Hello nForum,

I’ve been playing with something a little bit, and I think it’s impossible but I’d like to get another opinion.

Say C is a V-enriched category, and let’s suppose for simplicity that I want to carry out the small object argument with respect to a single map i. Then, the V-enriched small object argument should proceed by, for any morphism f, taking the pushout of the map Sq(i,f) \otimes i –> f. Here, Sq(i,f) is the “V-object of commuatative squares from i to f”, i.e. the hom-object in the arrow category of C. Note that this doesn’t really give the resulting left class of maps as transfinite compositions of pushouts of *coproducts* of i, but rather of *V-tensors* of i. If C is now an \infty-category, the same construction should be possible; one abstract way to describe the map Sq(i,f) \otimes i –> f is as the component at f of the natural transformation associated to a Kan extension of the functor i:pt –> Arr(C) along itself.

However, what I would like – and what I think is probably impossible – is to give a small object argument in an \infty-category in which the left class of maps is still a transfinite composition of pushouts of *coproducts* of i, but which is nevertheless functorial (i.e. it yields a section of a composition functor). Without the functoriality, this is easy – one can basically just use the classical argument, as is done e.g. in Prop. 1.4.7 of DAG X. Of course, right there is Warning 1.4.8, which claims that this cannot be made functorial in general.

My original (probably naive and foolish) hope was that, by choosing a bunch of sections to trivial fibrations (i.e. by choosing compositions, pushouts, direct limits, possibly et al.) I might be able to do it anyways. However, this seems to run into problems. For instance, given a map f –> f’ in Arr(C), after making such choices we can specify a map sq(i,f) –> sq(i,f’), where by sq(i,f) I now mean the *set* of maps \Delta^1 x \Delta^1 –> C restricting to i and f. But then, given a 2-simplex f –> f’ –> f” in Arr(C), making everything compatible would at the very least necessitate choosing an associator for my composition. I don’t feel confident enough in my ability to manipulate quasicategories to pursue this further. Maybe somebody else can see how to extend this, or can see why this has no choice but to fail.

Another half-baked idea I had was, at least in the case that my \infty-category is presentable, to choose a left-proper combinatorial simplicial model category presenting it, and then I could choose a representative of the map i that’s a cofibration between cofibrant objects (so that, by left-properness, pushouts of it would model homotopy pushouts, i.e. pushouts in the underlying \infty-category). But once again it seems like if I ignore the simplicial enrichment, I’m going to destroy the simplicial structure just the same.

So: can anyone confirm this suspicion? I certainly believe the heuristic that if you ignore the enrichment in whatever construction you’re making then you’re probably going to destroy it, but I’d be interested to see a more precise no-go theorem (or perhaps a construction of the functorial small object argument in an \infty-category that I was originally after!).

Thanks,

Aaron

P.S. I wasn’t sure what I was supposed to put under “tags”; anyone should feel free to re-tag this as appropriate.

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