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Stream: theory: algebraic topology

Topic: resolutions, simplicial and otherwise

James Deikun (Sep 13 2021 at 12:44):

I've noticed that even though resolutions of objects tend to be unique up to a nice notion of weak equivalence, the constructions I've seen for building them (e.g. at the nLab and Jonas Dallendörfer's bachelor's thesis ) seems to be kind of ad hoc, relying on the fact that you can build whatever diagram you're building (exact sequence or (co)simplicial object) one object at a time so you can interleave that with factorizations of objects. This doesn't help with doing, say, simplicial resolutions in bicategories, where it seems like a reasonable translation would use codescent objects, which are 2 objects deep and would break everything.

Is there a nice characterization of at least, say, P-projective and P-injective resolutions as a kind of homotopy (co)limit or Kan extension or something similar that would allow computation in more general contexts?

Dmitri Pavlov (Oct 08 2021 at 16:32):

One way to systematically construct resolutions in a nice, functorial way is to use the bar and cobar constructions: https://ncatlab.org/nlab/show/bar%20construction, https://ncatlab.org/nlab/show/bar+and+cobar+construction

Mike Shulman (Oct 08 2021 at 17:42):

I wouldn't expect anything to break in a bicategory if you do everything bicategorically. What exactly are you envisioning, and where do you think codescent objects would come in and break things?

James Deikun (Oct 08 2021 at 19:22):

I would expect to start by taking a codescent object instead of a kernel pair, factor the arrows in it, and continue by taking two-arrow codescent object instead of two-arrow kernel triple, etc. But then I already have another object and a triple of arrows from the first codescent object and I can't see how to stick all that data together into something coherent.

Mike Shulman (Oct 08 2021 at 19:38):

I don't think I can follow that without more details.

James Deikun (Oct 08 2021 at 20:26):

In the 1-categorical case, https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/bachelor_dallendoerfer.pdf#section.3.2 discusses the construction of semisimplicial resolutions as a first step of constructing simplicial resolutions. The two key constructions are a sort of lesser weak factorization system that only promises to factorize initial arrows, and something called a "simplicial kernel". A "simplicial kernel" is a limit-like construction that takes $n$ parallel arrows $x_i : Y \rightarrow X$ and returns an object $Z$ and $n+1$ parallel arrows $y_i : Z \rightarrow Y$ . The case $n = 1$ is an ordinary kernel pair.

The construction of a simplicial resolution is a sandwich, you start the algorithm with $n=0$ and $Y=Z=X_0$ and $y_i = id_Y$ . Then at each step you factor $!_Z : 0 \rightarrow Z$ to get an arrow $p : P \rightarrow Z$ which you then precompose with all your $y_i$ to get a set of $x_i$. At this point you increment $n$ and set $X$ to the old $Y$ and $Y$ to $P$ and then you take the simplicial kernel to get a new $Z$ and set of $y_i$ and you can repeat forever.

In a bicategory I would expect to use a codescent object instead of a kernel pair for the first step. A codescent object is illustrated at https://golem.ph.utexas.edu/category/2014/06/codescent_objects_and_coherenc.html under the heading "lax codescent objects" ... I would expect to need a "wide" version of this similar to the "simplicial kernel" but it's pretty easy to see what that would be. The trouble, however, already begins at the first application, since now instead of $Z$ being at the tip of the whole diagram I've constructed so far, it's somewhere in the middle, with three arrows $p$ $q$ and $r$ behind it, and I can't see how to tie those three arrows and the object $X_3$ attached to them into the rest of the process.

James Deikun (Oct 08 2021 at 20:27):

I hear you can draw diagrams on here; this would probably be a bit clearer if I could figure out how ...

John Baez (Oct 08 2021 at 20:27):

By the way, you need to use two dollar signs each time you're used to using one.

John Baez (Oct 08 2021 at 20:28):

You can edit your comment to fix those... yeah, it's annoying.

James Deikun (Oct 08 2021 at 20:37):

Sorry, keep learning that and then forgetting it...

Mike Shulman (Oct 08 2021 at 20:45):

It's hard to say for sure without knowing what you're trying to achieve -- just wanting to "categorify something" without a concrete goal in mind doesn't always lead to meaningful results -- but my guess would be that the process would look exactly the same except that the simplicial kernel would be a codescent-like-object involving not just the last family of parallel arrows but the also ones that came before them.

James Deikun (Oct 08 2021 at 20:52):

Ah, that would make it harder to get the process started, though. As for what I'm trying to do ... I'm trying to find a general process for constructing a canonical simplicial nerve. In particular I am trying to construct a fully faithful simplicial nerve for virtual equipments because general abstract considerations around the Duskin nerve make it seem like there should be one.

Mike Shulman (Oct 08 2021 at 20:55):

Well, at the first step there's no extra data to worry about, so there's nothing wrong with just taking the ordinary kernel pair.

James Deikun (Oct 08 2021 at 20:56):

(In particular, it seems like a Duskin nerve based on a particular orientation of the simplices should be the composition of the inclusion of bicategories as vertically trivial virtual equipments with all composites, and the nerve i'm looking for ...)

James Deikun (Oct 08 2021 at 21:00):

Maybe that could work. I might need to make it a comma object though ...

James Deikun (Oct 08 2021 at 21:05):

Perhaps the general formula is to repeatedly extend the entire resolution constructed so far by one level, and it just happens in 1-category theory that the rest of it gets screened off by the most recent level.