There seem to be two definitions of what a saturated class should be:

  1. A class of morphisms closed under retracts, pushouts and transfinite composition.
  2. A class of monomorphisms containing all isomomorphisms, closed under retracts, pushouts, arbitrary coproducts and countable composition.

My question is, does it make a difference which definition one choses in the context of the cofibrantly generated modelstructures on

  • $\mathbf{Top}$, with respect to the Quillen model structure
  • $\mathbf{sSet}$, with respect to the Quillen (or Kan) Model structure
  • $\mathbf{Cat}$, with respect to the Thomason model structure

(I'm perfectly happy with partial answers, so if someone's got an answer for at least one of the listed categories, feel free to post it)


1 Answer 1


It doesn't make a difference as long as we restrict attention to compactly generated saturated classes, i.e. cofibrantly generated ones where generators can be chosen to have $\aleph_0$-small domains. This is the case in all your examples. (In the case of topological spaces, compact spaces are not $\aleph_0$-small with respect to all maps, but they are with respect to sufficiently many maps for the small object argument to go through.)

Condition 1 always implies condition 2 regardless of any smallness assumptions. If you have an arbitrary coproduct you can well-order the indexing set and represent it as a transfinite composite where you attach the summands one step at a time.

Condition 2 implies condition 1 under the smallness assumption I stated in the first paragraph. Indeed, by the small object argument a class satisfying condition 2 is characterized by lifting properties and hence it is also closed under transfinite composition.

Clearly, if domains of the generators are only $\kappa$-small, then the argument works if we replace countable composites in condition 2 by transfinite composites of length $\kappa$.

  • 1
    $\begingroup$ Only after writing my response I noticed that you assumed that all maps are monomorphisms in your condition 2. I don't think I have ever seen this condition in the definition of a saturated class. Clearly, this will not follow from condtion 1. However, I believe that the question you have in mind is independent of whether cofibrations in your model categories happen to be monomorphisms. $\endgroup$ Jun 23, 2016 at 11:39
  • $\begingroup$ Goerss and jardine define saturated class that way in "Simplicial homotopy theory" (with simplicial sets in mind), which is why I included it. But you're correct with your assumption. $\endgroup$ Jun 23, 2016 at 11:46

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