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1. • CommentRowNumber1.
   • CommentAuthoramg
   • CommentTimeMar 26th 2014
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   Author: amg
   Format: MarkdownItexHello 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 27th 2014
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   Author: Mike Shulman
   Format: MarkdownItexWhy do you want to avoid tensors?

   Why do you want to avoid tensors?

3. • CommentRowNumber3.
   • CommentAuthoramg
   • CommentTimeMar 27th 2014
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   Author: amg
   Format: MarkdownItexFor computational reasons. One of the main applications I have in mind is to carry out something akin to the cofibrant resolutions given by the Dwyer--Kan--Stover [E2-model structure](https://www3.nd.edu/~wgd/Dvi/EtwoModelCategory.pdf), which in the case of based simplicial spaces are given inductively by latching on wedge-sums of spheres. The point is to apply homotopy/homology to the resulting resolution, and these functors behave well with respect to coproducts but not with respect to tensors against arbitrary spaces. I think I can get away with non-functorial resolutions, but it'd surely be much more convenient if they were functorial.

   For computational reasons. One of the main applications I have in mind is to carry out something akin to the cofibrant resolutions given by the Dwyer–Kan–Stover E2-model structure, which in the case of based simplicial spaces are given inductively by latching on wedge-sums of spheres. The point is to apply homotopy/homology to the resulting resolution, and these functors behave well with respect to coproducts but not with respect to tensors against arbitrary spaces. I think I can get away with non-functorial resolutions, but it’d surely be much more convenient if they were functorial.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 28th 2014
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   Author: Mike Shulman
   Format: MarkdownItexSo, in classical model category theory, the small object argument produces things that have an $\infty$-categorical universal property --- for instance, for localizations, they are the reflections into a subcategory --- and hence are automatically $\infty$-categorically functorial. The issue is only that they might not be enriched-functorial on the point-set level unless you use tensors instead of coproducts. But there, because of the universal property, the construction using coproducts is weakly equivalent to the construction using tensors, so you could use the first for computations and the second for general theory. Are you looking at some kind of $\infty$-categorical SOA that produces objects *not* having a universal property?

   So, in classical model category theory, the small object argument produces things that have an $\infty$-categorical universal property — for instance, for localizations, they are the reflections into a subcategory — and hence are automatically $\infty$-categorically functorial. The issue is only that they might not be enriched-functorial on the point-set level unless you use tensors instead of coproducts. But there, because of the universal property, the construction using coproducts is weakly equivalent to the construction using tensors, so you could use the first for computations and the second for general theory. Are you looking at some kind of $\infty$-categorical SOA that produces objects not having a universal property?

5. • CommentRowNumber5.
   • CommentAuthoramg
   • CommentTimeMar 28th 2014
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   Author: amg
   Format: TextRight, I don't necessarily want to be constructing something with an \infty-categorical universal property. Here's another example where I'd like to apply this, which should make clearer just how evil I want to be. Let \pi_0 : sSpaces <==> sSets : disc be the evident adjunction of \infty-categories (where sSets is just a 1-category considered as an \infty-category with essentially discrete hom-spaces), and let I_Quillen be the usual set of generating cofibrations for the Quillen model structure on sSets. I've shown that in sSpaces, any map with rlp(disc(I_Quillen)) becomes an equivalence upon geometric realization (i.e. upon application of the functor colim : sSpaces --> Spaces). So for instance, given any object Y \in sSpaces, we can factor the map \emptyset --> Y via a non-functorial SOA with respect to disc(I_Quillen) to get \emptyset --> Y' --> Y, where Y' is a levelwise-discrete sspace (i.e. it's in the essential image of disc : sSets --> sSpaces) and the map Y' --> Y becomes an equivalence upon geometric realization. It's this sort of operation that I'd like to make functorial inside of (some fixed choice of model for) the \infty-category sSpaces, despite the fact that obviously it won't *actually* be natural in any real sense.

   Right, I don't necessarily want to be constructing something with an \infty-categorical universal property.
   
   Here's another example where I'd like to apply this, which should make clearer just how evil I want to be. Let
   
   \pi_0 : sSpaces <==> sSets : disc
   
   be the evident adjunction of \infty-categories (where sSets is just a 1-category considered as an \infty-category with essentially discrete hom-spaces), and let I_Quillen be the usual set of generating cofibrations for the Quillen model structure on sSets. I've shown that in sSpaces, any map with rlp(disc(I_Quillen)) becomes an equivalence upon geometric realization (i.e. upon application of the functor colim : sSpaces --> Spaces). So for instance, given any object Y \in sSpaces, we can factor the map \emptyset --> Y via a non-functorial SOA with respect to disc(I_Quillen) to get
   
   \emptyset --> Y' --> Y,
   
   where Y' is a levelwise-discrete sspace (i.e. it's in the essential image of disc : sSets --> sSpaces) and the map Y' --> Y becomes an equivalence upon geometric realization. It's this sort of operation that I'd like to make functorial inside of (some fixed choice of model for) the \infty-category sSpaces, despite the fact that obviously it won't *actually* be natural in any real sense.
6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMar 29th 2014
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   Author: Mike Shulman
   Format: MarkdownItexOkay, then I think I'm out of ideas. My guess is that what you want may not be possible.

   Okay, then I think I’m out of ideas. My guess is that what you want may not be possible.

7. • CommentRowNumber7.
   • CommentAuthoramg
   • CommentTimeMar 29th 2014
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   Author: amg
   Format: TextAlright. Thanks anyways for taking a look, I appreciate it!

   Alright. Thanks anyways for taking a look, I appreciate it!