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Stream: theory: algebraic topology

Topic: realisation-nerve comonad, nondegenerate cells

Naso (Dec 19 2022 at 02:05):

Earlier this year I asked a question about the following functor:

$nd : \mathsf{sSet} \to \mathsf{Set}$ where $nd(X) = \text{set of nondegenerate simplices of } X$, and $nd(f) = x \mapsto y$ where $y$ is the unique nondegenerate simplex of $f(X)$ such that there exists a surjection $\phi \in \Delta$ with $f(x) = y \circ \phi$ (this comes from the 'Eilenberg-Zilber lemma').

We recently used this functor $nd : \mathsf{SSet}_+ \to \mathsf{Set}$ (but from augmented simplicial sets to simplicial sets) in a submitted paper, and a reviewer informed that actually $nd$

"is in fact a composition of two functors: a realization functor from simplicial sets to preordered sets, followed by the forgetful functor from preorders to sets. The realization functor is left adjoint to the nerve functor, thus it means that you implicitly work with some comonad structure. Perhaps it's worth checking."

This is because we postcomposed $nd$ with the (augmented) simplicial nerve functor $N : \mathsf{Pro} \to \mathsf{SSet}_+$ from prosets (preordered sets) to augmented simplicial sets.

I have two questions about this:

1. How can I verify that $D = U \circ \tau$? I think the formula for $\tau$ is $\tau X = colim_{\Delta^n \to X} [n]$ but I'm not sure how to compute it in practice. I think another name for $\tau$ is the 'fundamental category' functor.
   It seems reasonable, since the realization is gluing simplices together and the degenerate ones with the same generator should get identified in the colimit (?)

2. More vaguely... what could this comonad structure possibly do for me? Are there some examples of interesting uses of this particular comonad $\tau \circ N$?

By the way, my paper is on arxiv and this composition appears on page 5 (there $D = nd$). The topic of the paper is distributed computing.

Thank You

Paolo Capriotti (Dec 19 2022 at 16:21):

naso said:

1. How can I verify that $D = U \circ \tau$? I think the formula for $\tau$ is $\tau X = colim_{\Delta^n \to X} [n]$ but I'm not sure how to compute it in practice. I think another name for $\tau$ is the 'fundamental category' functor.
   It seems reasonable, since the realization is gluing simplices together and the degenerate ones with the same generator should get identified in the colimit (?)

If you mean $\tau: \mathbf{sSet}_+ \to \mathbf{Pro}$ given by that colimit formula, I don't think this is correct. For example $\tau(\Delta[n]) = [n]$, and $U([n])$ has $n + 1$ points, while $\mathsf{nd}(\Delta[n])$ should have $2^{n+1}$.

I think you can get $\mathsf{nd}$ as the composition of the restriction functor from simplicial sets to presheaves on the wide subcategory of $\Delta$ consisting of surjective maps, followed by the colimit functor. Not sure if there really is a comonad hiding in all this.