Timeline for Are simplicial finite CW complexes and simplicial finite simplicial sets equivalent?
Current License: CC BY-SA 4.0
18 events
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| Aug 20, 2019 at 21:07 | comment | added | Ben Wieland | The point of the original question is that a model does not support every morphism. Tyler's example of a functor from $B\mathbb N$ is just the choice of an object and a morphism. If you allow Kan complexes or finite CW complexes, then they are cofibrant-fibrant and thus they support arbitrary morphisms. In the purely homotopical setting, the object supports the morphism. Tyler's example was $S^1$, which is as nicely cellular as you can imagine: represent it as the constant simplicial object. Maybe having to spell out the cells allows counterexamples, but not that one. | |
| Aug 20, 2019 at 4:44 | comment | added | Gregory Arone | @BenWieland I am feeling a little slow. Why it is not a counterexample? | |
| Aug 19, 2019 at 23:24 | comment | added | Tyler Lawson | @BenWieland yes, absolutely. you can replace each object in a diagram with any weakly equivalent object and get an equivalent diagram at the cost of working coherently. | |
| Aug 19, 2019 at 21:28 | comment | added | Ben Wieland | If the "more general" version is purely homotopical, Tyler's example isn't a counterexample, is it? (a counterexample to an even more general version) And a nitpick, just a problem for objects, if a "finite homotopy colimit" includes retracts, then you run into Wall finiteness problems, but @R.vanDobbendeBruyn's version doesn't. | |
| Aug 19, 2019 at 17:32 | comment | added | R. van Dobben de Bruyn | A possible more general finiteness condition could be the assumption that every object in the diagram has finitely many maps out of it. (This has popped up in some inductive constructions I ran into; examples include finite diagrams and semisimplicial objects.) | |
| Aug 19, 2019 at 17:23 | history | edited | Gregory Arone | CC BY-SA 4.0 |
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| Aug 19, 2019 at 16:39 | history | edited | Gregory Arone | CC BY-SA 4.0 |
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| Aug 19, 2019 at 16:32 | history | edited | Gregory Arone | CC BY-SA 4.0 |
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| Aug 19, 2019 at 16:28 | comment | added | Tyler Lawson | @AchimKrause That is, indeed, a great deal simpler than what I had in mind. | |
| Aug 19, 2019 at 16:20 | comment | added | Achim Krause | In Tyler's example, can't we just observe that for any finite simplicial set $K$, only finitely many maps on $\pi_1$ are induced by endomorphisms $K\to K$? This is because there are only finitely many maps $K\to K$ in total. | |
| Aug 19, 2019 at 15:17 | history | edited | Gregory Arone | CC BY-SA 4.0 |
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| Aug 19, 2019 at 15:12 | comment | added | Gregory Arone | Hi Tyler, thanks. This sounds right and suggests I should impose a local finiteness condition on the indexing diagram (such as, there can be only finitely many morphisms between any two objects). What I really wanted to know was about simplicial objects: can any simplicial object in finite CW complexes be approximated by a simplicial object in finite simplicial sets? I will edit the question. | |
| Aug 19, 2019 at 15:10 | history | edited | Gregory Arone | CC BY-SA 4.0 |
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| Aug 19, 2019 at 14:07 | comment | added | Tyler Lawson | Hi Greg, I suspect that the map $B\Bbb N \to CW$, representing $S^1$ with the endomorphisms $2^k$ for $k > 0$, can't be realized by a diagram of finite simplicial sets, but to prove it I'd want to come up with some explicit growth condition on the number of edges needed to represent elements of $\pi_1$ for a fixed finite model. This may be a little harder for s.sets than for $\infty$-categories because the mapping complexes aren't Kan complexes. | |
| Aug 19, 2019 at 9:15 | history | edited | Gregory Arone | CC BY-SA 4.0 |
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| Aug 19, 2019 at 8:08 | history | edited | Gregory Arone | CC BY-SA 4.0 |
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| Aug 19, 2019 at 7:58 | history | edited | Gregory Arone | CC BY-SA 4.0 |
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| Aug 19, 2019 at 7:48 | history | asked | Gregory Arone | CC BY-SA 4.0 |