# A discrete approach to the prescribed Gaussian curvature problem

Tianqi Wu<EMAIL_ADDRESS>
###### Abstract.

We propose a discrete approach to approximate the solution for the prescribed
Gaussian curvature problem in 2-dimensional manifolds, based on the notion of
discrete conformality. Our approach gives a convenient way to numerically
compute the solution by minimizing a convex functional.

##  1\. Introduction

###  1.1. A theorem on the prescribed Gaussian curvature problem

Suppose $(M,g)$ is a connected oriented closed smooth Riemannian surface with
genus greater than 1. The following Theorem on the prescribed Gaussian
curvature problem is well-known (see [1] and [2] for example).

######  Theorem 1.1.

Given a smooth function $\kappa(x)<0$ on $(M,g)$, there exists a unique
function $u$ on $M$ such that $e^{2u}g$ has the Gaussian curvature $\kappa$.
Furthermore, such $u$ is smooth on $M$.

In this paper we will describe a discrete approach to compute the conformal
factor $u$ in Theorem 1.1. For other related results on the prescribed
Gaussian curvature problem, one may refer [2].

###  1.2. Discrete conformality for polyhedral surfaces

Assume $T$ is a simplicial _triangulation_ of $M$. Let $V(T),E(T),F(T)$ be the
set of vertices, edges, and triangles of $T$ respectively. Furthermore, $T$ is
called a _geodesic triangulation_ if any edge in $T$ is a geodesic arc. Given
a triangulation $T$, if an edge length $l\in\mathbb{R}^{E(T)}_{>0}$ satisfies
the triangle inequalities, we can construct a Euclidean polyhedral surface
$(T,l)$ by isometrically gluing the Euclidean triangles with the edge lengths
defined by $l$ along the pairs of edges.

######  Definition 1.2.

Given a triangulation $T$, a _discrete conformal factor_ $u$ is a real-valued
function on $V(T)$. Two Euclidean polyhedral surfaces $(T,l)$ and
$(T,l^{\prime})$ are _discretely conformally equivalent_ if for some discrete
conformal factor $u$,

(1) $l^{\prime}_{ij}=e^{\frac{1}{2}(u_{i}+u_{j})}l_{ij}$

for any edge $ij\in E(T)$. We denote $l^{\prime}=u*l$ if equation (1) holds.

###  1.3. Triangulations with mixed background constant curvatures

Suppose $\kappa\in\mathbb{R}^{F(T)}_{<0}$ and $l\in\mathbb{R}^{E(T)}_{>0}$.
Whenever well-defined, for any $\sigma\in F(T)$ we denote

(a) $g(\sigma,\kappa,l)$ as a Riemmanian metric on $\sigma$ such that
$(\sigma,g(\sigma,\kappa,l))$ is a geodesic triangle of the constant Gaussian
curvature $\kappa(\sigma)$ such that the lengths of an edge $e$ is

$\frac{2}{-\kappa(\sigma)}\sinh^{-1}\left(\frac{-\kappa(\sigma)}{2}l(e)\right),$

(b) $\theta^{i}_{jk}=\theta^{i}_{jk}(\kappa,l)$ as the inner angle at the
vertex $i$ in the geodesic triangle

$(\triangle ijk,g(\triangle ijk,\kappa,l)),$

and

(c) $K=K(\kappa,l)\in\mathbb{R}^{V}$ as a generalized discrete curvature such
that

$K_{i}=2\pi-\sum_{jk\in E:\triangle ijk\in F}\theta^{i}_{jk}.$

As observed in Bobenko-Pinkall-Springborn [3], if $\kappa(\sigma)=-1$, then
$(\sigma,g(\sigma,\kappa,u*l))$ is discretely conformal to
$(\sigma,g(\sigma,\kappa,l))$ as hyperbolic geodesic triangles with the
discrete conformal factor $u$, in the notion of hyperbolic discrete
conformality introduced in [3]. We can naturally extend their notion of
hyperbolic discrete conformality to geodesic triangles of constant negative
curvature $\kappa(\sigma)$. For fixed $\kappa,l$, we denote $K=K(\kappa,u*l)$
by $K(u)$ for convenience. Then by the variational principle developed by Luo
[4] and Bobenko-Pinkall-Springborn [3]

$\mathcal{F}(u)=\int\sum_{i\in V}K_{i}(u)du_{i}$

is well-defined and locally strictly convex on the domain where $K(u)$ is
well-defined. Furthermore, Bobenko-Pinkall-Sprinborn [3] gave a simple and
explicit formula extending $\mathcal{F}$ to a globally convex functional
$\tilde{\mathcal{F}}$ on $\mathbb{R}^{V}$. So $K(u)=0$ has at most one
solution, and can be computed efficiently by minimizing a globally convex
functional $\tilde{\mathcal{F}}$ on $\mathbb{R}^{V}$.

If $K(u)=0$, one may view $\cup_{\sigma}(\sigma,g(\sigma,\kappa,u*l))$ roughly
as an approximation of a Riemannian surface with the prescribed Gaussian
curvature $\kappa$. So if $\kappa$ is a discrete approximation of a smooth
function $\bar{\kappa}(x)<0$ on $M$, $u$ may be viewed as a discrete
approximation of the smooth conformal factor $\bar{u}$ satisfying that
$e^{2\bar{u}}g$ has the Gaussian curvature $\bar{\kappa}$.

Other related work on discrete conformality and geodesic triangulations could
be found in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].

###  1.4. Discrete approximation of the prescribed curvature problem

Suppose $T$ is a geodesic triangulation of $(M,g)$ with the edge length
$l\in\mathbb{R}^{E(T)}_{>0}$ measured in $(M,g)$, we view $(T,l)$ as a
polyhedral surface approximation of $(M,g)$. Suppose $\bar{\kappa}(x)<0$ is a
smooth function on $M$. By Theorem 1.1 there exists a smooth function
$\bar{u}$ such that $e^{2\bar{u}}g$ has the Gaussian curvature $\bar{\kappa}$.
We will find a discrete approximation of $\bar{u}$ using the information of
$(T,l)$ and $\bar{\kappa}$. For all $\sigma\in F(T)$, let
$\kappa(\sigma)=\bar{\kappa}(x)$ for some $x\in\sigma$. Then by minimizing a
convex functional on $\mathbb{R}^{V}$, we expect to find a unique discrete
conformal factor $u\in\mathbb{R}^{V}$ such that $K(\kappa,u*l)=0$. Such a
discerete conformal factor $u$ should be an approximation of $\bar{u}$

We need some regularity of the triangulation to prove the convergence.

######  Definition 1.3.

A polyhedral surface $(T,l)$ is called _$\epsilon$ -acute_ if

$\theta^{i}_{jk}\leq\frac{\pi}{2}-\epsilon$

for any inner angle.

In this paper, if $x\in\mathbb{R}^{A}$ is a vector for some finite set $A$, we
use $|x|$ to denote the infinity norm of $x$, i.e.,
$|x|=|x|_{\infty}=\max_{i\in A}|x_{i}|$. Here is our main theorem.

######  Theorem 1.4.

Suppose $(M,g)$ is a connected closed orientable smooth Riemannian surface
with genus $>1$, $\bar{\kappa}(x)<0$ is a smooth function on $M$, and
$\bar{u}$ is the smooth function on $M$ such that $e^{2\bar{u}}g$ has Gaussian
curvature $\bar{\kappa}(x)$. Assume $T$ is a geodesic triangulation of
$(M,g)$, and $l\in\mathbb{R}^{E(T)}_{>0}$ denotes the edge length in $(M,g)$.
For any triangle $\sigma$ in $F(T)$, let $\kappa(\sigma)=\bar{\kappa}(x)$ for
some $x\in\sigma$. Then for any $\epsilon>0$, there exist constants
$\delta=\delta(M,g,\bar{\kappa},\epsilon)>0$ and
$C=C(M,g,\bar{\kappa},\epsilon)$ such that if $(T,l)$ is $\epsilon$-acute and
$|l|<\delta$, then

  1. (a)

there exists a unique discrete conformal factor $u\in\mathbb{R}^{V(T)}$, such
that $K(u)=0$, and

  2. (b)

$\big{|}u-\bar{u}|_{V(T)}\big{|}\leq C|l|.$

The main idea of the proof of Theorem 1.4 follows [14]. We describe a key
discrete elliptic estimate in Section 2 and the formula of $\partial
K/\partial u$ in Section 3. The proof of Theorem 1.4 is given in Section 4.

##  2\. Discrete calculus on graphs

Assume $G=(V,E)$ is an undirected connected simple graph. Let $ij$ denote an
edge in $E$ with endpoints $i$ and $j$, and $i\sim j$ represent that the
vertices $i$ and $j$ are connected by an edge $ij\in E$. Let $\mathbb{R}^{E}$
and $\mathbb{R}^{E}_{A}$ be vector spaces of dimension $|E|$ such that

  1. (a)

a vector $x\in\mathbb{R}^{E}$ is represented symmetrically, i.e.,
$x_{ij}=x_{ji}$, and

  2. (b)

a vector $x\in\mathbb{R}^{E}_{A}$ is represented anti-symmetrically, i.e.,
$x_{ij}=-x_{ji}$.

A vector $x\in\mathbb{R}^{E}_{A}$ is called a _flow_ on $G$. An _edge weight_
$\eta$ on $G$ is a vector in $\mathbb{R}^{E}$. Given an edge weight $\eta$,
the _gradient_ $\nabla f=\nabla_{\eta}f$ of a vertex function
$f\in\mathbb{R}^{V}$ is a flow in $\mathbb{R}^{E}_{A}$ defined as

$(\nabla f)_{ij}=\eta_{ij}(f_{j}-f_{i})$

where $f_{i}=f(i).$ Given a flow $x\in\mathbb{R}^{E}_{A}$, its _divergence_
$div(x)$ is a vector in $\mathbb{R}^{V}$ such that

$div(x)_{i}=\sum_{j\sim i}x_{ij}.$

Given an edge weight $\eta$, the associated _Laplacian_
$\Delta=\Delta_{\eta}:\mathbb{R}^{V}\rightarrow\mathbb{R}^{V}$ is defined as
$\Delta f=div(\nabla_{\eta}f)$, i.e.,

$(\Delta f)_{i}=\sum_{j\sim i}\eta_{ij}(f_{j}-f_{i}).$

A Laplacian $\Delta_{\eta}$ is a linear transformation on $\mathbb{R}^{V}$,
and can be identified as a $|V|\times|V|$ symmetric negative semi-definite
matrix.

Now we introduce the notion of $C$-isoperimetry for a graph $G=(V,E)$
associated with a positive vector $l\in\mathbb{R}^{E}_{>0}$. Given any
$V_{0}\subset V$, denote

$\partial V_{0}=\\{ij\in E:i\in V_{0},j\notin V_{0}\\},$

and then define the $l$-_perimeter_ of $V_{0}$ and the $l$-_area_ of $V_{0}$
as

$|\partial V_{0}|_{l}=\sum_{ij\in\partial V_{0}}l_{ij}\quad\text{ and
}\quad|V_{0}|_{l}=\sum_{i,j\in V_{0},ij\in E}l_{ij}^{2}$

respectively.

For a constant $C>0$, such a pair $(G,l)$ is called $C$-_isoperimetric_ if for
any $V_{0}\subset V$

$\min\\{|V_{0}|_{l},|V|_{l}-|V_{0}|_{l}\\}\leq C\cdot|\partial
V_{0}|_{l}^{2}.$

We will see, from part (b) of Lemma 4.2, that a uniform $C$-isoperimetric
condition is satisfied by regular geodesic triangulations approximating a
closed smooth surface. The following discrete elliptic estimate is a key
ingredient of the proofs of our main theorems.

######  Lemma 2.1 (Wu-Zhu [14]).

Assume $(G,l)$ is $C_{1}$-isoperimetric, and
$x\in\mathbb{R}^{E}_{A},\eta\in\mathbb{R}^{E}_{>0},C_{2}>0,C_{3}>0$ are such
that

  1. (i)

$|x_{ij}|\leq C_{2}l_{ij}^{2}$ for any $ij\in E$, and

  2. (ii)

$\eta_{ij}\geq C_{3}$ for any $ij\in E$.

Then

$|\Delta^{-1}_{\eta}\circ
div(x))|\leq\frac{4C_{2}\sqrt{C_{1}+1}}{C_{3}}|l|\cdot|V|_{l}^{1/2}.$

Furthermore, if $y\in\mathbb{R}^{V}$ and $C_{4}>0$ and
$D\in\mathbb{R}^{V\times V}$ is a non-zero diagonal matrix such that

$|y_{i}|\leq C_{4}D_{ii}|l|\cdot|V|_{l}^{1/2}$

for any $i\in V$, then

$|(D-\Delta_{\eta})^{-1}(div(x)+y)|\leq\left(C_{4}+\frac{8C_{2}\sqrt{C_{1}+1}}{C_{3}}\right)|l|\cdot|V|_{l}^{1/2}.$

##  3\. Differential of the discrete curvatures and angles

The following proposition can be easily derived from Proposition 6.1.7 in [3]
or Lemma 3.4 in [14].

######  Proposition 3.1.

Given $(T,l)$, $\kappa\in\mathbb{R}^{F(T)}_{<0}$, and $u\in\mathbb{R}^{V(T)}$
such that $K(u)$ is well-defined, denote

$\tilde{\theta}^{i}_{jk}(u)=\frac{1}{2}(\pi+\theta^{i}_{jk}(u)-\theta^{j}_{ik}(u)-\theta^{k}_{ij}(u))$

and

$\lambda_{ij,k}=\frac{\kappa(\triangle
ijk)^{2}\cdot(u*l)_{ij}^{2}}{\kappa(\triangle ijk)^{2}\cdot(u*l)_{ij}^{2}+4}$

for all triangle $\triangle ijk\in F(T)$. Then

$\frac{\partial K}{\partial u}(u)=D(u)-\Delta_{\eta(u)}$

where

$\eta_{ij}(u)=\frac{1}{2}\cot\tilde{\theta}^{k}_{ij}(u)(1-\lambda_{ij,k})+\frac{1}{2}\cot\tilde{\theta}^{k^{\prime}}_{ij}(u)(1-\lambda_{ij,k^{\prime}}),$

and $D=D(u)$ is a diagonal matrix such that

$D_{ii}(u)=\sum_{\triangle ijk\in
F}(\cot\tilde{\theta}^{k}_{ij}(u)\lambda_{ij,k}+\cot\tilde{\theta}^{k^{\prime}}_{ij}(u)\lambda_{ij,k^{\prime}}).$

##  4\. Proof of the Main Theorems

In this section we first introduce two geometric lemmas, and then prove
Theorem 1.4 in Subsections 4.2.

###  4.1. Geometric Lemmas

The following Lemma was first proved in [21] and was also proved in [14].

######  Lemma 4.1.

Suppose $(M,g)$ is a closed Riemannian surface, and $u\in C^{\infty}(M)$. Then
there exists $C=C(M,g,u)>0$ such that for any $x,y\in M$,

$|d_{e^{2u}g}(x,y)-e^{\frac{1}{2}(u(x)+u(y))}d_{g}(x,y)|\leq Cd_{g}(x,y)^{3}.$

The following Lemma 4.2 is similar to Lemma 4.4 in [14] and can be proved
almost in the same way.

######  Lemma 4.2.

Suppose $(M,g)$ is a closed Riemannian surface, and $T$ is a geodesic
triangulation of $(M,g)$. Let $l\in\mathbb{R}^{E(T)}$ be the geodesic lengths
of the edges and assume $(T,l)$ is $\epsilon$-acute.

  1. (a)

Given $u\in C^{\infty}(M)$, there exists a constant
$\delta=\delta(M,g,u,\epsilon)>0$ such that if $|l|<\delta$ then there exists
a geodesic triangulation $T^{\prime}$ in $(M,e^{2u}g)$ such that
$V(T^{\prime})=V(T)$, and $T^{\prime}$ is homotopic to $T$ relative to $V(T)$.
Furthermore, $(T,\bar{l})$ is $\frac{1}{2}\epsilon$-acute where
$\bar{l}\in\mathbb{R}^{E(T^{\prime})}$ denotes the geodesic lengths of the
edges of $T^{\prime}$ in $(M,e^{2u}g)$.

  2. (b)

There exist constants $\delta=\delta(M,g,\epsilon)$ and $C=C(M,g,\epsilon)$
such that if $|l|<\delta$, then $(T,l)$ is $C$-isoperimetric.

###  4.2. Proof of Theorem 1.4

For a constant $\epsilon>0$, we assume that $(T,l)$ is $\epsilon$-acute and
$|l|\leq\delta$ where $\delta=\delta(M,g,\bar{\kappa},\epsilon)<1$ is a
sufficiently small constant to be determined. By Lemma 4.2, if $\delta$ is
sufficiently small there exists a geodesic triangulation $T^{\prime}$ of
$(M,e^{2\bar{u}}g)$ homotopic to $T$ relative to $V(T)=V(T^{\prime})$. Let
$\bar{l}\in\mathbb{R}^{E(T)}\cong R^{E(T^{\prime})}$ denote the geodesic
lengths of edges of $T^{\prime}$ in $(M,e^{2\bar{u}}g)$.

For simplicity, we will frequently use the notion $a=O(b)$ to denote that if
$\delta=\delta(M,g,\bar{\kappa},\epsilon)$ is sufficiently small, then
$|a|\leq C\cdot b$ for some constant $C=C(M,g,\bar{\kappa},\epsilon)$. For
example, we have that

  1. (a)

$l_{ij}=O(l_{jk})$ for any $\triangle ijk\in F(T)$,

  2. (b)

$(\bar{u}*l)_{ij}=O(l_{ij})$,

  3. (c)

$\bar{l}_{ij}=O(l_{ij})$, and

  4. (d)

$l_{ij}=O(\frac{2}{-\kappa(\triangle ijk)}\sinh^{-1}(\frac{-\kappa(\triangle
ijk)}{2}l_{ij}))$.

The remaining of the proof is divided into two steps.

  1. (1)

Firstly we show that

$(\bar{u}*l)_{ij}-\bar{l}_{ij}=O(l_{ij}^{3})$

and

$K(\bar{u})=div(x)+y$

for some $x\in\mathbb{R}^{E}_{A}$ and $y\in\mathbb{R}^{V}$ such that
$x_{ij}=O(l_{ij}^{2})$ and $y_{i}=O(l_{ij}^{3})$.

  2. (2)

Secondly, we construct a smooth path $u(t):[0,1]\rightarrow\mathbb{R}^{V}$
with $u(0)=\bar{u}$ such that the following identity holds

$K(u(t))=(1-t)K(\bar{u}).$

Furthermore, we show that $|u^{\prime}(t)|=O(|l|)$, and then $K(u(1))=0$ and
$u(1)-\bar{u}=O(|l|)$.

Therefore the vertex function $u(1)$ is the wanted discrete conformal factor
for $(T,l)$ in Theorem 1.4.

####  4.2.1. Part 1

By Lemma 4.2, $(T,\bar{l})$ is $\frac{1}{2}\epsilon$-acute if $\delta$ is
sufficiently small. For simplicity we denote $\bar{u}|_{V(T)}$ by $\bar{u}$.
By Lemma 4.1, we have that

$\bar{l}_{ij}-(\bar{u}*l)_{ij}=O(l_{ij}^{3}).$

Notice that

$\bar{\kappa}(x)-\kappa(\triangle ijk)=O(l_{ij})$

for all $\triangle ijk\in F(T)$ and $x\in\triangle ijk$. Denote
$\bar{\theta}^{i}_{jk}$ as the inner angle of the geodesic triangle
$(\triangle ijk,e^{2\bar{u}}g)$ in $T^{\prime}$. By standard estimates on
geodesic triangles with bounded Gaussian curvature, we have that

(2)
$\alpha^{i}_{jk}:=\bar{\theta}^{i}_{jk}-\theta^{i}_{jk}(\bar{u})=O(l_{ij}^{2}),$

and

(3) $\alpha^{i}_{jk}+\alpha^{j}_{ik}+\alpha^{k}_{ij}=O(l^{3}_{ij}).$

So $(T,\bar{u}*l)$ is $\frac{1}{3}\epsilon$-acute if $\delta$ is sufficiently
small. Let $x\in\mathbb{R}^{E}_{A}$ and $y\in\mathbb{R}^{V}$ be such that

(4)
$x_{ij}=\frac{\alpha^{i}_{jk}-\alpha^{j}_{ik}}{3}+\frac{\alpha^{i}_{jk^{\prime}}-\alpha^{j}_{ik^{\prime}}}{3}\quad\text{
and }\quad y_{i}=\frac{1}{3}\sum_{jk:\triangle ijk\in
F(T)}(\alpha^{i}_{jk}+\alpha^{j}_{ik}+\alpha^{k}_{ij})$

where $\triangle ijk$ and $\triangle ijk^{\prime}$ are adjacent triangles.
Then we have the following identity

$div(x)_{i}+y_{i}=K_{i}(\bar{u}).$

By (2), (3), (4) and the fact that any vertex $i\in V(T^{\prime})$ has at most
$2\pi/(\epsilon/2)$ neighbors, we have

(5) $x_{ij}=O(l_{ij}^{2}),$

and

(6) $y_{i}=O(l_{ij}^{3}).$

####  4.2.2. Part 2

Let

$\tilde{\Omega}=\\{u\in\mathbb{R}^{V}:u*l\text{ satisfies the triangle
inequalities and $(T,u*l)$ is $\frac{\epsilon}{4}$-acute}\\}$

and

$\Omega=\\{u\in\tilde{\Omega}:|u-\bar{u}|\leq 1\text{, $(T,u*{l})$ is
$\frac{\epsilon}{5}$-acute}\\}.$

Since $(T,\bar{u}*l)$ is $\frac{1}{3}\epsilon$-acute, $\bar{u}$ is in the
interior of $\Omega$. Now consider the following ODE on $int(\tilde{\Omega})$,

(7)
$\left\\{\begin{array}[]{lcl}u^{\prime}(t)=(\Delta_{\eta(u)}-D(u))^{-1}K(\bar{u})=(\Delta_{\eta(u)}-D(u))^{-1}(div(x)+y)\\\
u(0)=\bar{u}\end{array}\right.,$

where $D(u)$ and $\eta(u)$ are defined as in Proposition 3.1. For any triangle
$\triangle ijk$ and $u\in\tilde{\Omega}$, we have

(8) $D_{ii}(u)\geq\epsilon^{\prime}l_{ij}^{2},\quad\text{ and
}\quad\eta_{ij}(u)\geq\epsilon^{\prime}$

for some constant
$\epsilon^{\prime}=\epsilon^{\prime}(M,g,\bar{\kappa},\epsilon)>0$.

The right-hand side of equation (7) is a smooth function of $u$, so the ODE
(7) has a unique solution $u(t)$ satisfying

$\frac{dK(u(t))}{dt}=\frac{\partial K}{\partial
u}(u)u^{\prime}(t)=(D(u)-\Delta_{\eta(u)})(\Delta_{\eta(u)}-D(u))^{-1}K(\bar{u})=-K(\bar{u}).$

Therefore

$K(u(t))=(1-t)K(\bar{u}).$

Assume the maximum existence interval of $u(t)$ in $\Omega$ is $[0,T_{0})$
where $T_{0}\in(0,\infty]$. By Lemma 4.2, when $\delta$ is sufficiently small,
$(T,l)$ is $C$-isoperimetric for some constant $C=C(M,g,\epsilon)$. Then for
any $u\in\Omega$, $(T,u*l)$ is $(e^{4(|\bar{u}|+1)}C)$-isoperimetric. It is
not difficult to see

$|V|_{l}=O(1)$

and

$1=O(|V|_{l}).$

Then by Lemma 2.1 and equation (5),(6),(8), for any $t\in[0,T_{0})$

(9) $|u^{\prime}(t)|=O(|l|\cdot|V|_{l}^{1/2})=O(|l|).$

If $T_{0}\leq 1$,

(10) $|u(T_{0})-\bar{u}|=O(|l|)\quad\text{ and
}\quad\theta^{i}_{jk}(u(T_{0}))-\theta^{i}_{jk}(\bar{u})=O(|l|),$

and then $u(T_{0})\in\text{int}(\Omega)$ if $\delta$ is sufficiently small.
But this contradicts with the maximality of $T_{0}$. So $T_{0}>1$ and
$K(u(1))=0$ and $|u(1)-\bar{u}|=O(|l|).$

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# Completions and Terminal Monads

Emmanuel Dror Farjoun<EMAIL_ADDRESS>and  Sergei O. Ivanov
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider the terminal monad among those preserving the objects of a
subcategory $\mathpzc{D}\subseteq\mathpzc{C},$ and in particular preserving
the image of a monad over the category $\mathpzc{C}.$

Several common monads $\mathpzc{C}\to\mathpzc{C}$ are shown to be uniquely
characterized by the property of being terminal objects in the category of co-
augmented endo-functors. Once extended to infinity categories, this gives, for
example, a complete characterization of the well-known Bousfield-Kan
$R$-homology completion $R_{\infty}.$ In addition, we note that an idempotent
pro-completion tower $M_{\bullet}$ can be associated with any co-augmented
endo functor $M,$ whose limit $M_{\infty}$ is the terminal monad that
preserves the closure of $ImM,$ the image of $M,$ under finite limits. We
conclude that some basic properties of the homological completion tower
$R_{\bullet}X$ of a space can be formulated and proved for general monads over
any category with limits, and characterized as universal.

The second named author is supported by BIMSA

##  1\. Introduction and main results

Many well-known and extremely useful constructions, mostly known as
”completions”, such as the profinite completion $G\to\widehat{G}$ or the
Bousfield-Kan homology completion $X\to R_{\infty}X,$ are usually constructed
directly, without specifying what universal property, if any, determines them
up to equivalence. Here, using a notion that we call ”terminal monad,” many of
these are shown to be completely determined by the property of being terminal
objects in an appropriate category of co-augmented functors $X\to F(X)$ over
the given underlying category $\mathpzc{C}.$

Our first observation is that the category

$Fun(\mathpzc{C},\mathpzc{C})_{Id/-}$

of co-augmented functors over a category $\mathpzc{C}$ with limits, is itself
closed under limits. Moreover, consider any collection $D$ of objects in
$\mathpzc{C};$ and denote by ${\mathcal{F}}ix_{D}(\mathpzc{C}),$ the
subcategory of the above functor category, consisting of functors that
preserve each object of $D,$ namely with $d\to F(d)$ an equivalence for every
$d\in D.$ Then this category, ${\mathcal{F}}ix_{D}(\mathpzc{C}),$ of co-
augmented functors, is also closed under limits. In particular, it has a
terminal object $Id\to M_{D},$ which is easily shown to be a monad over
$\mathpzc{C}.$

For a given small full subcategory $\mathpzc{D}\subseteq\mathpzc{C},$ a
construction of the terminal monad, associated with the set of its objects,
can be done by re-considering the well-known co-density monad, denoted here by
$T{D}:\mathpzc{C}\to\mathpzc{C},$ associated with a full subcategory
$\mathpzc{D}\subseteq\mathpzc{C},$ of a category $\mathpzc{C}$ which is always
assumed to be closed under limits.

Hence, as above, this functor $T{D}$ can be characterized as the terminal
monad on $\mathpzc{C},$ among all co-augmented functors $Id\to F$ on
$\mathpzc{C},$ that ”preserves the objects of $\mathpzc{D},$” i.e. with
$d\cong F(d)$ for all $d\in\mathpzc{D}.$ This $d$ is ”fixpoint” in the
terminology of Adamek, [2] definition 2.5, see also [16]. It turns out that a
terminal monad can be associated with more general, not necessarily fully
faithful, functors. In particular, we consider terminal monads associated with
a given monad, or more generally with co-augmented endo-functors
$M:\mathpzc{C}\to\mathpzc{C}.$ Notice, as in the references above, that the
category of monads over $\mathpzc{C}$ is also closed under all limits (—but,
in general, not under colimits.)

This allows one to characterize, by a universal property, common
constructions, such as ”completions,” as terminal monads with respect to an
appropriate subcategory. There is an infinity-categorical extension of this
observation [9]. It leads, for example, to a seemingly new characterization by
a universal property of the well-known Bousfield-Kan homological completion
$R_{\infty}$ as an infinity monad on topological spaces or a simplicial sets
$X.$ Compare [6]: The completion $R_{\infty}$ is shown to be the terminal
$\infty-$monad associated with the monad $X\to R(X),$ among all co-augmented
functors that preserve, up to homotopy, the essential image of $R,$ the free
$R-$ module functor. Similar characterization of the pro-finite completion of
a group, an algebraic variety, or a topological space, and other ”completion”
are examples. In addition, following and elaborating on Fakir [8], and
Casacuberta-Frei [7], one can associate to a monad (or any co-augmented
functor) a terminal idempotent monad, i.e. a terminal localization functor,
$L_{M},$ projecting $\mathpzc{C}$ to the smallest subcategory of $\mathpzc{C}$
that contains the image of $M$ and is closed under limits. The above
definitions and constructions can be evidently dualized to get analog ones
associated with a co-monad $N\to Id.$ In [18], L. Yanovski constructs for
quite general $\infty-$categories a transfinite tower of co-monads with
similar (implicit) properties, and strong transfinite convergence results.

###  1.1. Examples:

To begin, consider some quite well-known examples. Recall that any
localization (or so-called reflection)functor $L:\mathpzc{C}\to\mathpzc{C}$ is
a terminal monad see [7]. It is the terminal that preserves all the $L-$ local
objects. Next, the canonical set $U(X)$ of all ultra-filters on a set $X$ is a
special case, see [10]. Namely, the monad $U$ now appears as the terminal
monad among all co-augmented functors on sets that preserve every finite set.
Another well-known example is the double dual of a vector space functor $V\to
V^{**}.$ It is the terminal monad that preserves the one-dimensional spaces,
or alternatively all finite-dimensional spaces. The last examples are clearly
related to theorems 1.2 and 1.3 below.

It turns out, see below, that even when $\mathpzc{D}$ is just the subcategory
spanned by a three-element set $3=d\in\mathpzc{C}=Sets$ in the category of
sets, then $T{D}=T_{3}(X)$ is again the underlying set  of the Stone-Čech
compactification of the set $X,$ i.e. $U(X)$ as above. Further, $T_{2}$ is a
canonical sub-monad of the ultrafilter monad, while $T_{n}$ for $n\geq 3$ is
again the ultrafilter monad. When $\mathpzc{D}\subseteq\mathpzc{C}$ is the
subcategory of finite groups in the category of (discrete) groups, then the
discrete profinite completion endo-functor, on the category of groups appears
as the terminal monad among all co-augmented ones $F,$ that preserve all
finite groups i.e. with $\Gamma\cong F(\Gamma)$ for every finite $\Gamma.$ Or
again, if $\mathpzc{D}\subseteq\mathpzc{C}$ is the subcategory of nilpotent
groups in the category of groups, the associated monad $T_{\mathpzc{D}}$ is
the (discrete) nilpotent completion functor in the category of groups. For a
ring $A,$ the (discrete) completion functor of an $A-$module, with respect to
an ideal $I\subseteq A,$ can be similarly expressed as a terminal monad. A
final, slightly stretched example, in an $\infty$-category, is the double dual
as in equation 6.2 section 6 below, and Theorem 6.2. This is very close to
Mandell’s functor, [15], see remark 6.2 below, that can be considered as the
terminal monad preserving certain GEM spaces expressed as a double dual monad.

####  1.1.1. Acknowlegements

This line of thought was a result of a private discussion with M. Hopkins
about the properties of the Bousfield-Kan $R$-completion. Our students Guy
Kapon and Shauly Regimov took an active part in the discussion leading to the
present paper. Their work led them to the corresponding formulations in the
context of $\infty$-algebras, see [9].

###  1.2. A sample of results

The results below regard the existence, basic properties, and explicit
formulas for the terminal monad in certain cases. Our first concern is to
guarantee the existence of terminal monads under certain rather weak
conditions. In any category, a terminal object can be considered as the limit
over the empty diagram. Hence the existence of a terminal co-augmented functor
in a given functor category would follow from its closure under limits.

In the following, the closure under limits and thus the existence of a
terminal object is guaranteed by the closure of the basic category
$\mathpzc{C}$ under limits. In the present case, the functor categories, coma
categories, and considered subcategories are clearly closed under limits.
Limits in the category of co-augmented endo-functors are taken in the
appropriate coma category under the identity functor.

The following gives a general construction of the terminal monad in quite a
general framework, see Proposition 2.4 below.

######  Proposition 1.1.

Let $\mathpzc{D}\subseteq\mathpzc{C}$ be a full subcategory of a category with
limits. The co-density functor, or the $\mathpzc{D}$-completion,
$T{D}:\mathpzc{C}\to\mathpzc{C}$ is the terminal object in the category of co-
augmented functors $Id\to F\in\mathpzc{C}^{\mathpzc{C}},$ such that the co-
augmentation map $d\to F(d)$ is an isomorphism for all $d\in\mathpzc{D}.$ This
functor has a unique canonical monad structure.

The following statements use the notation of 6.2, so they should be read with
caution: Although not treated here, they hold also in a complete monoidal
category with internal $hom(-,-)$ objects. In all cases, the notation
$hom_{O}(-,-)$ should be read as the appropriate equivariant maps, with
respect to the implied action on the monoid $End$ or operad $O,$ on the range
and domain.

######  Theorem 1.2.

(See equation 6.2) Let $d\in\mathpzc{C}$ be an element in a category with
limits. Denote by $End(d)$ the full subcategory generate by $d,$ namely the
endomorphism of $d.$ The terminal monad $Id\to T_{d}$ that preserves $d,$ is
given by the ”structured double dual” with respect to $d:$

$X\to T_{d}(x)=hom_{End\,(d)}(hom(X,d),d)$

Further, the terminal monad that preserves an element $d\in\mathpzc{C}$ and
all its cartesian powers $d^{n},$ is given by a similar expression as below,
where $\sf O={\sf O}_{d}$ denote the full endomorphism operad of an object
$d,$ given by all the morphisms $d^{i}\to d$ with $i>0.$

In the notation of 1.2 one has:

######  Theorem 1.3.

The terminal monad $Id\to T_{d^{\bullet}}$ that preserves $d^{i},$ for all
$i,$ is given by the ”operadic double dual” with respect to $d:$

$X\to T_{d}(X)=hom_{\sf O}(hom(X,d),d)$

#### Terminal monads associated to a given monad $M$

For a given monad $Id\to M:\mathpzc{C}\to\mathpzc{C},$ ( or, more generally, a
co-augmented endo-functor,) one has an associated terminal monad
$T_{M}:\mathpzc{C}\to\mathpzc{C},$ which is the terminal endo-functor among
all those that preserve the image of $M,$ namely with $M\to FM$ an
isomorphism. As an example, of such a monad $M$ one can take any of the monads
discussed above or even the terminal monad $T{D}$ as above. The terminal monad
associated with the (discrete) profinite completion functor, $G\to
M(G)=\widehat{G}\equiv proG$ is a functor $T_{pro},$ that preserves all groups
of the form $\widehat{G},$ i.e. groups that are the discrete profinite
completion of some group $G.$ Next, if $U$ is the ultrafilter monad discussed
above then its associated terminal monad can be seen to be the identity monad,
$T_{U}=Id,$ which is clearly the only monad that preserves all possible sets
of the form $U(X)$, since the latter have arbitrarily high cardinality.

The terminal monad associated to $M$ can be expressed explicitly as follows:
Compare [8]:

######  Theorem 1.4.

Let $M$ be a monad on a category $\mathpzc{C}.$ The associated terminal monad
is given as the equalizer

$T_{M}\cong Equal(M\rightrightarrows M^{2}).$

In addition, we may consider, following Fakir above, the category of
idempotent monads, which are often called localizations or reflections. Fakir
constructs for every monad $M$ a naturally associated idempotent monad K(M).
Casacuberta et el observed in [7] that this idempotent monad is terminal among
all idempotent monads $F,F\cong F^{2},$ with the property $M(f)$ is an
isomorphism if and only if $K(M)(f)$ is one. Note the difference between
$T_{M}$ and $K(M).$ The latter is discussed shortly in the last section below.
The $\infty$-category analog is clearly the totalization of the co-simplicial
monad $M^{\bullet}$ discussed in [9].

###  1.3. Outline of the rest of the paper

We begin with recalling the general concept of completion i.e. co-density with
respect to a subcategory, such as the subcategory of compact objects. This is
done by considering the right Kan extension of a subcategory over itself. This
gives many known examples of terminal monads. We then consider the terminal
monad associated with a given object in a category and one associated with a
given monad. The last example gives a functor from monads to terminal monads
on the category $\mathpzc{C}.$ The paper goes on to consider some known
special cases such as the category of sets and groups where the general
construction gives some well-known constructions as a terminal monad, this
characterizes them uniquely by a property. The last section deals with the
pro-idempotent monad associated with a co-augmented endo-functor, vastly
generalizing the classical Bousfield-Kan completion tower $R_{\bullet},$ here
only for a discrete category, but paving the way for a similar result for an
$\infty$-category.

##  2\.  $\mathpzc{D}$-Completions

Let $\mathpzc{D}$ be a full subcategory of a category $\mathpzc{C}$. Denote by
$I:\mathpzc{D}\to\mathpzc{C}$ the embedding and assume that the right Kan
extension of $I$ by $I$ exists and denote it by

(2.1) $T=T{D}={\sf Ran}_{I}(I):\mathpzc{C}\to\mathpzc{C}.$

So by the definition of the right Kan extension, $T$ is a functor together
with a natural transformation $\varepsilon:TI\longrightarrow I$

(2.2) ${\mathpzc{C}}$${\mathpzc{D}}$
${\mathpzc{C}}$$\leftarrow$$\rightarrow$$\scriptstyle{T}$$\Leftarrow$$\Rightarrow$$\scriptstyle{\varepsilon}$$\leftarrow$$\rightarrow$$\scriptstyle{I}$$\leftarrow$$\rightarrow$$\scriptstyle{I}$

such that for any functor $F:\mathpzc{C}\to\mathpzc{C}$ and any natural
transformation $\varepsilon^{\prime}:FI\to I$ there exists a unique natural
transformation $\delta:F\to T$ such that $\varepsilon\circ(\delta
I)=\varepsilon^{\prime},$ where $\delta I:FI\to TI$ is the whiskering of
$\delta$ and $I$. The equation $\varepsilon\circ(\delta
I)=\varepsilon^{\prime}$ can be rewritten as follows: for any
$d\in\mathpzc{D}$

(2.3) $\varepsilon_{d}\circ\delta_{d}=\varepsilon^{\prime}_{d}.$

Note that the universal property implies that for two natural transformations
$\delta,\delta^{\prime}:F\to T$ the equation $\delta I=\delta^{\prime}I$
implies $\delta=\delta^{\prime}.$ Thus $T$ appears already here as a terminal
functor, in a somewhat different sense from the above. Compare [11].

The functor $T$ will be called the functor of $\mathpzc{D}$-completion. Any
right Kan extension can be presented as a limit over a comma category [14, Ch.
X, §3, Th.1]. In our case, it is just the limit of the projection functor

(2.4) $T(c)={\sf lim}(c\downarrow\mathpzc{D}\to\mathpzc{D}).$

######  Lemma 2.1.

The morphism $\varepsilon:TI\to I$ is an isomorphism. In particular, for any
$d\in{\sf Ob}(\mathpzc{D})$ we have an isomorphism

(2.5) $\varepsilon_{d}:T(d)\cong d.$

###### Proof.

Since the functor $I:\mathpzc{D}\to\mathpzc{C}$ is full and faithful, by [14,
Ch. X, §3, Cor.3] we obtain that $\varepsilon$ is an isomorphism. ∎

######  Lemma 2.2.

There exists a unique natural transformation

(2.6) $\eta^{T}:{\sf Id}{C}\longrightarrow T$

such that $\eta^{T}_{d}=\varepsilon_{d}^{-1}$ for any $d\in{\sf
Ob}(\mathpzc{D}).$

###### Proof.

Take $F={\sf Id}{C}$ and $\varepsilon^{\prime}={\sf id}_{I}$ and use the
universal property of the right Kan extension. ∎

Further we will treat $T$ as an co-augmented functor $T=(T,\eta^{T}).$ For any
co-augmented functor $F=(F,\eta^{F})$ we set

(2.7) ${\sf Inv}(F)=\\{c\in{\sf Ob}(\mathpzc{C})\mid\eta^{F}_{c}\text{ is
iso}\\}.$

Note that $\mathpzc{D}\subseteq{\sf Inv}(T).$

######  Lemma 2.3.

The class ${\sf Inv}(F)$ is closed under retracts.

###### Proof.

Because a retract of an isomorphism is an isomorphism. ∎

The following is a basic observation that follows from the above, that
justifies the term ”terminal monad,” compare [3.7.3] in [5].

######  Proposition 2.4.

The co-augmented functor $T_{\mathpzc{D}}:\mathpzc{C}\to\mathpzc{C}$ of the
$\mathpzc{D}$-completion is a terminal object in the category of co-augmented
functors $F$ for which $\mathpzc{D}\subseteq{\sf Inv}(F).$ Moreover,
$T_{\mathpzc{D}}$ is a monad in functor category $\mathpzc{C}^{\mathpzc{C}}.$

###### Proof.

Take a co-augmented functor $F=(F,\eta^{F})$ such that
$\mathpzc{D}\subseteq{\sf Inv}(F).$ Note that $\eta^{F}I:I\to FI$ is an
isomorphism. We use the universal property of $T$ and take
$\varepsilon^{\prime}=(\eta^{F}I)^{-1}:FI\to I.$ Then there exists a unique
natural transformation $\delta:F\to T$ such that $\varepsilon\circ(\delta
I)=(\eta^{F}I)^{-1}.$ Since $\varepsilon=(\eta^{F}I)^{-1},$ we obtain that the
equation $\varepsilon\circ(\delta I)=(\eta I)^{-1}$ is equivalent to the
equation $(\delta\circ\eta^{F})I=\eta^{T}I.$ And the equation
$(\delta\circ\eta^{F})I=\eta^{T}I$ is equivalent to
$\delta\circ\eta^{F}=\eta^{T}$ by the universal property of $T.$

The monad structure of this terminal $T{D}$ follows immediately from the fact
that its square preserves all objects of $\mathpzc{D},$ giving a unique
natural transformation $T_{\mathpzc{D}}^{2}\to T{D}.$ The monadic equations
are satisfied since they all involve equality of natural transformations from
powers of $T{D}$ to $T{D}$ itself, but there is a unique such transformation
for each power since $T{D}$ is terminal among these co-augmented functors, all
of which preserve the objects of $\mathpzc{D}.$ ∎

Remark: Note that the above characterization shows that the terminal monad
$T_{\mathpzc{D}},$ associated with a subcategory
$\mathpzc{D}\subseteq\mathpzc{C},$ can be identified using solely its effect
on the objects in $\mathpzc{D},$ being the terminal co-augmented functor
$\mathpzc{C}\to\mathpzc{C},$ that ”preserves the objects” of this subcategory.
Of course, its usual construction, as above, does employ morphisms in
$\mathpzc{D}$ and $\mathpzc{C}.$

##  3\. Terminal monads associated with a functor $\mathpzc{D}\to\mathpzc{C}$

More generally, consider a general functor $F:\mathpzc{D}\to\mathpzc{C}.$ Now,
consider the subcategory $\mathpzc{C}^{\mathpzc{C}}_{F},$ of the category of
end-functors $\mathpzc{C}^{\mathpzc{C}},$ consisting of all co-augmented
functors G, $Id\to G:\mathpzc{C}\to\mathpzc{C},$ that preserve the image of
$F,$ i.e. with

$(Id\to G)(F(x))=F(x)\to GF(x)$

is an isomorphism in $\mathpzc{C}$ for any object $x\in\mathpzc{D}.$ The
subcategory $\mathpzc{C}^{\mathpzc{C}}_{F},$ of the full functor category, is
a category of co-augmented functors $G,$ which is evidently closed under all
limits. Hence it has a terminal object which is the terminal monad $T_{F}$
associated with the given functor $F,$ namely, preserving the image of $F.$

We note that this terminal object has a natural monad structure:

######  Proposition 3.1.

Let $M$ be an co-augmented endo-functor in $\mathpzc{C}^{\mathpzc{C}.}$ The
terminal object, $T_{M},$ in the category $\mathpzc{C}^{\mathpzc{C}}_{M}$ of
co-augmented endo-functors preserving the image of $M,$ is naturally a monad.

###### Proof.

Denote the terminal co-augmented functor by $T_{M}$ as above. Since the
composition: $T_{M}\circ T_{M}$ clearly preserves the image of $M,$ and
$T_{M}$ is terminal among those preserving $M,$ there is a unique map
$\mu:T_{M}T_{M}\to T_{M}.$ The conditions, on a co-augmented functor with this
$\mu$ as a structure map, of being a monad, involve equality among various
maps from compositions of $T_{M}$ with itself to $T_{M}.$ Each such
composition preserves the image of $M,$ therefore there is a unique map, from
any self-composition of $T_{M},$ to the terminal object $T_{M}.$ Recall all
the conditions on a co-augmented to be a monad involve equality between
various maps to the monad itself. It follows that all the needed equalities
are satisfied by $T_{M}.$ ∎

We conclude that the above basic properties of $T{D}$ holds when one replaces
the inclusion $I:\mathpzc{D}\subseteq\mathpzc{C}$ with any functor
$M:\mathpzc{D}\to\mathpzc{C}.$ In this case, the right Kan extension $T_{M}$
is the terminal monad on $\mathpzc{C}$ that preserves the image subcategory of
the given functor $M.$ In case $\mathpzc{D}=\mathpzc{C}$ and where the functor
$M:\mathpzc{C}\to\mathpzc{C}$ is a co-augmented functor, we got the terminal
monad $T_{M}$ among those that preserve the image of $M.$

Consider the special, well-known case, where $M$ is an idempotent localization
functor $Id\to M\cong M^{2}.$ Namely, a projection onto a subcategory of
$\mathpzc{C}.$ In that case, $m:T_{M}\to M$ is an equivalence. Namely, $M$ is
its own terminal monad. (Compare: [17])

For the sake of completeness, we state:

######  Proposition 3.2.

Let $L:\mathpzc{C}\to\mathpzc{C}$ be a co-augmented idempotent functor, i.e.
localization- projection onto a full subcategory of local object. Then $L$ is
its own terminal monad i.e. $L\cong T_{L}.$

###### Proof.

First note that $L$ has associated monad structure since $L\cong L^{2}$ by the
two natural maps. Second, for every monad that preserves the image of $L,$
namely, with the natural map $L\to ML$ an equivalence (isomorphism) one gets a
map

$M\to ML\to LML\cong L^{2}\cong L$

The monad structure of $M$ forces the uniqueness of the map of monads $M\to L$
since any map of monads

$f:M\to L$ is a retract of $Mf:M^{2}\to ML\cong L,$ being a monad map, which
is a retract of $M(id\to L.)$

Starting with the map of monads 3.1,

(3.1) ${id}$${M}$
${L}$$\leftarrow$$\rightarrow$$\scriptstyle{i_{L}}$$\leftarrow$$\rightarrow$$\scriptstyle{i_{M}}$$\leftarrow$$\rightarrow$$\scriptstyle{f}$

Applying $M$ to this triangle of maps we see immediately that $f$ is uniquely
determined by the monad

∎

######  Remark 3.3.

If the subcategory $\mathpzc{D}$ as above is closed under all limits then it
is localizing and the $\mathpzc{D}$-completion $T{D}$ is the localization
$L{D}:\mathpzc{C}\to\mathpzc{D},$ projecting $\mathpzc{C}$ to the subcategory
$\mathpzc{D}.$

###  3.1. $T_{M}$ in terms of $M$

It turns out that there is a direct formula expressing the terminal monad
$T_{M},$ associated with $M,$ in terms of $M.$ Generally, given a monad $Id\to
M,$ consider a co-augmented functor $Id\to F$ that preserves the image of $M,$
i.e. $M\cong FM.$ Applying such an $F$ to $Id\to M$ we get a canonical map
$F\to M,$ for every such functor. In particular, for any monad $M,$ one gets a
natural map $T_{M}\to M,$ from the terminal monad to $M,$ giving rise to the
augmentation $T\to Id$ of the functor $T.$

In the following this last map is identified with the natural map to $M,$ of
the equalizer of the natural diagram: $M\rightrightarrows M^{2}.$ In addition
this map $T_{M}\to M$ is shown to be a map of monads.

Let us start with two basic properties:

######  Proposition 3.4.

Let $f:(M,\mu_{{}_{M}})\to(N,\mu_{{}_{N}})$ be a map of monads. Then the monad
$N$ is naturally an $M-$algebra. In particular, $T_{M}(N)\cong N,$ thus $f$
induces a map of monads $Tf:T_{M}\to T_{N}.$ Hence $T:Mon\mathpzc{C}\to
Mon\mathpzc{C},$ has a natural structure of augmented endo-functor $T\to Id$
on the category $Mon\mathpzc{C}$ of monads over $\mathpzc{C}.$

###### Proof.

Since $M$ is an algebra over itself, we need to show that the natural map
$N\to M(N),$ gotten by applying the co-augmentation $\iota_{M}:Id\to M$ to
$N,$ has a left inverse i.e., that $N$ is a retract of $M(N)\equiv M\circ N.$
Since $T_{M}$ preserves $M,$ it preserves also any retract of $M.$ The natural
left inverse is given by the composition:

$MN\xrightarrow{f\circ N}NN\xrightarrow{\mu_{{}_{N}}}N$

Hence, $N(x)$ has been shown to be a retract of $M(N(x)),$ for all objects
$x\in obj\mathpzc{C}.$ Thus $T_{M}(N(x))\cong N(x).$ But $T_{N}$ is the
terminal monad that preserves all objects of the form $N(x).$ Therefore, there
is a unique map $T_{M}\to T_{N},$ as needed. ∎

Second, an interesting closure property

######  Proposition 3.5.

Let $X:I\to\mathpzc{C}$ be a functor with $I$ being a small (indexing)
category and $M$ a monad over $\mathpzc{C}.$ Assume that for each $i\in I$ the
object $X_{i}\in obj\mathpzc{C},$ is a retract of $M(X_{i}.)$ Then the object
$Y=lim_{I}X_{i}$ is a retract of $T_{M}(Y),$ hence $T_{{T_{M}}}(Y)\cong Y.$
Similarly, any such limit $Y$ of $M-$algebras is naturally a $T_{M}$\-
algebra.

Note: there is no assumption here about relations among the various
retractions, namely the retract structures on different $X_{i}.$ Thus the
limit is not in general an $M$-retract but it is a $T_{M}$ retract. For
example, (in $Cat_{\infty}$) in the infinity category of spaces, if $M=R$ is
the free $R$-module spanned by a space $X$, then a limit of any diagram of
such $R$-GEMs is not, in general, a $R$-algebra but rather a
$R_{\infty}=T_{R}$-algebra.

###### Proof.

Consider the composition:

$lim_{i}X_{i}\to T_{M}(lim_{i}X_{i})\overset{a}{\to}lim_{i}T_{M}(X_{i})\cong
lim_{i}X_{i}$

which is clearly the identity map.

The right-hand side map $a$ is the assembly map for limits, and the equality
on the right is a consequence of $T_{M}(X_{i})\cong X_{i},$ since the latter
is a retract of $M(X_{i})$ by assumption and hence also preserved by $T_{M}.$

The right-hand side map $a,$ is directly seen to equip $lim_{i}X_{i}$ with an
$T_{M}-$ algebra structure.

∎

###  3.2. The equalizer as the terminal monad.

Given a co-augmented functor $Id\to M,$ denote by $Eq_{M}$ the equalizer of
the two natural maps $M\rightrightarrows M^{2}$ coming form the co-
augmentation structure $Id\to M.$

First, we note the following:

######  Lemma 3.6.

Let $Id\to M$ be an co-augmented functor $\mathpzc{C}\to\mathpzc{C}.$ Let $F$
be any co-augmented functor $F:\mathpzc{C}\to\mathpzc{C}$ that preserves $M,$
i.e. with $(Id\to F)(M)=M\overset{\cong}{\to}F(M)$ an equivalence. Then there
is a natural map, of co-augmented functors, $F\to Eq_{M}$ from $F$ to the
equalizer of $M\rightrightarrows M^{2}.$

###### Proof.

Apply $F$ to the commutative diagram $Id\to M\rightrightarrows M^{2},$ to get
the desired factorization to the equalizer: observing that we get commutative:

$F\to F(M)\rightrightarrows F(M^{2})$

which is equivalent to:

$F\to M\rightrightarrows M^{2}$

by the assumption on $F.$ Hence there is a well-defined factorization of the
left-hand side map through the equalizer $F\to Eq_{M}.$ This map clearly
respects the co-augmentation $Id\to F\to M.$ ∎

For the rest of the discussion, we will mostly assume, sometimes for
convenience only, that $M$ is a monad on $\mathpzc{C}.$ We noticed that every
functor that preserves $M$ maps naturally to the functor $Eq_{M}.$ The same is
true in particular to the terminal functor that preserves $M.$ But $Eq_{M}$
preserves $M$ thus, by definition, it maps uniquely to the terminal $T_{M}.$
This brings us to the following:

######  Proposition 3.7.

Let $M$ be a monad in $Mon\mathpzc{C},$ the category of monads over
$\mathpzc{C}.$ The terminal monad preserving the image of $M$ is naturally
equivalent to the equalizer:

$T_{M}\cong Eq(M\rightrightarrows M^{2}.)$

Equalizer that is, with respect to the two natural transformations given by
the augmentation. In particular, the equalizer itself has a natural structure
of a monad.

###### Proof.

Observe that the diagram of maps $M\rightrightarrows M^{2}$ is not, in
general, a diagram of monads. The maps in it are natural transformations of
co-augmented functors. Thus the equalizer is a co-augmented functor, but it is
not immediately clear why it is a monad. Fakir states this without proof. It
does follow below from the observation that the equalizer is naturally
isomorphic to the terminal monad $T_{M}.$

To prove that, note that $Eq_{M}$ preserves $M,$ i.e. $Eq_{M}(M)\cong M,$ were
here $M$ denotes the underlying co-augmented functor of the monad $M.$ The
reason is that clearly as co-augmented functors, there is an equivalence
$M\cong totM^{\bullet}(M)=lim_{\Delta}(M^{\bullet}(M)),$ since the latter co-
simplicial functor has an extra co-degeneracy map. But in 1-category,
$totM^{\bullet}\cong Eq_{M}.$ Therefore there is a unique map of co-augmented
functors $Eq_{M}\to T_{M}.$

First, we prove that this map is an equivalence of functors. This will endow
the equalizer with a monad structure.

The monad structure on $T_{M}$ comes from the natural map $T_{M}T_{M}\to
T_{M}$ given by the universal property of the range. Since the identity is
only self-map $T_{M}\to T_{M},$ the last map satisfies the monad conditions.

Now consider the following maps (=natural transformations) of co-augmented
functors:

$Eq_{M}\overset{q}{\longrightarrow}T_{M}\overset{t}{\longrightarrow}Eq_{M}\overset{q}{\longrightarrow}T_{M}$

The map $q$ is uniquely guaranteed by the equation $Eq_{M}\circ M\cong M,$
since $T_{M}$ is terminal co-augmented functor with this property.

The map $t$ is given by the universal property of $Eq_{M}.$ See lemma 3.6
above. Namely, since the monoid $T_{M}$ preserves $M,$ ($T_{M}\circ M\cong M$)
so it preserves the co-simplicial object $M^{\bullet}.$ When we apply $T_{M}$
to $Id\to M^{\bullet}$ we get a map of $T_{M}$ to the limit of $M^{\bullet}$,
which in our case is the equalizer $Eq_{M}.$

In the above composition of three natural transformations, the induced self-
map of $T_{M}$ is the identity since $T_{M}$ is a terminal object. We claim
that the composition $t\circ q$ is equivalent to the identity.

To see that, consider the diagram:

$\textstyle{Eq_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{T_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\textstyle{Eq_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Eq_{M}\circ
M\cong
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\cong}$$\textstyle{T_{M}\circ
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\cong}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Eq_{M}\circ
M^{2}\cong T_{M}\circ
M^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{T_{M}\circ
M^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{M^{2}}$

gotten by applying the above maps: $Eq_{M}\to T_{M}\to Eq_{M}$ to the

natural transformations:

$Id\to M\rightrightarrows M^{2}.$

Since all the horizontal maps in diagram that involve $M,M^{2},$ and denoted
by $"\cong"$ are equivalences, the required self-map on the equalizer,
$Eq_{M}$ is also an equivalence as needed. Thus we conclude that the maps
$t,s$ are equivalences of co-augmented functors. It follows that $Eq_{M}$ has
a structure of a monad coming from that of $T_{M},$ and the two are equivalent
as a monad, as stated. ∎

The terminal monad $T_{M},$ has the following properties:

######  Theorem 3.8.

For any monad $M,$ the above map $T_{M}\to M$, 3.8, is a map of monads. The
assignment $M\mapsto T_{M}$ gives an endo-functor $MonC\to MonC,$ together
with a natural transformation $T\to Id,$ namely, an augmented endo-functor $T$
on $MonC.$

###### Proof.

By 3.4 above, $T_{M}$ is natural in the variable $M.$ Or, since $Eq_{M}$ is
functorial in $M,$ so $T_{M}$ is by the above theorem 3.7. Hence the
assigment: $T:M\mapsto T_{M}$ defines an endo-functor on $Mon\mathpzc{C}:$ For
a given map of monads $M\to M^{\prime}$ one gets a map of co-simplicial
resolutions $M^{\bullet}\to M^{\prime\bullet}.$ Or, since $T_{M}$ is naturally
equivalent to the equalizer of $M\rightrightarrows M^{2},$ we get a well-
defined map on the terminal monads with a natural map $m_{M}:T_{M}\to M,$ as
needed.

We now prove that this map $T_{M}\to M$ is a map of monads, namely, respect
the monad structure $M^{2}\to M.$

Now since $Eq_{(-)}$ is a co-augmented functor we get a commutative diagram
involving $Eq_{M^{2}}$ by applying this functor to $M\rightrightarrows M^{2}.$
The natural map $Eq_{M}Eq_{M}\to Eq_{M^{2},}$ completes the argument, giving
the necessary commutation of the monad structures.

In more detail, for any map of co-augmented functors, such as $m_{M}:T_{M}\to
M,$ the corresponding co-faces maps $M\rightrightarrows M^{2}$ etc. commutes
with $m_{M}$ and $m_{T_{M}.}$

Consider the natural diagram:

$\textstyle{T_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{T_{M}T_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{3}}$$\textstyle{{T_{M}\cong
Eq_{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{M}=\iota_{M}}$$\textstyle{Eq_{M^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\iota_{M^{2}}}$$\textstyle{M}$$\textstyle{M^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$

To prove that $T_{M}\to M$ is a map of monads, one only needs to show that the
monad structures of $M$ and $T_{M},$ written as $\mu,\mu_{1},$ are compatible
with the natural map $T_{M}\to M.$ Namely, that the two composition arrows,
involving $\mu$ and $\mu_{1},$ of $T_{M}T_{M}\to M$ in the diagram below are
equal. Namely, that the outer square of the maps below commutes.

Consider the diagram below: The arrows $\mu_{1},\mu_{3}$ are defined by the
universal properties of $T_{M}$ and the equalizer, correspondingly, since
their common domain preserves $M$ and $M^{2},$ using 6.5 above. The map
${\iota_{M^{2}}}$ is the inclusion of the equalizer.

Note that the bottom square commutes since here the equalizer $Eq_{(-)}$ is
considered here as a co-augmented functor,$Eq_{(-)}\to Id,$ from the category
of co-augmented functors over $\mathpzc{C}$ to itself.

The top square below commutes by the terminal property of its bottom left
corner $T_{M},$ admitting only one map from the functor $T_{M}T_{M}$ at the
top right corner.

Thus the whole diagram commute as needed.

∎

####  3.2.1. Example

Consider the ultrafilter monad $X\to U(X)$ discussed elsewhere in this paper.
The associated terminal monad $T_{U}$ preserves the image of $U$ which
includes sets of arbitrary cardinality. Hence it preserves all sets and must
be the identity monad. In fact, the equalizer $Eq_{U}$ is easily seen to be
the identity functor on sets, as it includes only principal ultrafilters. On
the other hand, the terminal monad associated to the profinite completion
$G\to pro-G\equiv\widehat{G}$ in the category of groups is not the identity
functor since on a finitely presented group $T_{pro}(G)\cong pro-G$ since the
completion is idempotent on these groups so that the equalizer of
$pro-G\rightrightarrows pro-(pro-G)$ is $pro-G$ itself, the completion of $G.$

##  4\. Explicit expressions for $T_{d}$

Here we explicitly express the terminal monad, associated with an object
$d\in\mathpzc{C},$ as a structured double dual, see below, as opposed to the
usual one as in [3], section 2.

For a set $S$ and an object $c\in{\sf Ob}(\mathpzc{C})$ we denote by $c^{S}$
the product of copies of $c$ indexed by $S$

(4.1) $c^{S}=\prod_{s\in S}c.$

The projections will be denoted by ${\sf pr}_{s}:c^{S}\to c.$ A morphism
$f:c^{\prime}\to c^{S}$ is defined by a family of morphisms $f_{s}={\sf
pr}_{s}\circ f:c^{\prime}\to c$ which are called components of $f.$ The object
$c^{S}$ is contravariant by $S.$ More precisely, this defines a functor

(4.2) $\mathpzc{C}\times{\sf Set}^{\sf op}\to\mathpzc{C},\hskip
28.45274pt(c,S)\mapsto c^{S}$

such that, if $f:S\to S^{\prime}$ is a function, then the morphism
$c^{f}:c^{S^{\prime}}\to c^{S}$ is defined so that $(c^{f})_{s}={\sf
pr}_{f(s)}.$

######  Proposition 4.1.

Let $\mathpzc{C}$ be a category with limits and $\mathpzc{D}$ be a small full
subcategory of $\mathpzc{C}.$ Then the functor of $\mathpzc{D}$-completion
exists and it is given by the end

(4.3) $T(c)=\int_{d\in\mathpzc{D}}d^{\mathpzc{C}(c,d)}.$

It can be also presented as an equalizer

(4.4) $T(c)={\sf
eq}\left(\prod_{d\in\mathpzc{D}}d^{\mathpzc{C}(c,d)}\rightrightarrows\prod_{\alpha:d_{1}\to
d_{2}}d_{2}^{\mathpzc{C}(c,d_{1})}\right),$

where the first morphism is induced by
$d_{1}^{\mathpzc{C}(c,d_{1})}\xrightarrow{\alpha^{\mathpzc{C}(c,d_{1})}}d_{2}^{\mathpzc{C}(c,d_{1})}$
and the second morphism is induced by
$d_{2}^{\mathpzc{C}(c,d_{2})}\xrightarrow{d_{2}^{\mathpzc{C}(c,\alpha)}}d_{2}^{\mathpzc{C}(c,d_{1})}.$

###### Proof.

Follows from the interpretation of right Kan extensions in terms of ends [14,
Ch. X, §4, Th.1] and the characterisation of ends in terms of equalisers [13,
Remark 1.2.4]. ∎

######  Corollary 4.2.

Let $\mathpzc{C}$ be a complete category and $\mathpzc{D}=\\{d\\}$ the full
subcategory consisting of one object. Then $T_{d}=T_{\mathpzc{D}}$ exists and

(4.5) $T_{d}(c)\cong{\sf
eq}\left(d^{\mathpzc{C}(c,d)}\rightrightarrows\left(d^{\mathpzc{C}(c,d)}\right)^{{\sf
End}{C}(d)}\right).$

####  4.0.1. Remark.

There is an alternative description of $T_{d}$ using the double dual monad.
Denote by $DD_{d}(c)$ the ”naive double dual” monad $c\to
d^{\mathpzc{C}(c,d)}=\Pi_{c\to d}d.$ Then it is not hard to see using the
arguments in 3.2 that $T_{d}$ is equivalent to the terminal monad $T_{DD_{d}}$
associated with $DD_{d}.$ Notice that $DD_{d}(d)$ is isomorphic to $d^{l}$ for
some $l\geq 1.$ Hence, since the terminal $T_{DD_{d}}$ preserves the image of
${DD_{d}},$ $T_{DD_{d}}$ also preserves its retract $d,$ and one has a unique
map of monads $Eq_{DD_{d}}\cong T_{DD_{d}}\to T_{d}.$ Similarly there is a map
in the other direction: Namely, the desired element in the monad category
$Mon\mathpzc{C}$:

$Mon(T_{d},DD_{d})=Mon(T_{d},d^{\mathpzc{C}(c,d)})$

is determined by the composition of maps in a $\mathpzc{C}$:

$T_{d}(c)\times\mathpzc{C}(c,d)\to
T_{d}(c)\times\mathpzc{C}(T_{d}(c),T_{d}(d)=d)\overset{eval}{\longrightarrow}d.$

Or, stated otherwise, the map $T_{d}(c)\to\Pi_{c\to d}d$ is give factor-wise
by: $T_{d}(c)\to d=T_{d}(c\to d),$ since $T_{d}(d)\cong d.$ This also can
serve to prove the crucial property of $T_{d}$ namely, for any map $x\to y$ in
$\mathpzc{C}$ with $\mathpzc{C}(y,d)\cong\mathpzc{C}(x,d)$ satisfies
$T_{d}(x)\cong T_{d}(y);$ which is of course evident from the explicit
expression for $T_{d}$ above. It is clear, though not expanded here, that, in
the case where the category $\mathpzc{C}$ is enriched over itself, the above
approach works well, using internal hom objects $hom(hom(-,d),d).$

##  5\. Examples in the category of sets and groups

###  5.1. Variations on the set of ultrafilters.

For any set $X$ we denote by $\mathpzc{P}(X)$ the set of all subsets of $X$.
We treat $\mathpzc{P}$ as a (contravariant) functor

(5.1) $\mathpzc{P}:{\sf Set}^{\sf op}\longrightarrow{\sf Set},$

where, for a map $f:X\to X^{\prime},$ the map
$\mathpzc{P}(f):\mathpzc{P}(X^{\prime})\to\mathpzc{P}(X)$ is defined as
$\mathpzc{P}(f)(Y)=f^{-1}(Y).$ Note that the characteristic function defines
an isomorphism

(5.2) $\chi:\mathpzc{P}(X)\cong{\sf Set}(X,2).$

The composition

(5.3) $\mathpzc{P}\mathpzc{P}:{\sf Set}\longrightarrow{\sf Set}$

is a (covariant) functor that has a natural coaugmentation

(5.4) $\eta_{X}:X\longrightarrow\mathpzc{P}(\mathpzc{P}(X))$

such that $\eta_{X}(x)$ is the set of all sets $Y\subseteq X$ containing $x.$
If we denote by $\mathpzc{UF}(X)$ the set of ultrafilters on $X,$ we obtain
that $\mathpzc{UF}$ is an co-augmented sub-functor of $\mathpzc{P}\mathpzc{P}$

(5.5) $\mathpzc{UF}\subseteq\mathpzc{P}\mathpzc{P}.$

Recall that $\mathpzc{UF}(X)$ can be thought of as the underlying set of the
Stone-Čech compactification of the set $X.$

Let us define another co-augmented sub-functor of $\mathpzc{P}\mathpzc{P}.$ An
element $A\in\mathpzc{P}(\mathpzc{P}(X))$ is called _ultraset_ , if

  * (US1)

$\emptyset\notin A;$

  * (US2)

for any $Y\subseteq X$ one and only one of the sets $Y,X\setminus Y$ is an
element of $A.$

Note that the axiom (US1) can be equivalently replaced by

  * (US1’)

$X\in A.$

######  Example 1.

Any ultrafilter is an ultraset.

######  Example 2.

Let $X$ is a finite set of an odd cardinality $|X|=2n+1$ and

(5.6) $A=\\{Y\subseteq X\mid|Y|\geq n+1\\}.$

Then $A$ is an ultraset on $X$ which is not an ultrafilter for $n\geq 1.$

######  Lemma 5.1.

Let $A$ be an ultraset on a set $X.$ Then $A$ is an ultrafilter if and only if
for any partition into three disjoint subsets $X=P_{0}\sqcup P_{1}\sqcup
P_{2}$, there exists a unique $i\in\\{0,1,2\\}$ such that $P_{i}\in A.$

###### Proof.

Assume that $A$ is an ultrafilter and $X=P_{0}\sqcup P_{1}\sqcup P_{2}$ is a
partition. If there exists $i$ such that $P_{i}\in A,$ then it is obviously
unique because $A$ is closed under finite intersections. Let us prove that it
exists. Assume the contrary that $P_{i}\notin A$ for any $i.$ Then $X\setminus
P_{i}\in A,$ and hence $P_{0}=(X\setminus P_{1})\cap(X\setminus P_{2})\in A,$
which is a contradiction.

Now assume that for any partition into three disjoint subsets $X=P_{0}\sqcup
P_{1}\sqcup P_{2}$, there exists a unique $i\in\\{0,1,2\\}$ such that
$P_{i}\in A.$ In order to prove that $A$ is an ultrafilter, we need to prove
that: (1) $Y\in A$ and $Y\subseteq Y^{\prime}\subseteq X$ implies
$Y^{\prime}\in A;$ (2) $Y,Y^{\prime}\in A$ implies $Y\cap Y^{\prime}\in A.$

Let us prove (1). Take $P_{0}=Y,$ $P_{1}=Y^{\prime}\setminus Y$ and
$P_{2}=X\setminus Y^{\prime}.$ Then $P_{0}\in A,$ and hence, $P_{2}\notin A.$
By (US2) we obtain $X\setminus P_{2}=Y^{\prime}\in A.$

Let us prove (2). In the proof, we use that we already proved (1). Take
$P_{0}=Y\cap Y^{\prime},$ $P_{1}=Y\setminus Y^{\prime},$ $P_{2}=X\setminus Y.$
Since $Y\in A,$ we have $P_{2}\notin A.$ Therefore either $P_{0}\in A,$ or
$P_{1}\in A.$ We need to prove that $P_{0}\in A.$ Assume the contrary that
$P_{1}\in A.$ Since $P_{1}\subseteq X\setminus Y^{\prime},$ using (1), we
obtain $X\setminus Y^{\prime}\in A.$ It follows that $Y^{\prime}\notin A,$
which is a contradiction. Hence $P_{0}\in A.$ ∎

The set of all ultrasets on $X$ is denoted by $\mathpzc{US}(X).$ It is easy to
check that $\mathpzc{US}$ is a co-augmented sub-functor of
$\mathpzc{P}\mathpzc{P}$

(5.7) $\mathpzc{UF}\subseteq\mathpzc{US}\subseteq\mathpzc{P}\mathpzc{P}.$

Let $n$ be a natural number, taken as an ordinals $n=\\{0,\dots,n-1\\}.$ We
denote by $T_{n}:{\sf Set}\to{\sf Set}$ the functor of $\\{n\\}$-completion
i.e. it is the terminal co-augmented functor with the property that $n\to
T_{n}(n)$ is an isomorphism $n.$

######  Lemma 5.2.

Let ${\sf Fin}_{\leq n}$ denotes the class of finite sets of cardinality at
most $n.$ Then

(5.8) $T_{n}=T_{{\sf Fin}_{\leq n}}.$

###### Proof.

Since any set of cardinality at most $n$ is a retract of $n,$ this follows
from Lemma 2.3. ∎

######  Proposition 5.3.

The co-augmented functor of $2$-completion on the category of sets is
isomorphic to $\mathpzc{US}$

(5.9) $T_{2}\cong\mathpzc{US}.$

###### Proof.

By Corollary 4.2 we see

(5.10) $T_{2}(X)={\sf eq}\Big{(}{\sf Set}({\sf
Set}(X,2),2)\rightrightarrows{\sf Set}({\sf Set}(X,2),2)^{{\sf
End}(2)}\Big{)}.$

The characteristic function defines a bijection $\mathpzc{P}(X)\cong{\sf
Set}(X,2).$ There are four maps $2\to 2:$ (1) the identity map ${\sf
id}=e_{1};$ (2) the map $e_{2}$ sending all to $0;$ (3) the map $e_{3}$
sending all to $1;$ (4) the permutation $e_{4}.$ The composition with them
correspond to four maps on $f_{i}^{X}:\mathpzc{P}(X)\to\mathpzc{P}(X):$ (1)
$f_{1}^{X}(Y)=Y$; (2) $f_{2}^{X}(Y)=\emptyset;$ (3) $f_{3}^{X}(Y)=X;$ (4)
$f_{4}^{X}(Y)=X\setminus Y.$ Consider the isomorphism

(5.11) $\mathpzc{P}(\mathpzc{P}(X))\cong{\sf Set}({\sf Set}(X,2),2).$

So we need to prove that

(5.12) $\mathpzc{US}(X)={\sf
eq}\Big{(}\mathpzc{P}(\mathpzc{P}(X))\rightrightarrows\mathpzc{P}(\mathpzc{P}(X))^{{\sf
End}(2)}\Big{)}$

The equaliser consists of such elements $A\in\mathpzc{P}(\mathpzc{P}(X))$ that
the equation $f_{i}^{\mathpzc{P}(X)}(A)=\mathpzc{P}(f^{X}_{i})(A)$ is
satisfied for any $i.$ For $i=1$ it is satisfied for any $A$. For $i=2$ we
have $f_{2}^{\mathpzc{P}(X)}(A)=\emptyset$ and

(5.13)
$\mathpzc{P}(f^{X}_{2})(A)=(f_{2}^{X})^{-1}(A)=\begin{cases}\mathpzc{P}(X),&\emptyset\in
A;\\\ \emptyset,&\emptyset\notin A.\end{cases}$

Then it is satisfied for $i=2$ iff $\emptyset\notin A$ (axiom (US1)).
Similarly, we obtain that the equation is satisfied for $i=3$ iff $X\in A$
(axiom (US1’)). For $i=4$ we have that
$f_{4}^{\mathpzc{P}(X)}(A)=\mathpzc{P}(X)\setminus A$ and
$\mathpzc{P}(f_{4}^{X})(A)=\\{X\setminus Y\mid Y\in A\\}.$ Then the equation
is satisfied for $i=4$ iff the axiom (US2) is satisfied. ∎

######  Proposition 5.4.

Let ${\sf Fin}$ denote the full subcategory of ${\sf Set}$ consisting of
finite sets. Then $T_{\sf Fin}$ is isomorphic $T_{3}$ and isomorphic to
$\mathpzc{UF}$

(5.14) $T_{\sf Fin}\cong T_{3}\cong\mathpzc{UF}.$

###### Proof.

It is well-known that $\eta:K\to\mathpzc{UF}(K)$ is an isomorphism for any
finite $K.$ So it is enough to prove that $\mathpzc{UF}$ is the terminal among
all co-augmented functors $(F,\eta^{F})$ such that $\eta^{F}:K\to F(K)$ is an
isomorphism for any set $K$ such that $|K|\leq 3.$ By the universal property
of $\mathpzc{US}=T_{2}$ (Proposition 5.3) we see that there is a unique
morphism of co-augmented functors $\varphi:F\to\mathpzc{US}.$ So we just need
to prove that for any set $X$ the image of $\varphi_{X}$ is in
$\mathpzc{UF}(X).$ Denote by $F^{\prime}$ the image of $\varphi.$ Note that
$F^{\prime}$ is a co-augmented sub-functor of $\mathpzc{US}.$ So we need to
prove that $F^{\prime}(X)\subseteq\mathpzc{UF}(X).$

Since $\eta:K\to F(K)$ is an isomorphism for finite any $K$ such that $|K|\leq
3$ we see that

(5.15) $\mathpzc{UF}(K)=F^{\prime}(K).$

for any $K$ such that $|K|\leq 3.$

Let us prove that $F^{\prime}(X)\subseteq\mathpzc{UF}(X).$ Take an ultraset
$A\in F^{\prime}(X).$ Consider a partition $X=P_{1}\sqcup P_{2}\sqcup P_{3}.$
Define a map $\alpha:X\to 3$ such that $\alpha^{-1}(i)=P_{i}.$ The map
$F^{\prime}(\alpha):F^{\prime}(X)\to F^{\prime}(3)=\mathpzc{UF}(3)$ sends $A$
to $\eta(i_{0})\in\mathpzc{UF}(3)$ for some $i_{0}\in\\{0,1,2\\}.$ Since
$\\{i_{0}\\}\in\eta(i_{0})$ and for $i\neq i_{0}$ we have
$\\{i\\}\notin\eta(i_{0}),$ we obtain that $P_{i_{0}}=\alpha^{-1}(i_{0})\in A$
and $P_{i}=\alpha^{-1}(i)\notin A$ for $i\neq i_{0}.$ Therefore the assumption
of Lemma 5.1 is satisfied, and hence, $A$ is an ultrafilter. ∎

###  5.2. Examples: Groups and modules

Here we briefly consider the examples alluded to in the first section. The
examples below are proved by applying the expression 2.4 above. It is rather
immediate to see that by taking $\mathpzc{D}\subset\mathpzc{C}$ to be the
subcategory of finite groups in the category of all groups, the
$\mathpzc{D}-$completion $T{D}$ is canonically isomorphic to the (discrete!)
pro-finite completion functor on groups.

Similarly, when $\mathpzc{D}\subset\mathpzc{C}$ is the subcategory of
nilpotent groups in the category of all groups. Similarly, for the completion
of an $A-$module $M,$ with respect to an ideal $I\subseteq A$ in a ring $A:$
Namely, $M\to limM/I^{k}M.$

In the above example, the fact that $\mathpzc{D}$ is a large category can be
dealt with by noticing that for each $A-$module $M,$ the tower of quotients
$M\to(M/I^{k}M)_{k}$ is co-final in the category $M\downarrow\mathpzc{D},$
appearing in 2.4.

In case the ring $A=K$ is a field the usual double dual functor of a $K-$
vector space $V\to V^{**}$ appears as a terminal monad $T_{K},$ since the
double dual in 6.2 above is reduce here to $V^{**}.$

##  6\. Completions and operads

In this section, we continue to assume that $\mathpzc{C}$ is closed under
limits.

###  6.1. Objects with an action of a monoid

Let $M$ be a monoid. An $M$-object in $\mathpzc{C}$ is an object $c$ endowed
by a homomorphism of monoids $f^{c}:M\to{\sf End}(c).$ If $X$ is an $M$-set
and $c$ is an $M$-object, we define the _hom-object_ over $M$ as an equalizer

(6.1) ${\sf hom}_{M}(X,c)={\sf
eq}(\sigma,\tau:c^{X}\rightrightarrows(c^{X})^{M}),$

where $\sigma_{m}=c^{f^{X}(m)}$ and $\tau_{m}=(f^{c}(m))^{X}$ for any $m\in
M.$ If $\mathpzc{C}$ is a category of sets, $\mathsf{hom}_{M}(X,c)$ coincides
with the ordinary hom-set in the category of $M$-sets.

For any two objects $c,d$ from $\mathpzc{C}$ the hom-set $\mathpzc{C}(c,d)$
has a natural structure of ${\sf End}(d)$-set defined by the composition. Then
Corollary 4.2 can be reformulated as

(6.2) $T_{d}\cong\mathsf{hom}_{{\sf End}(d)}(\mathpzc{C}(-,d),d).$

###  6.2. Objects with an action of an operad

Let $O$ be an operad (of sets). For an object $c$ we denote by ${\sf O}(c)$
the endomorphism operad of $c,$ whose $n$-th component is ${\sf
O}(c)_{n}=\mathpzc{C}(c^{n},c).$ An $O$-algebra in $\mathpzc{C}$ is an object
$c$ endowed by a morphism $f^{c}:O\to{\sf O}(c).$ If $X$ is an $O$-algebra in
the category of sets and $c$ is an $O$-algebra in $\mathpzc{C},$ we defined
the hom-object over $O$ as an equaliser

(6.3) $\mathsf{hom}_{O}(X,c)={\sf
eq}(\sigma,\tau:c^{X}\rightrightarrows\prod_{n=0}^{\infty}(c^{X^{n}})^{O_{n}}),$

where $\sigma$ and $\tau$ are defined so that
$\sigma_{n,o}=c^{f^{X}_{n}(o)}:c^{X}\to c^{X^{n}}$ and

(6.4) $\tau_{n,o,x_{1},\dots,x_{n}}=f^{c}_{n}(o)\circ({\sf
pr}_{x_{1}},\dots,{\sf pr}_{x_{n}}):c^{X}\to c$

for any $n\geq 0,$ $o\in O_{n}$ and $x_{1},\dots,x_{n}\in X.$ Here we denote
by $({\sf pr}_{x_{1}},\dots,{\sf pr}_{x_{n}}):c^{X}\to c^{n}$ the morphism
with components ${\sf pr}_{x_{i}}.$ For the special case $n=0$ we have
$X^{n}=1=\\{0\\},$ $c^{n}=1,$ and $\tau_{0,o}:c^{X}\to c$ is the composition
of $c^{X}\to 1$ and $f_{0}^{X}(o):1\to c.$ Note that

(6.5) $\sigma_{n,o,x_{1},\dots,x_{n}}={\sf
pr}_{f^{X}_{n}(o)(x_{1},\dots,x_{n})}:c^{X}\to c.$

For any $n\geq 0$ we also consider

(6.6) $\mathsf{hom}^{n}_{O}(X,c)={\sf
eq}(\sigma_{n},\tau_{n}:c^{X}\rightrightarrows(c^{X^{n}})^{O_{n}})$

and

(6.7) $\mathsf{hom}^{\leq n}_{O}(X,c)={\sf eq}(\sigma_{\leq n},\tau_{\leq
n}:c^{X}\rightrightarrows\prod_{i=0}^{n}(c^{X^{i}})^{O_{i}}).$

The projection
$\prod_{i=0}^{n}(c^{X^{i}})^{O_{i}}\to\prod_{i=0}^{n-1}(c^{X^{i}})^{O_{i}}$
induces a morphism

(6.8) $\mathsf{hom}^{\leq n}_{O}(X,c)\longrightarrow\mathsf{hom}^{\leq
n-1}_{O}(X,c).$

Since $\prod_{i=0}^{\infty}(c^{X^{i}})^{O_{i}}=\underset{n}{\varprojlim}\
\prod_{i=0}^{n}(c^{X^{i}})^{O_{i}},$ using that limits commute with limits, we
obtain

(6.9) $\mathsf{hom}_{O}(X,c)=\underset{n}{\varprojlim}\ \mathsf{hom}_{O}^{\leq
n}(X,c).$

###  6.3. Completion with respect to a power

For an object $d$ of $\mathpzc{C}$ we denote by ${\sf O}^{1}(d)$ the suboperad
of the endomorphism operad ${\sf O}(d)$ such that

(6.10) ${\sf O}^{+}(d)_{0}=\emptyset,\hskip 28.45274pt{\sf O}^{+}(d)_{n}={\sf
O}(d)_{n}$

for $n\geq 1.$

For any two objects $c,d$ of $\mathpzc{C}$ there is a natural structure of
${\sf O}^{1}(d)$-algebra on the set $\mathpzc{C}(c,d):$ for any
$\alpha:d^{n}\to d$ we consider the map

(6.11)
$f(\alpha):\mathpzc{C}(c,d)^{n}\cong\mathpzc{C}(c,d^{n})\xrightarrow{\mathpzc{C}(c,\alpha)}\mathpzc{C}(c,d).$

For any object $d$ of $\mathpzc{C}$ we consider the functor $T_{d^{n}}$ of
$d^{n}$-completion. We also consider

(6.12) $T_{d^{+}}=T_{\\{d^{n}\mid n\geq 1\\}},\hskip
28.45274ptT_{d^{\bullet}}=T_{\\{d^{n}\mid n\geq 0\\}}.$

This subsection is devoted to the proof of the following theorem.

######  Theorem 6.1.

Let $\mathpzc{C}$ be a complete category and $d$ be its object. Then for
$n\geq 1$, there are isomorphisms of co-augmented functors

(6.13) $T_{d^{n}}\cong\mathsf{hom}_{{\sf
O}^{+}(d)}^{n}(\mathpzc{C}(-,d),d)\cong\mathsf{hom}_{{\sf O}^{+}(d)}^{\leq
n}(\mathpzc{C}(-,d),d),$ (6.14) $T_{\\{1,d^{n}\\}}\cong\mathsf{hom}_{{\sf
O}(d)}^{\leq n}(\mathpzc{C}(-,d),d),$ (6.15) $T_{d^{+}}\cong\mathsf{hom}_{{\sf
O}^{+}(d)}(\mathpzc{C}(-,d),d),$ (6.16)
$T_{d^{\bullet}}\cong\mathsf{hom}_{{\sf O}(d)}(\mathpzc{C}(-,d),d),$

where the augmentations of the right-hand functors are induced by morphism to
the element $d$ raised to the power ${\mathpzc{C}(c,d)}$, given by:
$\tilde{\eta}_{c}:c\to d^{\mathpzc{C}(c,d)}$ with components
$(\tilde{\eta}_{c})_{\alpha}=\alpha.$

######  Remark 6.2.

An analog and potentially a special case of this formula, within the
$\infty$-category of simplicial sets, appears in Mandell’s theorem, [15] and
[4], Proposition 4.4. Here, the operadic double-dual appears as a version of
homological p-completion. It gives the terminal functor that preserves certain
$p$-adic Eilenberg-MacLane spaces.

In order to prove this theorem, we need to prove several lemmas.

######  Lemma 6.3.

For $n\geq 0$ and a morphism $\varphi:e\to d^{\mathpzc{C}(c,d)},$ the diagram

(6.17)
$e\xrightarrow{\varphi}d^{\mathpzc{C}(c,d)}\underset{\tau_{n}}{\overset{\sigma_{n}}{\rightrightarrows}}\Big{(}d^{\mathpzc{C}(c,d)^{n}}\Big{)}^{\mathpzc{C}(d^{n},d)}$

is commutative ($\tau_{n}\varphi=\sigma_{n}\varphi$) if and only if for any
morphism $\alpha:d^{n}\to d$ and any morphism $\beta:c\to d^{n}$ we have

(6.18)
$\varphi_{\alpha\circ\beta}=\alpha\circ(\varphi_{\beta_{1}},\dots,\varphi_{\beta_{n}}).$

###### Proof.

The components of the morphisms $\tau_{n,\alpha},\sigma_{n,\alpha}$ are
$\sigma_{n,\alpha,\beta_{1},\dots,\beta_{n}}={\sf pr}_{\alpha\circ\beta}$ and
$\tau_{n,\alpha,\beta_{1},\dots,\beta_{n}}=\alpha\circ({\sf
pr}_{\beta_{1}},\dots,{\sf pr}_{\beta_{n}}).$ The assertion follows. ∎

######  Lemma 6.4.

For $n\geq 0$ the diagram

(6.19)
$c\xrightarrow{\tilde{\eta}_{c}}d^{\mathpzc{C}(c,d)}\underset{\tau_{n}}{\overset{\sigma_{n}}{\rightrightarrows}}\Big{(}d^{\mathpzc{C}(c,d)^{n}}\Big{)}^{\mathpzc{C}(d^{n},d)}$

is commutative ($\sigma_{n}\tilde{\eta}_{c}=\tau_{n}\tilde{\eta}_{c}$), where
$(\tilde{\eta}_{c})_{\alpha}=\alpha.$

###### Proof.

It follows from Lemma 6.3. ∎

######  Lemma 6.5.

For $n\geq 0$ the diagram

(6.20)
$d^{n}\xrightarrow{\tilde{\eta}_{d^{n}}}d^{\mathpzc{C}(d^{n},d)}\underset{\tau_{n}}{\overset{\sigma_{n}}{\rightrightarrows}}\Big{(}d^{\mathpzc{C}(d^{n},d)^{n}}\Big{)}^{\mathpzc{C}(d^{n},d)}$

is an equalizer.

###### Proof.

By Lemma 6.4 we have $\sigma_{n}\tilde{\eta}=\tau_{n}\tilde{\eta}.$ Let
$\varphi:e\to d^{\mathpzc{C}(d^{n},d)}$ be a map that equalizes $\sigma_{n}$
and $\tau_{n}.$ Lemma 6.3 implies that for any $\alpha:d^{n}\to d$ and
$\beta:d^{n}\to d^{n}$ we have
$\varphi_{\alpha\circ\beta}=\alpha\circ(\varphi_{\beta_{1}},\dots,\varphi_{\beta_{n}}).$
In particular, if we take $\beta={\sf id}_{d^{n}},$ we get

(6.21) $\varphi_{\alpha}=\alpha\circ(\varphi_{{\sf
pr}_{1}},\dots,\varphi_{{\sf pr}_{n}}).$

So, if we take $\psi=(\varphi_{{\sf pr}_{1}},\dots,\varphi_{{\sf pr}_{n}}),$
we obtain $\tilde{\eta}\circ\psi=\varphi.$ Let us prove that such $\psi$ is
unique. Assume that $\psi:e\to d^{n}$ is a morphism such that
$\tilde{\eta}\circ\psi=\varphi.$ Then $\varphi_{\alpha}=\alpha\circ\psi.$ It
follows that $\psi_{i}=\varphi_{{\sf pr}_{i}}.$ ∎

######  Lemma 6.6.

For $1\leq n^{\prime}\leq n,$ if $\varphi:e\to d^{\mathpzc{C}(c,d)}$ is a
morphism such that $\sigma_{n}\varphi=\tau_{n}\varphi$, then
$\sigma_{n^{\prime}}\varphi=\tau_{n^{\prime}}\varphi.$

###### Proof.

It is enough to prove for $n^{\prime}=n-1\geq 1.$ Take morphisms
$\alpha:d^{n-1}\to d$ and $\beta:c\to d^{n-1}.$ Since $d^{n}=d^{n-1}\times d$
we have a projection ${\sf pr}_{\leq n-1}:d^{n}\to d^{n-1}$ and we can take a
map $\delta:d^{n-1}\to d^{n}$ such that ${\sf pr}_{\leq n-1}\circ\delta={\sf
id}_{d^{n-1}}$ and ${\sf pr}_{n}\circ\delta={\sf pr}_{n-1}$. We set
$\alpha^{\prime}=\alpha\circ{\sf pr}_{\leq n-1}:d^{n}\to d$ and
$\beta^{\prime}=\delta\circ\beta:c\to d^{n}.$ Then
$\alpha\circ\beta=\alpha^{\prime}\circ\beta^{\prime};$
$\beta^{\prime}_{i}=\beta_{i}$ for $1\leq i\leq n-1$ and
$\beta^{\prime}_{n}=\beta_{n-1}.$ Then by the assumption and Lemma 6.3 we have

(6.22)
$\varphi_{\alpha\circ\beta}=\varphi_{\alpha^{\prime}\circ\beta^{\prime}}=\alpha^{\prime}\circ(\varphi_{\beta^{\prime}_{1}},\dots,\varphi_{\beta^{\prime}_{n}})=\alpha\circ(\varphi_{\beta_{1}},\dots,\varphi_{\beta_{n-1}}).$

The assertion follows. ∎

######  Remark 6.7.

Generally the equation $\sigma_{n}\varphi=\tau_{n}\varphi$ for $n\geq 1$ does
not imply $\sigma_{0}\varphi=\tau_{0}\varphi.$ The assumption $n^{\prime}\geq
1$ of Lemma 6.6 is essential.

###### Proof of Theorem 6.1.

Lemma 6.6 implies that

(6.23) $\mathsf{hom}_{{\sf
O}^{+}(d)}^{n}(\mathpzc{C}(-,d),d)\cong\mathsf{hom}_{{\sf O}^{+}(d)}^{\leq
n}(\mathpzc{C}(-,d),d).$

Let us prove $T_{d^{n}}\cong\mathsf{hom}_{{\sf O}^{+}(d)}^{\leq
n}(\mathpzc{C}(-,d),d).$ By Lemma 6.5 we have

(6.24) $\eta:d^{n}\cong\mathsf{hom}_{{\sf O}^{+}(d)}^{\leq
n}(\mathpzc{C}(d^{n},d),d).$

So we need to prove the universal property. Consider a co-augmented functor
$F$ such that $d^{n}\in{\sf Inv}(F).$ Since $d$ is a retract of $d^{n},$ we
have $d\in{\sf Inv}(F).$ Therefore there is a unique morphism of co-augmented
functors $\theta:F\to\mathsf{hom}_{{\sf End}(d)}(\mathpzc{C}(-,d),d).$ Taking
the composition with the morphism $\mathsf{hom}_{{\sf
End}(d)}(\mathpzc{C}(-,d),d)\to d^{\mathpzc{C}(-,d)},$ we obtain a morphism
$\varphi:F\to d^{\mathpzc{C}(-,d)}.$ Then, in order to prove that
$T_{d^{n}}\cong\mathsf{hom}_{{\sf O}^{+}(d)}^{n}(\mathpzc{C}(-,d),d)$ it is
sufficient to prove that
$\sigma_{n^{\prime}}\varphi_{c}=\tau_{n^{\prime}}\varphi_{c}$ for any $c,$ any
$1\leq n^{\prime}\leq n$ and any morphism of co-augmented functors
$\varphi:F\to d^{\mathpzc{C}(-,d)}.$ By Lemma 6.6 it is enough to prove that
$\sigma_{n}\varphi_{c}=\tau_{n}\varphi_{c}.$

Let us prove that $\sigma_{n}\varphi_{c}=\tau_{n}\varphi_{c}$ for any $c.$
Take $\alpha:d^{n}\to d$ and $\beta:c\to d^{n}.$ Note that
$\varphi\eta^{F}=\tilde{\eta}.$ The commutative diagram

(6.25)
${c}$${F(c)}$${d^{\mathpzc{C}(c,d)}}$${d}$${d^{n}}$${F(d^{n})}$${d^{\mathpzc{C}(d^{n},d)}}$${d}$$\leftarrow$$\rightarrow$$\scriptstyle{\beta}$$\leftarrow$$\rightarrow$$\scriptstyle{\eta_{c}^{F}}$$\leftarrow$$\rightarrow$$\scriptstyle{F(\beta)}$$\leftarrow$$\rightarrow$$\scriptstyle{\varphi_{c}}$$\leftarrow$$\rightarrow$$\scriptstyle{d^{\mathpzc{C}(\beta,d)}}$$\leftarrow$$\rightarrow$$\scriptstyle{{\sf
pr}_{\alpha\circ\beta}}$$\leftarrow$$\rightarrow$$\scriptstyle{\sf
id}$$\leftarrow$$\rightarrow$$\scriptstyle{\alpha}$$\leftarrow$$\rightarrow$$\scriptstyle{\eta^{F}_{d^{n}}}$$\scriptstyle{\cong}$$\leftarrow$$\rightarrow$$\scriptstyle{\varphi_{d^{n}}}$$\leftarrow$$\rightarrow$$\scriptstyle{{\sf
pr}_{\alpha}}$

shows that

(6.26)
$(\varphi_{c})_{\alpha\circ\beta}=\alpha\circ(\eta_{d^{n}}^{F})^{-1}\circ
F(\beta).$

And the diagram

(6.27)
${c}$${F(c)}$${d^{\mathpzc{C}(c,d)}}$${d}$${d^{n}}$${F(d^{n})}$${d^{\mathpzc{C}(d^{n},d)}}$${d}$$\leftarrow$$\rightarrow$$\scriptstyle{\beta}$$\leftarrow$$\rightarrow$$\scriptstyle{\eta_{c}^{F}}$$\leftarrow$$\rightarrow$$\scriptstyle{F(\beta)}$$\leftarrow$$\rightarrow$$\scriptstyle{\varphi_{c}}$$\leftarrow$$\rightarrow$$\scriptstyle{d^{\mathpzc{C}(\beta,d)}}$$\leftarrow$$\rightarrow$$\scriptstyle{{\sf
pr}_{\beta_{i}}}$$\leftarrow$$\rightarrow$$\scriptstyle{\sf
id}$$\leftarrow$$\rightarrow$$\scriptstyle{{\sf
pr}_{i}}$$\leftarrow$$\rightarrow$$\scriptstyle{\eta^{F}_{d^{n}}}$$\scriptstyle{\cong}$$\leftarrow$$\rightarrow$$\scriptstyle{\varphi_{d^{n}}}$$\leftarrow$$\rightarrow$$\scriptstyle{{\sf
pr}_{{\sf pr}_{i}}}$

implies that

(6.28) $((\eta_{d^{n}}^{F})^{-1}\circ
F(\beta))_{i}=(\varphi_{c})_{\beta_{i}}.$

Therefore, we have

(6.29)
$(\varphi_{c})_{\alpha\circ\beta}=\alpha\circ((\varphi_{c})_{\beta_{1}},\dots,(\varphi_{c})_{\beta_{n}}).$

Then Lemma 6.3 implies that $\sigma_{n}\varphi_{c}=\tau_{n}\varphi_{c}.$ This
implies that $T_{d^{n}}=\mathsf{hom}^{n}_{{\sf
O}^{+}(d)}(\mathpzc{C}(-,d),d).$

Now we prove that $T_{\\{1,d^{n}\\}}=\mathsf{hom}^{\leq n}_{{\sf
O}(d)}(\mathpzc{C}(-,d),d).$ The proof is similar. We just need to note that
if $\eta^{F}_{1}:1\to F(1)$ is an isomorphism, then for any natural
transformation $\varphi:F\to d^{\mathpzc{C}(-,d)},$ any $\alpha:1\to d$ and
$\beta:c\to 1$ we have a diagram similar to (6.25).

(6.30)
${c}$${F(c)}$${d^{\mathpzc{C}(c,d)}}$${d}$${1}$${F(1)}$${d^{\mathpzc{C}(1,d)}}$${d}$$\leftarrow$$\rightarrow$$\scriptstyle{\beta}$$\leftarrow$$\rightarrow$$\scriptstyle{\eta_{c}^{F}}$$\leftarrow$$\rightarrow$$\scriptstyle{F(\beta)}$$\leftarrow$$\rightarrow$$\scriptstyle{\varphi_{c}}$$\leftarrow$$\rightarrow$$\scriptstyle{d^{\mathpzc{C}(\beta,d)}}$$\leftarrow$$\rightarrow$$\scriptstyle{{\sf
pr}_{\alpha\circ\beta}}$$\leftarrow$$\rightarrow$$\scriptstyle{\sf
id}$$\leftarrow$$\rightarrow$$\scriptstyle{\alpha}$$\leftarrow$$\rightarrow$$\scriptstyle{\eta^{F}_{1}}$$\scriptstyle{\cong}$$\leftarrow$$\rightarrow$$\scriptstyle{\varphi_{1}}$$\leftarrow$$\rightarrow$$\scriptstyle{{\sf
pr}_{\alpha}}$

This diagram implies that

(6.31) $(\varphi_{c})_{\alpha\circ\beta}=\alpha\circ(),$

where $():F(c)\to 1.$ Then Lemma 6.3 implies that
$\sigma_{0}\varphi_{c}=\tau_{0}\varphi_{c}$ and the rest of the proof is the
same as for $T_{d^{n}}.$

The fact that $T_{d^{+}}=\mathsf{hom}_{{\sf O}^{+}(d)}(\mathpzc{C}(-,d),d)$
follows from the equations

$T_{d^{+}}=\varprojlim T_{d^{n}}$

and

$\mathsf{hom}_{{\sf
O}^{+}(d)}(\mathpzc{C}(-,d),d)=\varprojlim\mathsf{hom}^{\leq n}_{{\sf
O}^{+}(d)}(\mathpzc{C}(-,d),d).$

Similarly we have

$T_{d^{\bullet}}=\varprojlim T_{\\{1,d^{n}\\}}=\mathsf{hom}_{{\sf
O}(d)}(\mathpzc{C}(-,d),d).$

∎

##  7\. An idempotent pro-completion tower

We end with a few comments on a pro-idempotent monad $M_{\bullet}$ associated
with a given monad $M.$ Recall from [1] that the Bousfield-Kan $R$-homology
completion tower $R_{\bullet}X,$ associated with a topological space $X,$ is
pro-idempotent. In addition, and as consequence, its $R-$ homology
$H_{*}(R_{\bullet}X,R)$ is naturally pro-isomorphic to the homology
$H_{*}(X,R)$ of $X.$

One would like to have a similar result for a general monad $M.$ This is
possible, with the price being the replacement of the tot-tower
$tot_{\bullet}X,$ with a slightly more involved tower defined inductively.
This line was considered to with clear results for a a general co-augmented
functor in the homotopy category of spaces, by A. Libman, compare [12].

For a general subcategory $\mathpzc{D}\subseteq\mathpzc{C}$ of a nice category
$\mathpzc{C},$ one can construct the right Kan extension functor
$T{D}:\mathpzc{C}\to\mathpzc{C}.$ as above. This functor is not idempotent.
However, we can consider a ”refined” right extension

$T{D}^{pro}:\mathpzc{C}\to pro-\mathpzc{D}.$

This last functor associate, as usual to each $X\in\mathpzc{C}$ a diagram of
objects in $\mathpzc{D}$ indexed by the coma category $\mathpzc{D}_{X/}.$ The
limit, in $\mathpzc{C},$ of this diagram of objects in $\mathpzc{D}$ is the
value of right Kan extension, $T{D}(X),$ on the object $X,$ see equation 2.4
above.

Now for the diagram $X/\mathpzc{D}\to\mathpzc{D},$ to define a pro-object it
must be filtering. Therefore if we assume that $D$ is closed under finite
limits, i.e. pullbacks. In this case, the diagram
$X/\mathpzc{D}\to\mathpzc{D}$ of objects in $\mathpzc{D}$ is filtering. Hence,
the the above define diagram $T{D}^{pro}$ is a pro-object in
$pro-\mathpzc{C}.$ Moreover, the functor can be directly prolonged to a
functor:

$T^{pro}{D}:pro-\mathpzc{C}\to pro-\mathpzc{C}$

which deserves the name ”tautological pro-completion” with respect to the
inclusion $\mathpzc{D}\subseteq\mathpzc{C}.$ As such it is clearly pro-
idempotent. For example, if $\mathpzc{D}$ is the full subcategory of Groups
consisting of finite groups, one gets the usual diagram $G\to(\Gamma_{i})_{i}$
of finite all finite groups under a given group $G.$

Our aim is to show that in case $\mathpzc{D}$ is the closure under finite
limits of the image of a monad $M,$ there is a small variant on the Bousfield-
Kan tower associated with $M,$ which is pro-equivalent to this canonical pro-
object $T^{pro}{D}.$ Moreover, that pro-object is the terminal pro-monad among
those that preserve the closure of $ImD$ under finite limits. Note that the
image of $M$ is not, in general, closed under finite limits. Therefore, the
following construction, which is valid for any $M,$ will be shown to be
equivalent to the above $T^{pro}_{\overline{ImM}}$ where the subscript
$\overline{ImM}$ denotes the closure of the image of the monad $M$ under
finite limits.

Consider the inductively defined tower of injective maps of terminal monads:

$M_{0}=M;M_{i+1}:=T_{M_{i}}\to M_{i}\cdots\to M.$

To continue, we notice that under no additional assumptions on $\mathpzc{C}$,
one has associated with $M$ an idempotent pro-monad tower,
$(M_{i})_{i<\omega}.$ The limit of this tower of monads is precisely the
terminal monad that preserves the closure of $ImM,$ the image of our monad
$M,$ under all finite limits.

By 3.5, each $M_{i}$ in this tower not only preserves $M_{j}$ for $j<i,$ but
also any finite limit of objects of the form $M_{j}(Y_{j})$ for
$Y_{j}\in\mathpzc{C}.$ It follows that if $X\in\mathpzc{C}$ is in the closure
under finite limits of the image of $M,$ then for $i$ large enough, there is
an equivalence $X\cong M_{i}(X).$ Since each $M_{i}(X)$ is, by construction,
an element in the above finite limits closure of $ImM$ in $\mathpzc{C}.$
Therefore we have a pro-idempotent tower:

$M_{\bullet}\cong M_{\bullet}\circ M_{\bullet}.$

In addition, using the argument in [6] and [7] it follows that for every monad
$M$ one has a pro-equivalence:

$M\cong M(M_{\bullet}).$

In other words, the tower $M_{\bullet}$ satisfies some of the basic properties
of the classical Bousfield-Kan $R$-homology completion tower as given by [1].

The limit, $M_{\infty}=lim_{i}M_{i},$ of the tower $M_{i}$ is the terminal
monad among all co-augmented functors that preserve $\overline{ImM},$ the
closure of the image of $M$ under finite limits.

######  Remark 7.1.

following Fakir, [8] one can continue to this tower of inclusions, see 3.7
above, trasfinitely. Under suitable rather weak assumption on $\mathpzc{C}$
this tower converges to an idempotent monad $L_{M}.$ This idempotent monad is
easily seen to be the terminal monad $T_{\overline{\mathpzc{D}}}$ where now
$\overline{\mathpzc{D}}$ denotes the closure of the image of $M$ under all
limits. In the infinity category of spaces the classical example is the map
$L_{HR}\to R_{\infty}$ from the idempotent Bousfield homological localization
to the $R-$ completion functor on spaces, compare [7].

Examples: In the category of groups we can consider the terminal monad $T_{G}$
associated with a group $G,$ so that $T_{G}(G)\cong G.$ It is given as above
by the double dual

$T_{G}(\Gamma)\cong map_{End\,G}(map(\Gamma,G),G).$

This is a subgroup of $G^{l}$ for $l=|map(\Gamma,G)|,$ the cardinality of the
set of homomorphisms. For the group of integers $\Gamma=\mathbb{Z},$ we have
$T_{G}(\mathbb{Z})\subseteq G^{|G|}.$ The transfinite tower of Fakir
stabilizes at $\mathbb{Z}\to L_{G}(\mathbb{Z})\cong C,$ where $C$ is a cyclic
group whose order is the LCM of the orders of all elements of $G,$ since the
image of the generator of $\mathbb{Z}\to T_{G}(\mathbb{Z},)$ is the diagonal
element $(g_{g})_{g\in G}.$ Namely, $L_{G}(\mathbb{Z})$ is the image subgroup
of $\mathbb{Z}\to T_{G}(\mathbb{Z}).$ This $L_{G}$ localization, or reflexive
functor can be characterized as terminal among those that preserve the closure
in the category of groups of $\\{G\\}$ under all limits, mapping $\mathpzc{C}$
into that closure, or the initial idempotent monad that turns every
$map(-,G)$-equivalence (i.e. a sort of ’G-cohomology equivalence’) into an
equivalence.

An infinity categorical example see [9]. The classical $R_{\infty}$ of
Bousfield and Kan comes from the monad $R$ on the $\infty$-category of
topological spaces. It preserves not only spaces of the form $RX,$ i.e.
$R-$GEMs but also $R$-polyGEM spaces i.e. the closure of $R-$GEMs under finite
limits. In this special case the construction of the terminal $R_{\infty}$ is
somewhat simpler than the above inductive tower $(M_{i})_{i}.$
The precise meaning or value of this tower for the (discrete) pro-finite
completion of groups, considered as a monad $\mathpzc{M}(G)=\widehat{G},$ is
not immediately clear. For every group $G,$ the monad $\mathpzc{M}_{\infty}G$
is a natural subgroup of the pro-finite completion $\widehat{G}$ of $G;$ with
the property that it is idempotent
($\mathpzc{M}_{\infty}\mathpzc{M}_{\infty}(G)\cong\mathpzc{M}_{\infty}(G))$ if
$G$ is a finite limit of (discrete) profinite groups. In fact, since
$\mathpzc{M}_{\infty}$ preserves all finite limits in $Im\mathpzc{M},$ see
above, we have that it preserves any finite limit of profinite groups. Note,
however, that for a finitely presented group $G,$ one has an isomorphism,
$\mathpzc{M}_{\infty}G=\widehat{G},$ since for such a group one has an
isomorphism $\widehat{G}\cong\widehat{\widehat{G}},$ namely, the completion
$\mathpzc{M}=\widehat{(-)}$ is idempotent on this subcategory of groups. The
transfinite intersection or limit of $lim_{\alpha}\mathpzc{M}_{\alpha}$ on all
ordinals is also an interesting subgroup of the pro-finite completion, which
is just $\widehat{G}$ itself if $G$ is finitely presented.

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# An intuition for physicists: information gain from experiments

Johannes Buchner Max Planck Institute for Extraterrestrial Physics,
Giessenbachstrasse, 85741 Garching, Germany

###### Abstract

How much one has learned from an experiment is quantifiable by the information
gain, also known as the Kullback-Leibler divergence. The narrowing of the
posterior parameter distribution $P(\theta|D)$ compared with the prior
parameter distribution $\pi$($\theta$), is quantified in units of bits, as:

$D_{\mathrm{KL}}(P|\pi)=\int\log_{2}\left(\frac{P(\theta|D)}{\pi(\theta)}\right)\,P(\theta|D)\,d\theta$

This research note gives an intuition what one bit of information gain means.
It corresponds to a Gaussian shrinking its standard deviation by a factor of
three.

##  1 Gaussian update

Figure 1: The more the Gaussian shrinks (x-axis), the higher the information
gain (y-axis, in bits). The inset illustrates a shrinkage factor $k=3$ between
prior (dashed) and posterior (solid curve).

Consider for the posterior a Gaussian with standard deviation $\sigma$, and
for the prior a Gaussian at the same mean, but wider by a factor $k$,
$\sigma^{\prime}=k\sigma$. The inset of Figure 1 illustrates this scenario.
After some computation we find:

$\displaystyle D_{\mathrm{KL}}$
$\displaystyle=\int\left(2\pi\sigma{{}^{2}}\right)^{-\frac{1}{2}}e^{-\frac{\theta{{}^{2}}}{2\sigma^{2}}}\times\left(-\frac{1}{2}\log_{2}\left(2\pi\sigma{{}^{2}}\right)-\frac{\theta{{}^{2}}}{2\sigma^{2}}+\frac{1}{2}\log_{2}\left(2\pi\sigma{{}^{2}}k^{2}\right)+\frac{\theta{{}^{2}}}{2\sigma^{2}k^{2}}\right)d\theta$
$\displaystyle=\log_{2}k+\left(k^{-2}-1\right)/2/\ln 2$

This relation between the more intuitive “Gaussian shrinkage factor $k$” and
bits is illustrated in Figure 1.

The update of a flat prior distribution to a flat posterior distribution
narrowed by a shrinkage factor $k$ gives $D_{\mathrm{KL}}(P|\pi)=\log_{2}k$.
That is, two (eight) bits corresponds to $k=4$ ($256$). This follows the usual
computer science interpretation that removing $b$ bits restricting the
possibility space by a factor of $2^{b}$, which here is the shrinkage factor
of the posterior volume $k$.

##  2 Estimation from posterior samples

The information gain can be computed easily from one-dimensional histograms of
posterior samples as:

$D_{\mathrm{KL}}(P|\pi)\approx\sum_{i}\frac{n_{i}}{N}\log_{2}\frac{n_{i}/N}{m_{i}/M},$

where $n_{i}$ ($m_{i}$) are posterior (prior) samples within bin $i$, out of
$N$ ($M$) total samples. If the bin widths are sized to contain equal prior
mass (e.g., equal size for uniform priors), the formula simplifies to:
$D_{\mathrm{KL}}(P|\pi)\approx\sum_{i=1}^{B}n_{i}/N\log_{2}\left(n_{i}B/N\right)$.
Empty bins should be skipped. Negative $D_{\mathrm{KL}}$ are numerical
artifacts and should be set to zero.

The general-purpose Bayesian inference package UltraNest (Buchner, 2021)
computes the information gain overall (using the method of Skilling, 2004) and
for each parameter.

## References

  * Buchner (2021) Buchner, J. 2021, The Journal of Open Source Software, 6, 3001
  * Skilling (2004) Skilling, J. 2004, AIP Conference Proceedings, 735, 395