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http://www.math.ucr.edu/home/baez/trimble/bar.html I am reading this interesting note by Todd Trimble.

I will put the context first and put my question at the end.

Let $\Delta$ denote the category of finite ordinals (including the empty ordinal) and $\oplus$ denotes ordinal addition, which we regard as a monoidal bifunctor. We will adopt the convention that $[n]$ is the finite ordinal containing $0,\dots, n-1$, but not $n$, which conflicts with the usual convention in topology. We define the category of simplicial sets to be the functor category $[\Delta^{op},\mathbf{Sets}]$; again, this conflicts with the usual terminology, where simplicial sets refers to contravariant functors on the category of nonempty finite ordinals, and what we are calling simplicial sets are called augmented simplicial sets.

Define the decalage comonad on $\mathbf{SSet}$ to be the functor which sends $X$ to $X(-\oplus [1])$. It is endowed with a comonad structure as follows: the unique map $e : [0]\to [1]$ in $\Delta$ induces a natural transformation from the functor $-\oplus [0]$ to $-\oplus [1]$ in $[\Delta, \Delta]$; or equivalently a natural transformation $-\oplus [1]\to id_{\Delta^{op}}$ in $[\Delta^{op},\Delta^{op}]$. The functorial nature of functor composition induces, for each simplicial set $X$, a natural transformation $X\circ id_{\Delta^{op}}\to X\circ (-\oplus [1])$, or in other words a map of simplicial sets $X \to X(-\oplus [1])$, as desired.

The comultiplication is given by the identical process except that we start with the unique map $m : [2]\to [1]$.

This functor is of the form ``precompose with a fixed functor'' and so its left/right adjoint, if it exists, is given by finding the left/right Kan extension of a simplicial set $X$ along the functor $-\oplus [1]: \Delta^{op}\to \Delta^{op}$. Indeed, as $\Delta^{op}$ is small and $\mathbf{Sets}$ is (co)complete, the functor has both a left and a right Kan extension. We are interested in the left adjoint, which we call $C : \mathbf{SSet}\to \mathbf{SSet}$.

$C$ is called the cone monad. We explain the terminology as follows. First, that it is a monad follows from the Eilenberg-Moore theorem (see, for example, Sheaves in Geometry and Logic by Mac Lane and Moerdijk). Introduce, momentarily, the notation $\mathbf{S_+Set}$ to denote what topologists call the category of simplicial sets; that is,

$\Delta_+$ is the full subcategory of $\Delta$ consisting of nonempty finite ordinals.

There is an adjunction between $\mathbf{SSet}$ and $\mathbf{S_+Set}$, $\mathbf{S_+Set}\dashv \mathbf{SSet}$, where the right adjoint $\mathbf{SSet}\to \mathbf{S_+Set}$ is the obvious truncation functor which simply deletes $X_0$, and the left adjoint, which we can here call the augmentation functor $\mathbf{A}$, which appends to a given simplicial set $X_1\leftleftarrows X_2\dots$ the augmentation $X_0 = \Pi_0(X)$, where $\Pi_0$, the ``connected components'' functor, returns the coequalizer of the two maps $d_0,d_1: X_2\to X_1$.

I ask you to consider now the composition

$$\mathbf{S_+Set}\xrightarrow{\mathbf{A}}\mathbf{SSet}\xrightarrow{C}\mathbf{SSet}\xrightarrow{tr}\mathbf{S_+Set}\xrightarrow{|-|}\mathbf{Top}$$ which simply converts an unaugmented simplicial set into an augmented simplicial set, applies the cone monad, and returns the geometric realiation. Trimble claims that this is equivalent to taking the geometric realization to begin with, and then forming the mapping cone (he perhaps means mapping cylinder?) of the obvious map from $|X|$ into the set $\Pi_0(X)$ of connected components of the simplicial set $X$.

My question is about the last line. I don't know how to prove this. I am familiar with the basic properties of coends and Kan extensions, and I've been trying to fiddle with the "join" of simplicial sets by the Day convolution and get it to agree with this, but my coend-fu is not strong enough to solve this, and I have close to zero intuition how to show that constructions on simplicial sets actually agree with their topological analogues. (In particular I have been looking at the definitions of mapping cone, mapping cylinder etc. but I have no idea how to relate the definitions by homotopy pushouts or even ordinary pullback squares to a computation in coends or Kan extensions that I could use here.)

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  • $\begingroup$ Yes, I meant mapping cylinder (of the canonical map $X \to \pi_0(X)$ taking a point to its connected component). Let me know if this doesn't answer your question. $\endgroup$ – user43208 Feb 9 '20 at 14:36
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Patrick,

I think the article: https://arxiv.org/abs/1711.03451 should help you. Also some of the articles cited there.

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