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Stream: theory: category theory

Topic: Coskeletal Induction (seeking reference)

Noah Chrein (Feb 26 2024 at 19:38):

Consider the coskeleton adjunction:
$[\Delta_{\leq n-1}^\text{op},\text{Set}] \underset{i_!}{\overset{i_*}\leftrightarrows}[\Delta_{\leq n}^\text{op},\text{Set}]$ for $\Delta_{\leq n} \overset{i}\hookrightarrow \Delta_{\leq n+1}$ stacks
$i_!$ cotrivially creates n simpleces by pasting together n-1 simplecies. I'd like to associate data $A:\text{Set}$ to these "empty" n-simpleces:

given $\Sigma:[\Delta_{\leq n-1}^\text{op},\text{Set}]$, the coskeleton $i_!\Sigma:[\Delta_{\leq n}^\text{op},\text{Set}]$, and some $A\to i_!\Sigma_n:\text{Set}$ I would like a $\Sigma[A]:[\Delta_{\leq n}^\text{op},\text{Set}]$. $\Sigma[A]$ interprets the arbitrary data of A as having lower dimensional faces which the coskeleton provides. One can imagine doing this repeatedly to have some precise control in building a simplicial set from the ground up, hence "induction".

Here is a picture, where data $\in A$, $i_!\Sigma:[\Delta_{\leq 2}^\text{op},\text{Set}]$ and $A \to i_!\Sigma_2:\text{Set}$

One must be careful of degeneracies, I have also worked this out in higher generality, and can provide more detail. I'd rather not clutter the concept unless prompted. I have reached a point in the general pursuit that has stumped me, so I am just wondering if there is a reference for this kind of idea.

Thank you for any leads!

Kevin Arlin (Feb 26 2024 at 19:45):

If I follow your description correctly, $\Sigma[A]$ is simply the smallest subobject of $i_!\Sigma$ containing the $n$-simplices in $A.$ There are lots of ways to formalize this; one is to transpose the map $A\to i_!\Sigma_n$ into a map $\sqcup_{a:A} \Delta^n\to i_!\Sigma$ and then take its canonical image factorization to construct $\Sigma[A].$

Noah Chrein (Feb 26 2024 at 20:08):

Thank you, this kills a bit of the generality because it requires tensoring over sets, ($A\cdot \Delta^n$) which is fine for truncated simplicial sets as I presented here. Regardless the general situation (which I used fibered categories) does involve an image factorization as you say. I will post more when J return to my computer.

This does give me a bit to think about thank you.

James Deikun (Feb 26 2024 at 21:02):

In the case of simplicial sets, it seems like what you're doing here is following the way that $\Delta$ is built up as a Reedy category by iteratively gluing elements of the [[Isbell envelope]] to build up a simplicial set. You can even view them as the same process, gluing elements of the Isbell envelope that lie over the next simplex to build up the category of simplices inductively.

James Deikun (Feb 26 2024 at 21:05):

It also reminds me a bit of building up free structures on generalized computads, only here maybe all the things end up being free?

Noah Chrein (Feb 26 2024 at 21:52):

James Deikun said:

building up free structures on generalized computads

This sounds like what I am looking for, I will google around this concept. Do you have any suggested papers? I imagine there's some stuff by Riehl.

I've formalized the ideas, but my formalizations are a bit unwieldy as it stands so a standard construction like this will help a lot, thank you.

James Deikun (Feb 26 2024 at 22:43):

The concept is pretty new. The main thing I'm familiar with is Markakis. I don't know of any followup on it and the way you deal with degeneracies could be considered new work in the area AFAIK, though it's hard to tell for sure without more information on how you do it.

Noah Chrein (Feb 27 2024 at 02:10):

Thank you James!

Noah Chrein (Feb 28 2024 at 17:50):

I should say a bit more why I am considering this. I have not fully read through the computad literature, so I am not aware that the following is the typical use. For brevity, please excuse my informality in what follows:

We often only give the structure of objects of a category $\mathbb C$, with the implication that the (higher) morphisms are obvious. I think this is almost always due to the presence of some meta-theory $\mathbb C at_*$ and a schema $\Delta:\mathbb C at_*$ so that $\text{ob}(\mathbb C):\mathbb Cat_*(\Delta,\mathcal S)_0$ . The "obvious" morphisms belong to the full category typed $\mathbb C:\mathbb Cat_*(\Delta,\mathcal S)$. One extracts higher morphisms from the data of $\Delta$ via some form of yoneda lemma. This typically looks like $\text{mor}(\mathbb C) := \mathbb C^{\Delta^1} =\mathbb Cat_*(\Delta,\mathcal S)(\Delta^1, \mathbb C)$ .

Sometimes the "obvious" (induced) morphisms have the incorrect structure, hence we need a further specification for the morphisms, $\text{mor}(\mathbb C):\neq \mathbb C^{\Delta^1}$.

as a simple example, consider the free category fc(G).
we have
$\text{ob}(fc(G)) = G_0$
$\text{mor}(fc(G)) = \left [\sum_n(G^{\text{sk}_1(\Delta^n)})\right ]_0$
With coskeletal induction $\left [\sum_n(G^{\text{sk}_1(\Delta^n)})\right ]_0 \to \text{cosk}_1(G_0)$

It would be nice if $fc:\text{Graph}\to \text{Graph}$ were polynomial, but $fc(G) \neq \sum_n(G^{\text{sk}_1(\Delta^n)})$ is the wrong graph. So perhaps $fc$ is instead "inductively polynomial", i.e. each step of the coskeletal induction is polynomial.

Is this a standard method for formally specifying morphisms with different structure than their objects?

Noah Chrein (Feb 28 2024 at 18:00):

A more simple example:
$\text{ob}(\text{span}(\mathbb C)) = \mathbb C_0$
$\text{mor}(\text{span}(\mathbb C)) = \left (\mathbb C^{\leftarrow\cdot\rightarrow}\right)_0$