Homotopy Theory

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1:59 PM

Hi guys, I am dealing with homotopy coherent diagrams of chain complexes. I want to obtain a spectral sequence in the end.
Say you have a simplicial functor F:N(Delta) \to Ch and you want to compute the (cohomology of the) inifnity limit. If it was a strict functor Delta \to Ch, it is known that one can take the alternate sum of faces and get a double complex: its totalization computes the cohomology of its homotopy limit. Is there something similar one can do with homotopy coherent cosimplicial object? For example it would be cool if one could find a strict resolution involving the use of …

Thanks!!

I hope I have not said (too much) stupid things... :D

I think that Ch is a somewhat nice infinity category, probably it is a combinatorial simplicial model category (maybe if one restricts the attention to bounded below complexes). I guess Corollary 4.2.4.8. in HTT gives a hint on how to compute infinity limits via homotopy limits in the model category, but I am not confident enough to go through the details :(

Dylan Wilson

@AndreaMarino yes you can do this, you don't need to strictify. Apply Remark HA.1.2.4.4 to the opposite category

(of course, you have to be concerned about whether or not this sseq converges)

Andrea Marino

2:24 PM

Thank you very much Dylan!!! You saved me a lot of time.. :D yes, I will inspect the convergence, but the "infinity dold-kan correspondence" is exactly what I was searching for. I have to understand then how to get a spectral sequence from a functor N(Z>= 0) to Ch, but this seems to be well explained in the section of Lurie.

1 hour later…

Andrea Marino

3:27 PM

Let me ask another small thing. In the strict context, the totalization of the associated Dold-Kan complex is quasi isomorphic to the homotopy colimit of the simplicial chain complex. Then, if one wants to, can use spectral sequence to approximate the result. Do we have something similar in the non strict context, that gives an explicit construction for a complex quasi-isomorphic to the hocolimit of the (non strict) simplicial chain complex?

Adrian Clough

4:03 PM

A new diagram editor, Quiver, has been made available. I have played around with it a little bit, and I find it very easy to use. It can draw higher categorical diagrams up to level three. varkor.github.io/blog/2020/11/25/announcing-quiver.html

1 hour later…

Andrea Marino

5:14 PM

wow, that is a very cool tool. It can literally save hours of work. Added to my bookmarks