to this

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

The approach I have in mind proceeds ‘in the opposite direction’ from the usual one. The idea is to begin by putting a Hurewicz type model structure on a ‘structured version’ of what will afterwards be the category of fibrant (say) objects in the model structure one is trying to construct. For cubical sets, one could, for example, take the category of algebraic Kan complexes: I believe that one can put a Hurewicz type model structure on this category by the methods of my thesis.

One then explicitly constructs what will afterwards be a fibrant replacement functor. For cubical sets with connections, one should be able to, for example, work with a cubical analogue of Kan’s Ex Ex^\infty functor.

I see. This strategy seems to have some common points with another one I have recently come across. Namely, instead of considering algebraic Kan complexes, we consider just ordinary Kan complexes and make them into a fibration category, e.g. as in Simplicial Homotopy Theory by Goerss and Jardine. Then we use this fibration category to induce the structure of a cofibration category on the entire category of simplicial sets. This is sufficient to give an elementary and purely combinatorial verification of all the crucial properties of Ex Ex^\infty which can be then used to construct the model structure as you pointed out.

Posted by: Karol Szumiło on September 26, 2013 12:57 PM | Permalink | Reply to this

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

The argument you suggest doesn’t seem to be in [Goerss and Jardine], though? Ex Ex^{\infty} is only introduced after the Kan–Quillen model structure on all simplicial sets is established. Joyal and Tierney also gave a purely combinatorial construction of the Kan–Quillen model structure, using the theory of anodyne extensions and minimal fibrations.

Posted by: Zhen Lin on September 26, 2013 5:04 PM | Permalink | Reply to this

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

No, I meant Goerss and Jardine only as a reference for the fibration category of Kan complexes (though to my taste their formulation is slightly overcomplicated by the use of simplicial homotopy groups). The next step is to create the cofibration category of all simplicial sets from the fibration category of Kan complexes via the mapping space functors into Kan complexes. I don’t know a reference for this, but it is not difficult. Once we have this we can prove that Ex Ex^\infty is a fibrant replacement functor using an argument similar to the proof of Lemma 5.4 in Barwick, Kan Relative categories: Another model for the homotopy theory of homotopy theories. We just need to realize that this argument doesn’t really rely on the model structure, just on the cofibration structure. (Their lemma is more complicated than the version we need. I would expect that there should be a more direct reference for this, but I don’t know one.)

Posted by: Karol Szumiło on September 26, 2013 7:19 PM | Permalink | Reply to this

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

Ah, I see. That’s more-or-less what Joyal and Tierney do – including for the definition of weak homotopy equivalence – but they use the small object argument to generate fibrant replacements instead of Ex Ex^{\infty}.

Posted by: Zhen Lin on September 26, 2013 8:27 PM | Permalink | Reply to this

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

Not really, I think these arguments are rather different. Actually, they both use the small object argument to construct factorizations. The difference is in the identification of acyclic (co)fibrations. Joyal and Tierney do it via minimal fibrations. In the argument I sketched above we use Ex Ex^\infty essentially as described in the book by May and Ponto. A minor innovation here is to use (co)fibration categories to establish that Ex Ex^\infty is indeed a fibrant replacement functor.

Posted by: Karol Szumiło on September 26, 2013 8:55 PM | Permalink | Reply to this

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

Peter or Tobi might have something useful to say about why the mapping cylinder factorization gives a Hurewicz fibration for chain complexes but only a Dold fibration for spaces. I’ll direct them here.

In the paper with Tobi, we use a further generalization of the “algebraic” small object argument because the generating trivial cofibrations in that case are a class of maps indexed by all topological spaces.

The way I like to think about step zero of the small object argument is that it constructs a generic lifting problem — a solution to that single lifting problem precisely characterizes the fibrations. Then step-one factorization is then a re-expression of that lifting problem: a map lifts against its step-one left factor if and only if it is a fibration.

For the Strøm model structure on spaces, the step-one factorization is given by a construction of Cole, but it still has the property just mentioned. Garner’s algebraic description of the small object argument can then be used to produce the desired model theoretic factorization.

A priori the factorization Tobi and I describe is unenriched, but enrichment does play a role in the topological case as well. Hurewicz fibrations turn out to have the enriched right lifting property with respect to the maps i A:AA×Ii_A\colon A \to A \times I; see § 5.2. The point is the functor that sends AA to the space of lifting problems from i Ai_A to ff is represented by the mapping cocylinder NfN f. So a solution to the generic lifting problem defines a continuous solution to all lifting problems in the form of a continuous function from the space of commutative squares to the mapping space from A×IA \times I to the domain of ff.

Posted by: Emily Riehl on September 16, 2013 3:01 PM | Permalink | Reply to this

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

Tobi reminds me that the first factor of the mapping cylinder factorization can also fail to be a cofibration in examples of interest: the mapping cylinder of 0X0 \to X is just XX, which in general might not be cofibrant. The example we have in mind is differential graded modules over a differential graded algebra. More on this will appear in the forthcoming joint paper with Peter.

Posted by: Emily Riehl on September 23, 2013 1:30 PM | Permalink | Reply to this

Re: Mapping (Co)cylinder Factorizations via the Small Object Argument

Very nice! What about simplicial enrichment?

Posted by: Mike Shulman on September 18, 2013 6:51 AM | Permalink | Reply to this

Re: Mapping (Co)cylinder Factorizations via the Small Object Argument

What do you mean?

Posted by: Emily Riehl on September 18, 2013 7:27 PM | Permalink | Reply to this

Re: Mapping (Co)cylinder Factorizations via the Small Object Argument

E.g. what enriched awfs’s does the enriched algebraic small object argument generate on simplicial sets from the usual generators?

Posted by: Mike Shulman on September 19, 2013 12:24 AM | Permalink | Reply to this

Re: Mapping (Co)cylinder Factorizations via the Small Object Argument

Sadly, I don’t think it’s easy to say because I don’t expect the enriched algebraic small object argument to converge before ω\omega.

Posted by: Emily Riehl on September 19, 2013 8:59 PM | Permalink | Reply to this

Re: Mapping (Co)cylinder Factorizations via the Small Object Argument

Ah, too bad.

Posted by: Mike Shulman on September 21, 2013 6:22 PM | Permalink | Reply to this

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