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I recently rediscovered a classical theorem from Hatcher which states that simply-connected CW complexes have a 'minimal' cell structure, where the cells correspond to spheres and disks indexed by the free and torsion generators of homology respectively.

I was wondering if there is a general collection of results like this in (combinatorial) model categories, where the cells of a minimal object correspond to generators of an algebraic invariant.

A couple of examples that come to mind:

  1. The cells in the Sullivan minimal model in rational homotopy theory correspond to the generators of the rational homotopy groups of a space.
  2. The 'minimal model' in chain complexes over a PID should have a sphere-disk structure similar to the classical Hatcher theorem.
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    $\begingroup$ I'm really no sure there is any chance of having a general theory of thess - and in any case I'm almost certain it doesn't exist yet. Though maybe another things that can be considered an example is the theory of minimal Kan complex and minimal fibrations. This has been extended to a certain class of Reedy model structure in arxiv.org/abs/1509.01073 $\endgroup$
    – Simon Henry
    Commented Feb 7 at 17:09
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    $\begingroup$ I would say that Andre-Quillen homology is a fairly general machination satisfying what you want. The general idea is that Andre-Quillen homology measures the free cells of your objects, and this is usually enough to detect equivalences by a spectral sequence argument. $\endgroup$
    – Connor Malin
    Commented Feb 8 at 3:25
  • $\begingroup$ Thank you both for the suggestions! @SimonHenry is there something you had in mind as to why a general theory might not be possible? $\endgroup$
    – kelly maggs
    Commented Feb 8 at 8:44
  • $\begingroup$ @kellymaggs just the fact that there are many situation where I don't see how to do this, or the fact that the different example I can think off don't really look similar make me very doubtfull there is a general theory that apply to any model structure - but I could be wrong! $\endgroup$
    – Simon Henry
    Commented Feb 8 at 15:15

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If you want to generalize the intuition from "minimal cell structures" in topological spaces to model categories, you actually probably want to be thinking about cellular model categories rather than combinatorial ones. I think of combinatorial model categories as a good place to generalize arguments based on powerful results in category theory, like results that work well for simplicial sets. But, the category of topological spaces is not locally presentable, so arguments of combinatorial model categories don't work. But, the notion of a cellular model category was invented with topological spaces in mind.

The best source to read about cellular model categories is Hirschhorn's book, specifically chapters 11 and 12. Chapter 11 gives a whole theory of cell structures (known as cell complexes) and Chapter 12 explains how in a cellular model category, that structure lets you do homotopy theory. There's a theory of subcomplexes, intersections of subcomplexes, and bounding the size of a cellular presentation (section 12.5). The main motivation for cellular model categories is to do Bousfield localization, and section 4.5 explains the Bousfield-Smith cardinality argument, which is done first for spaces (in chapter 2) and then for cellular model categories in 4.5. Definition 4.5.3, Lemma 4.5.4, and Prop 4.5.5 are all about finding small cell structures, small enough to make the cardinality arguments work. The material in Section 10.6, about cellular presentations and subpresentations (smaller ones). The theory stops short of trying to get a minimal cell structure, but it does set up a poset of structures. So, if you had a model category that was both cellular (to have cells) and combinatorial, then you could probably get a minimal one. This kind of question reminds me of left-determined model structures, for some reason. But, since Hirschhorn tells you how to take intersections, maybe you don't even need local presentability.

Now let's think about your two examples. First, models for spectra (build on either spaces or simplicial sets) are cellular, so rational spectra are too. Model structures on chain complexes are also commonly cellular (and combinatorial) if they are cofibrantly generated. So, you certainly have a "sphere-disk" cellular presentation like you want, and can try to generate a minimal one via the intersection theory of cellular presentations from Hirschhorn's book.

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  • $\begingroup$ Thanks for the references! This is what I was looking for $\endgroup$
    – kelly maggs
    Commented Feb 27 at 8:44

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