# Cellular Cohomology in Homotopy Type Theory

Ulrik Buchholtz<EMAIL_ADDRESS>and  Kuen-Bang Hou
(Favonia<EMAIL_ADDRESS>
###### Abstract.

We present a development of cellular cohomology in homotopy type theory.
Cohomology associates to each space a sequence of abelian groups capturing
part of its structure, and has the advantage over homotopy groups in that
these abelian groups of many common spaces are easier to compute. Cellular
cohomology is a special kind of cohomology designed for cell complexes: these
are built in stages by attaching spheres of progressively higher dimension,
and cellular cohomology defines the groups out of the combinatorial
description of how spheres are attached. Our main result is that for finite
cell complexes, a wide class of cohomology theories (including the ones
defined through Eilenberg-MacLane spaces) can be calculated via cellular
cohomology. This result was formalized in the Agda proof assistant.

###### Key words and phrases:

homotopy type theory, cellular cohomology, mechanized reasoning

##  1\. Introduction

_Homotopy type theory_ (HoTT) [hott-as:book] is a new area exploring the
potential of type-theoretic presentations of homotopy theory, an area
originating from the study of topological spaces up to continuous
transformation. It facilitates computer checking of proofs, and in some case
results in the discovery of new proofs. (See for example [blakers-massey-in-
hott].) Many proofs have since been mechanized and checked by proof assistants
such as Agda [hott-in:agda], Lean [hott-in:lean] and Coq [hott-in:coq]. HoTT
sheds new light on how type theory can be understood and leads to new type
theories based on the resulting insights [cchm, chtt, ccctt-at:csl,
chtt.part1, chtt.part2, chtt.part3, chtt.part4, cubicaltt-hits].

One important concept in homotopy theory is homotopy groups, groups associated
with every (pointed) space that reveal important structures of that space.
However, it is difficult to calculate higher homotopy groups for many common
spaces, for example spheres. As a result, homology and cohomology theories
arise as an alternative way to study spaces.

The success of homotopy type theory begins with homotopy groups partially
because of their simple type-theoretic definition. For example, the homotopy
groups $\pi_{n}(\mathbb{S}^{n})$ were calculated in [licata-brunerie-pinsn].
Over time people have been extending the success in homotopy and cohomology
theories as well. This paper is yet another milestone we have achieved.

The basic idea of _cohomology theory_ is to study _functions from cycles_ in a
space. Functions from the cycles at a fixed dimension $n$ form a group, which
is called the $n$th cohomology group. There are several ways to define a
theory of such groups, some combinatorial and some axiomatic; amazingly, the
classical theory states that the two approaches are essentially equivalent.
Our goal is to recreate such an equivalence in HoTT.

In this paper we focus on a particular class of types, _CW complexes,_ which
come with an explicit description of how the space is built by iterated
attachment of disks. We will then introduce _cellular cohomology theory,_ a
combinatorial cohomology theory specifically defined for CW complexes. After
that, we will define _ordinary Eilenberg–Steenrod cohomology theory,_ an
axiomatic framework for cohomology theory.

This journal paper is an expansion of the conference version [cohomology-
in:hott] with more explanations and diagrams. A large part of the text and
Agda formulation presented in the conference paper already appeared in Hou
(Favonia)’s PhD thesis [favonia-as:thesis]. However, one critical lemma, Lemma
LABEL:lma:cochain, was stated as a conjecture and the main theorem was only
proved after the thesis. During the study, a critical component, _degree_ ,
was radically changed to facilitate the proving.

##  2\. Notation

We assume that readers are familiar with common type-theoretic expressions as
in [hott-as:book, favonia-as:thesis]. In particular, we work in Martin-Löf
type theory augmented with Voevodsky’s _univalence axiom_ and the _pushout_
type constructor for homotopy pushouts. From these ingredients we can define
cofibers, the propositional and higher dimensional truncations [prop-trunc,
joinconstruction], set quotients, and other higher inductive types used in
this work. Our constructions also work in the proposed cubical type theories
[cchm, chtt, ccctt-at:csl, chtt.part1, chtt.part2, chtt.part3, chtt.part4,
cubicaltt-hits]. The following is a brief review.

###  2.1. Basics

We often use the word “space” as a synonym for “type” in order to facilitate
the intuition of types as spaces. The identification type between two elements
$x,y$ of type $A$ is denoted by $x=y$.

A pointed type $X$ is equipped with a point $\mathrm{pt}_{X}$, and a pointed
arrow $f$ from a pointed type $X$ to another pointed type $Y$ is an arrow
equipped with an identification from $f\mskip 0.0mu\lparen\mskip
0.0mu\mathrm{pt}_{X}\mskip 0.0mu\rparen\mskip 0.0mu$ to $\mathrm{pt}_{Y}$.
Pointed arrow types are specially marked as $X\mathrel{{\cdot}{\to}}Y$ for
their importance in calculating degrees (Section 5).

###  2.2. Higher Inductive Types

Throughout this paper we are going to employ different higher inductive types
that are definable using only pushouts. The first one is the _suspension_ of
type $A$ as the following pushout:

$A$$\mathbb{1}$$\mathbb{1}$$\mathrm{susp}\mskip 0.0mu\lparen\mskip
0.0muA\mskip 0.0mu\rparen\mskip 0.0mu$$\ulcorner$

If $f$ is an arrow from $A$ to $B$, then $\mathrm{susp}\mskip
0.0mu\lparen\mskip 0.0muf\mskip 0.0mu\rparen\mskip 0.0mu$ is the arrow from
$\mathrm{susp}\mskip 0.0mu\lparen\mskip 0.0muA\mskip 0.0mu\rparen\mskip 0.0mu$
to $\mathrm{susp}\mskip 0.0mu\lparen\mskip 0.0muB\mskip 0.0mu\rparen\mskip
0.0mu$ arising from the functoriality of $\mathrm{susp}$. The $n$th _sphere_
$\mathbb{S}^{n}$ is defined to be the iterated suspension
$\mathrm{susp}^{n}\mskip 0.0mu\lparen\mskip 0.0mu\mathbb{2}\mskip
0.0mu\rparen\mskip 0.0mu$.

Another instance is the _homotopy cofiber_ of a pointed map
$f:X\mathrel{{\cdot}{\to}}Y$. It is defined to be the following pushout:

$X$$Y$$\mathbb{1}$$\mathrm{cofiber}\mskip 0.0mu\lparen\mskip 0.0muf\mskip
0.0mu\rparen\mskip 0.0mu$$f$$\mathrm{cfcod}$$\mathrm{cfbase}$$\ulcorner$

If the map $f$ is understood, we write $Y\mskip 2.0mu/\mskip-1.0mu\mskip
2.0muX:=\mathrm{cofiber}(f)$.

Finally, given a family of pointed types $\mskip 0.0mu\\{\mskip
0.0muX_{i}\mskip 0.0mu\\}\mskip 0.0mu_{i:I}$ indexed by some type $I$, the
_wedge_ $\bigvee_{i:I}X_{i}$ is defined to be this pushout:

$I$$\sum_{i:I}X_{i}$$\mathbb{1}$$\bigvee_{i:I}X_{i}$$i\mapsto\mskip
0.0mu\langle\mskip 0.0mui;\mathrm{pt}_{X_{i}}\mskip 0.0mu\rangle\mskip
0.0mu$$\ulcorner$

The $n$th truncation of type $A$, written $\mskip 0.0mu\lVert\mskip
1.0muA\mskip 1.0mu\rVert\mskip 0.0mu_{n}$, enjoys the universal property that
for any $n$-truncated type $B$, arrows from $A$ to $B$ uniquely factor through
$\mskip 0.0mu\lVert\mskip 1.0muA\mskip 1.0mu\rVert\mskip 0.0mu_{n}$.
Truncations are definable with (iterated) pushouts [prop-trunc,
joinconstruction]; so are set quotients.

###  2.3. Groups

The _quotient group_ $G\mskip 2.0mu/\mskip-1.0mu\mskip 2.0muH$ is defined for
any group $G$ and any normal subgroup $H$ of $G$. For any pointed space $X$,
the $n$th _homotopy group_ $\pi_{n}(X)$ is defined to be the $0$-truncation of
the $n$th iterated loop space of $X$. The _free abelian group_ of a set $A$ is
written as $\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA\mskip 0.0mu]\mskip
0.0mu$.111In the Agda formulation, we allowed $A$ to be an arbitrary type, but
then took the set quotient to force the carrier of the resulting group to be a
set.

##  3\. CW complexes

A CW complex (also known as a cell complex) is an inductively defined type
built by attaching cells, starting from points, lines, faces, and so on. The
description consists of $A_{n}$ as the index set of cells at dimension $n$,
along with functions $\alpha_{n}$ denoting how cells are attached. (We refer
to [hatcher-at, Chapter 0] for a discussion of CW complexes in a classical
context.) With this combinatorial description at hand, one may define the
cellular cohomology groups as shown in Section 4.

Many common types can be represented as CW complexes: the unit type, the
spheres, the torus, and even the real projective spaces [realprojective]. One
can also build the two-cell complexes obtained by attaching a cell of
dimension $m+1$ to a sphere of dimension $n$ with attaching map given by an
element of $\pi_{m}(\mathbb{S}^{n})$. A special case is that of the _Moore
spaces_ $M(\mathbb{Z}/q\mathbb{Z},n)$ given by the element of
$\pi_{n}(\mathbb{S}^{n})$ corresponding to $q\in\mathbb{Z}$ under the
isomorphism $\pi_{n}(\mathbb{S}^{n})\simeq\mathbb{Z}$. It follows from this
description that $M(\mathbb{Z}/q\mathbb{Z},n)$ is the cofiber of the degree
$q$ map $\mathbb{S}^{n}\to\mathbb{S}^{n}$. The two-cell complexes are the
simplest ones that are not spheres, and they form an important family of
examples and counter-examples, see Section 5 below for an instance of this.

Let $X_{0}$ be the index set $A_{0}$ itself as the start and in general
$X_{n}$ be the construction up to dimension $n$.222This definition was adapted
from Buchholtz’s work in Lean [cellular.complexes-in:utt-by:ulrik]. A cell of
index $a:A_{n+1}$ at dimension $n+1$ is specified by its boundary in $X_{n}$,
denoted by the function $\alpha_{n+1}\mskip 0.0mu\langle\mskip 0.0mua;-\mskip
0.0mu\rangle\mskip 0.0mu$ from $\mathbb{S}^{n}$ to $X_{n}$. The type $X_{n+1}$
is then the result after attaching all cells at dimension $n+1$ to $X_{n}$.
Formally, the function $\alpha_{n+1}$ is of type
$A_{n+1}\times\mathbb{S}^{n}\to X_{n}$ describing the boundary of each cell.
Inductively, the type $X_{n+1}$ is defined to be the following pushout: (Note
that every pushout in homotopy type theory is a homotopy pushout.)

$A_{n+1}\times\mathbb{S}^{n}$$A_{n+1}$$X_{n}$$X_{n+1}$$\alpha_{n+1}$$\operatorname{fst}$$\ulcorner$

In this paper, we only work with _finite_ CW complexes whose building process
stops at some finite dimension and for which every $A_{n}$ is a finite set.
(That is, every $A_{n}$ is the standard $k$-element set for some $k$.)
Pictorially, a (finite) CW complex is the following iterated pushout, starting
from the type $X_{0}\mathrel{\vcentcolon\equiv}A_{0}$ and ending at some
dimension.

$A_{n+1}\times\mathbb{S}^{n}$$A_{n+1}$$A_{n+2}\times\mathbb{S}^{n+1}$$A_{n+2}$…$X_{n}$$X_{n+1}$…$\alpha_{n+1}$$\operatorname{fst}$$\alpha_{n+2}$$\operatorname{fst}$

A _pointed_ CW complex additionally requires $A_{0}$ to be pointed (and hence
$X_{0}$ and all following pushouts).

##  4\. Cellular Cohomology

Cohomology theory concerns _functions from cycles,_ and one of the best ways
to introduce it is through its dual, _homology theory,_ which is about the
cycles themselves. With access to an explicit, combinatorial description of a
type, suitable algebraic structures can be defined for cycles.

To begin with, a one-dimensional cycle in homology theory (and cohomology
theory) is a _linear combination of oriented lines_ that at each any point has
equally many incoming and outgoing lines counted with multiplicity (the in-
degree equals the out-degree).

$x$
---
$y$
---
$z$
---

$a$
---

$b$
---

$c$
---
$d$
---

In the above diagram, for example, $a+b+d$, $c-d$ and $-c-a-b$ are all cycles.
We identify cycles consisting of lines in different orders so that $c-d$ and
$-d+c$ are the same cycle. The intuition is to capture traversals in the space
without worrying about the sequence of lines being traversed; in fact, lines
in a cycle need not be connected. Each coefficient in a cycle (as a linear
combination) tracks the number of occurrences of a line, where a negative
number signifies a traversal in the opposite direction. One can define the
addition, subtraction and negation on these cycles as those on linear
combinations.

Let $\tilde{\partial}_{1}$ be the function mapping a line from $x$ to $y$ to
the linear combination $x-y$, which represents the oriented boundary of the
line. For example, in the above diagram, we have

$\displaystyle\tilde{\partial}_{1}\mskip 0.0mu\lparen\mskip 0.0mua\mskip
0.0mu\rparen\mskip 0.0mu$ $\displaystyle=x-y$
$\displaystyle\tilde{\partial}_{1}\mskip 0.0mu\lparen\mskip 0.0mub\mskip
0.0mu\rparen\mskip 0.0mu$ $\displaystyle=y-z$
$\displaystyle\tilde{\partial}_{1}\mskip 0.0mu\lparen\mskip 0.0muc\mskip
0.0mu\rparen\mskip 0.0mu$ $\displaystyle=z-x$
$\displaystyle\tilde{\partial}_{1}\mskip 0.0mu\lparen\mskip 0.0mud\mskip
0.0mu\rparen\mskip 0.0mu$ $\displaystyle=z-x.$

The criterion of being a cycle, that is, whether the in- and out- degrees are
in balance, reduces to whether the summation of $\tilde{\partial}_{1}$ of
these lines is exactly zero. If we extend the function $\tilde{\partial}_{1}$
on lines to linear combinations of lines as $\partial_{1}$, then _a linear
combination of lines is a cycle if and only if it is in the kernel of
$\partial_{1}$._ Therefore, we know $a+b+d$ is a cycle because

$\partial_{1}\mskip 0.0mu\lparen\mskip 0.0mua+b+d\mskip 0.0mu\rparen\mskip
0.0mu=\partial_{1}\mskip 0.0mu\lparen\mskip 0.0mua\mskip 0.0mu\rparen\mskip
0.0mu+\partial_{1}\mskip 0.0mu\lparen\mskip 0.0mub\mskip 0.0mu\rparen\mskip
0.0mu+\partial_{1}\mskip 0.0mu\lparen\mskip 0.0mud\mskip 0.0mu\rparen\mskip
0.0mu=\mskip 0.0mu\lparen\mskip 0.0mux-y\mskip 0.0mu\rparen\mskip 0.0mu+\mskip
0.0mu\lparen\mskip 0.0muy-z\mskip 0.0mu\rparen\mskip 0.0mu+\mskip
0.0mu\lparen\mskip 0.0muz-x\mskip 0.0mu\rparen\mskip 0.0mu=0.$

In the context of CW complexes introduced in Section 3, the linear
combinations of lines and points are the free abelian groups $\mathbb{Z}\mskip
0.0mu[\mskip 0.0muA_{1}\mskip 0.0mu]\mskip 0.0mu$ and $\mathbb{Z}\mskip
0.0mu[\mskip 0.0muA_{0}\mskip 0.0mu]\mskip 0.0mu$, respectively:

$\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA_{1}\mskip 0.0mu]\mskip
0.0mu$$\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA_{0}\mskip 0.0mu]\mskip
0.0mu$$\partial_{1}$

We would like to identify cycles up to cells at higher dimensions. in
particular, if there is a two-dimensional cell filling the difference between
two cycles, then those two cycles should be regarded as the same. Note that
for two cycles $u$ and $v$, the difference between them, $u-v$, is still a
cycle. This means we can turn our attention to how a two-dimensional cell
fills a cycle. Intuitively, a cell fills a cycle if its boundary matches the
cycle. For instance, the cell $e$ below fills in the cycle $c-d$:

$x$
---
$y$
---
$z$
---

$a$
---

$b$
---

$c$
---
$d$
---
$e$
---

Similar to $\partial_{1}$, we will later define the boundary function
$\partial_{2}$ for any two-dimensional cell indexed by $w:A_{2}$ by summing up
lines traversed by $\alpha_{2}\mskip 0.0mu\langle\mskip 0.0muw;-\mskip
0.0mu\rangle\mskip 0.0mu$ where $\alpha_{2}$ is given in the description of
the CW complex. Here, $\partial_{2}\mskip 0.0mu\lparen\mskip 0.0mue\mskip
0.0mu\rparen\mskip 0.0mu$ is $c-d$. A cycle can be filled if and only if it is
in the image of $\partial_{2}$. Consider the following diagram:

$\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA_{2}\mskip 0.0mu]\mskip
0.0mu$$\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA_{1}\mskip 0.0mu]\mskip
0.0mu$$\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA_{0}\mskip 0.0mu]\mskip
0.0mu$$\partial_{2}$$\partial_{1}$

The subject of our interest, _cycles up to identifications,_ is exactly the
quotient of cycles (the kernel of $\partial_{1}$) by boundaries of cells at
the next dimension (the image of $\partial_{2}$). Concretely, in the above
example, the kernel of $\partial_{1}$ is generated by the cycles $a+b+c$ and
$c-d$, the image of $\partial_{2}$ is generated by the cycle
$\partial_{2}\mskip 0.0mu\lparen\mskip 0.0mue\mskip 0.0mu\rparen\mskip
0.0mu=c-d$, and thus the cycles up to identifications constitute the cyclic
group generated by the equivalence class $\mskip 0.0mu[\mskip 0.0mua+b+c\mskip
0.0mu]\mskip 0.0mu$. This quotient is called the _first cellular homology
group_ of type $X$ (with integer coefficients), and groups for higher
dimensions can be defined in a similar way. How exactly the boundary functions
$\partial_{n}$ at higher dimensions should be defined from $A_{n}$ and
$\alpha_{n}$ will be discussed later.

The sequence formed by the free abelian groups $\mathbb{Z}[A_{n}]$ and
boundary maps $\partial_{n}$ is a _chain complex_. Cellular _co_ homology
takes the dual of the sequence before calculating the quotients of kernels by
images; it applies the contravariant functor $\hom\mskip 0.0mu\lparen\mskip
0.0mu-,G\mskip 0.0mu\rparen\mskip 0.0mu$ for some given abelian group $G$ to
the entire sequence. The dualized sequence is a _cochain complex._ 333The
kernel-image quotienting is called _homology_ in the classical literature, so
the homology of a chain complex is homology, and the homology of a cochain
complex is cohomology. The resulting diagram is

$\hom\mskip 0.0mu\mathopen{\big{\lparen}}\mskip 0.0mu\mathbb{Z}\mskip
0.0mu[\mskip 0.0muA_{2}\mskip 0.0mu]\mskip 0.0mu,G\mskip
0.0mu\mathclose{\big{\rparen}}\mskip 0.0mu$$\hom\mskip
0.0mu\mathopen{\big{\lparen}}\mskip 0.0mu\mathbb{Z}\mskip 0.0mu[\mskip
0.0muA_{1}\mskip 0.0mu]\mskip 0.0mu,G\mskip
0.0mu\mathclose{\big{\rparen}}\mskip 0.0mu$$\hom\mskip
0.0mu\mathopen{\big{\lparen}}\mskip 0.0mu\mathbb{Z}\mskip 0.0mu[\mskip
0.0muA_{0}\mskip 0.0mu]\mskip 0.0mu,G\mskip
0.0mu\mathclose{\big{\rparen}}\mskip 0.0mu$$\hom\mskip 0.0mu\lparen\mskip
0.0mu\partial_{2},G\mskip 0.0mu\rparen\mskip 0.0mu$$\hom\mskip
0.0mu\lparen\mskip 0.0mu\partial_{1},G\mskip 0.0mu\rparen\mskip 0.0mu$

and the _first _co_ homology group_, denoted $H^{1}(X;G)$, is the quotient of
the kernel of $\hom\mskip 0.0mu\lparen\mskip 0.0mu\partial_{2},G\mskip
0.0mu\rparen\mskip 0.0mu$ by the image of $\hom\mskip 0.0mu\lparen\mskip
0.0mu\partial_{1},G\mskip 0.0mu\rparen\mskip 0.0mu$. Groups at higher
dimensions are defined in a similar way; we write $H^{n}(X;G)$ for the $n$th
cellular cohomology group with coefficients in $G$.

One reasonable definition of $\partial_{n+1}$ on a cell $b$ at dimension $n+1$
is to individually calculate the coefficient $\mathrm{coeff}(b,a):\mathbb{Z}$
of each cell $a$ within the boundary of the cell $b$; that is, the boundary
function is of the following form (where the $\sum$ below is the summation in
linear algebra, not sum types):

$\partial_{n+1}(b)\mathrel{\vcentcolon\equiv}\sum_{a:A_{n}}\mathrm{coeff}(b,a)\;a.$

If $A_{n}$ could be infinite (which is impossible in our current work but
possible in future generalization), in order to make sense of the summation,
it seems we have to assume $\mathrm{coeff}(b,-)$ always has finite support;
this corresponds to the _closure-finiteness_ condition in the classical
theory, which is part of the definition of CW complexes and in fact what the
“C” in the “CW” stands for. The classical condition says the boundary of each
cell should be covered by a finite union of cells at lower dimensions, and so
our assumption is well-motivated and may be necessary.

$a$$a$$a$$\alpha_{n+1}\mskip 0.0mu\langle\mskip 0.0mub;-\mskip
0.0mu\rangle\mskip 0.0mu$$\mathrm{proj}_{a}$ Figure 1. The function used to
define $\mathrm{coeff}(b,a)$.

Intuitively, the value $\mathrm{coeff}(b,a)$ should capture the number of
(signed) occurrences of $a$ in the boundary of $b$. Considering the CW complex

$a$
---
$b$
---

with a two-cell $b$ with boundary consisting of the line $a$ and two other
lines, the value $\mathrm{coeff}(b,a)$ should be $1$ under suitable
orientation. How should we define $\mathrm{coeff}(b,a)$? The trick is to
identify all points and obtain a rose

$a$

where the boundary of $b$ is now composed of loops at the center; the winding
number of the loop $a$ is then the coefficient we are looking for. More
precisely, the coefficient is the winding number of the function depicted in
Figure 1, or more formally,

${\mathbb{S}^{1}}$${X_{1}}$${X_{1}\mskip 2.0mu/\mskip-1.0mu\mskip
2.0muX_{0}\simeq\bigvee_{\underline{\hskip
3.5pt}:A_{1}}\mathbb{S}^{1}}$${\mathbb{S}^{1}}$$\scriptstyle{\alpha_{2}\mskip
0.0mu\langle\mskip 0.0mub;-\mskip 0.0mu\rangle\mskip
0.0mu}$$\scriptstyle{\mathrm{cfcod}_{\phantom{1}}}$$\scriptstyle{\mathrm{proj}_{a}}$

where $X_{1}\mskip 2.0mu/\mskip-1.0mu\mskip 2.0muX_{0}$ is the cofiber of the
inclusion from $X_{0}$ to $X_{1}$ and $\mathrm{proj}_{a}$ kills every loop
except the one indexed by $a$. The projection function $\mathrm{proj}_{a}$ is
definable whenever the index type $A_{1}$ has decidable equality. For
arbitrary dimension $n\geq 1$, we follow the same pattern to obtain a function
from $\mathbb{S}^{n}$ to $\mathbb{S}^{n}$,

${\mathbb{S}^{n}}$${X_{n}}$${X_{n}\mskip 2.0mu/\mskip-1.0mu\mskip
2.0muX_{n-1}\simeq\bigvee_{\underline{\hskip
3.5pt}:A_{n}}\mathbb{S}^{n}}$${\mathbb{S}^{n}}$$\scriptstyle{\alpha_{n+1}\mskip
0.0mu\langle\mskip 0.0mub;-\mskip 0.0mu\rangle\mskip
0.0mu}$$\scriptstyle{\mathrm{cfcod}_{\phantom{n}}}$$\scriptstyle{\mathrm{proj}_{a}}$

and then inspect its generalized winding number, namely its _degree_ given by
the function $\deg:(\mathbb{S}^{n}\to\mathbb{S}^{n})\to\mathbb{Z}$ that will
be defined in Section 5. This critically relies on the following lemma which
will be used repeatedly in our work:

######  Lemma 1.

For $n\geq 1$, the types $X_{n}\mskip 2.0mu/\mskip-1.0mu\mskip 2.0muX_{n-1}$
and $\bigvee_{\underline{\hskip 3.5pt}:A_{n}}\mathbb{S}^{n}$ are equivalent.

######  Proof 4.1.

This follows from a direct calculation. Another way to look at it is to
recognize the two readings of the following square consisting of two pushout
squares. The lower square gives $P\simeq X_{n}\mskip 2.0mu/\mskip-1.0mu\mskip
2.0muX_{n-1}$ and the whole square (which is also a pushout square) gives
$P\simeq\bigvee_{\underline{\hskip 3.5pt}:A_{n}}\mathbb{S}^{n}$, and thus they
must be equivalent.

$A_{n}\times\mathbb{S}^{n-1}$$A_{n}$$X_{n-1}$$X_{n}$$\mathbb{1}$$P$$\alpha_{n}$$\operatorname{fst}$$\ulcorner$$\ulcorner$

The intuition is that, if we shrink $X_{n-1}$ into a point, the remaining of
$X_{n}$ is the attachment of $A_{n}$ many cells at the $n$th dimension, which
results into a wedge of $\mathbb{S}^{n}$ indexed by $A_{n}$.

The coefficient is then defined as

$\mathrm{coeff}(b,a)\mathrel{\vcentcolon\equiv}\deg\mskip
0.0mu\mathopen{\big{\lparen}}\mskip 0.0mu\mathrm{proj}_{a}\circ
e\circ\mathrm{cfcod}\circ\lambda{x}.\alpha_{n+1}\mskip 0.0mu\langle\mskip
0.0mub;x\mskip 0.0mu\rangle\mskip 0.0mu\mskip
0.0mu\mathclose{\big{\rparen}}\mskip 0.0mu$

where $e$ is the equivalence between $X_{n}\mskip 2.0mu/\mskip-1.0mu\mskip
2.0muX_{n-1}$ and $\bigvee_{\underline{\hskip 3.5pt}:A_{n}}\mathbb{S}^{n}$.
This finishes our definition of boundary functions in HoTT.

The cellular cohomology theory that is relevant in our paper is the _reduced_
cellular cohomology theory for pointed types. A characteristic difference is
that a reduced theory will assign the trivial group instead of the group
$\mathbb{Z}$ as the zeroth cohomology group of the unit type. It is arguably
more stylish to have trivial groups for the unit type, the most trivial
pointed type. To achieve this, the _reduced_ homology theory augments the
chain complex with $\epsilon$ to $\mathbb{Z}$ to give

${\cdots}$${\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA_{1}\mskip 0.0mu]\mskip
0.0mu}$${\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA_{0}\mskip 0.0mu]\mskip
0.0mu}$${\mathbb{Z}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\epsilon}$

where $\epsilon$ sums up integer coefficients in $\mathbb{Z}\mskip
0.0mu[\mskip 0.0muA_{0}\mskip 0.0mu]\mskip 0.0mu$; its dual,

${\cdots}$${\hom\mskip 0.0mu\mathopen{\big{\lparen}}\mskip
0.0mu\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA_{1}\mskip 0.0mu]\mskip 0.0mu,G\mskip
0.0mu\mathclose{\big{\rparen}}\mskip 0.0mu}$${\hom\mskip
0.0mu\mathopen{\big{\lparen}}\mskip 0.0mu\mathbb{Z}\mskip 0.0mu[\mskip
0.0muA_{0}\mskip 0.0mu]\mskip 0.0mu,G\mskip
0.0mu\mathclose{\big{\rparen}}\mskip 0.0mu}$${\hom\mskip 0.0mu\lparen\mskip
0.0mu\mathbb{Z},G\mskip 0.0mu\rparen\mskip 0.0mu}$$\scriptstyle{\hom\mskip
0.0mu\lparen\mskip 0.0mu\partial_{1},G\mskip 0.0mu\rparen\mskip
0.0mu}$$\scriptstyle{\hom\mskip 0.0mu\lparen\mskip 0.0mu\epsilon,G\mskip
0.0mu\rparen\mskip 0.0mu}$

leads to reduced cohomology groups. The ending $\mathbb{Z}$ effectively kills
one degree of freedom in $\mathbb{Z}\mskip 0.0mu[\mskip 0.0muA_{0}\mskip
0.0mu]\mskip 0.0mu$. In general, the reduced and unreduced integral homology
groups only differ by a $\mathbb{Z}$ at the zeroth dimension.

##  5\. Degrees

The notion of the degree of an endo-arrow of spheres is the key tool for
defining and computing with cellular cohomology (and homology), as shown in
the previous section. We therefore need a definition of degrees in HoTT.

The requirements of the definition are that we get for each $n:\mathbb{N}$ a
function $\deg:(\mathbb{S}^{n}\to\mathbb{S}^{n})\to\mathbb{Z}$ satisfying at
least the following properties:

  1. (1)

$\deg(\mathrm{id})=1$, where $\mathrm{id}:\mathbb{S}^{n}\to\mathbb{S}^{n}$ is
the identity function,

  2. (2)

$\deg(g\circ f)=\deg(g)\deg(f)$, where $g\circ f$ is the composition of $f$
and $g$,

  3. (3)

$\deg(\mathrm{susp}(f))=\deg(f)$, where
$\mathrm{susp}(f):\mathbb{S}^{n+1}\to\mathbb{S}^{n+1}$ is the suspension of
$f:\mathbb{S}^{n}\to\mathbb{S}^{n}$.

  4. (4)

$\deg(f)$ is the usual winding number of $f$, for
$f:\mathbb{S}^{1}\to\mathbb{S}^{1}$.

These requirements suffice to uniquely determine the degree function, although
we have not formalized this fact.

One classical definition of degrees relies on homology theories, which are
only now becoming available in HoTT [homology-in-hott]. The cellular homology
theory for a CW complex can be defined by skipping the dualization on page 4,
and a more abstract version (in the style described in Section 6) can be
obtained by heavily using properties of the smash product, but in either case
it seems to be an unnecessary complication just for the definition of degrees.

Another approach is to apply the homotopy group functor $\pi_{n}$ to the
function in question. Hou (Favonia)’s thesis [favonia-as:thesis] employed
this. However, for our purposes where we need good control on the interaction
between the degree and a cohomology theory, we have found it expedient to use
yet another approach. More precisely, it seems difficult to write down a
generator of the cyclic group $\pi_{n}\mskip 0.0mu\lparen\mskip
0.0mu\mathbb{S}^{n}\mskip 0.0mu\rparen\mskip
0.0mu\mathrel{\vcentcolon\equiv}\mskip 0.0mu\lVert\mskip 1.0mu\Omega_{n}\mskip
0.0mu\lparen\mskip 0.0mu\mathbb{S}^{n}\mskip 0.0mu\rparen\mskip 0.0mu\mskip
1.0mu\rVert\mskip 0.0mu_{0}\simeq\mathbb{Z}$ and thus challenging to relate
the group homomorphisms showing up during the proving.

The first observation is that by property (3) we can always reduce the
computation of degree to the case of a pointed map because
$\mathrm{susp}\mskip 0.0mu\lparen\mskip 0.0muf\mskip 0.0mu\rparen\mskip 0.0mu$
is always pointed. Another manifestation of this is that the canonical map
from the set of pointed endo-arrows of the sphere $\mskip
0.0mu\mathopen{\big{\lVert}}\mskip
1.0mu\mathbb{S}^{n}\mathrel{{\cdot}{\to}}\mathbb{S}^{n}\mskip
1.0mu\mathclose{\big{\rVert}}\mskip 0.0mu_{0}$ to the set of endo-arrows
$\mskip 0.0mu\mathopen{\big{\lVert}}\mskip
1.0mu\mathbb{S}^{n}\to\mathbb{S}^{n}\mskip 1.0mu\mathclose{\big{\rVert}}\mskip
0.0mu_{0}$ given by forgetting the point is an equivalence for $n\geq 1$. This
means pointedness is free. For $n=0$, it is not the case: the identity on
$\mathbb{S}^{0}$ ($=\mathbb{2}$) has degree $1$, the map that swaps the points
has degree $-1$, while the remaining two maps have degree $0$, because they
suspend to maps of the circle with winding number zero.

The second idea is to observe that the degree function is for each $n\geq 1$ a
bijection on connected components. Since $\mathbb{Z}$ is a group, this
suggests that we should be able to endow the type of pointed endo-arrows of
$\mathbb{S}^{n}$ with a natural group structure (up to homotopy). Thus, we
want to define a group-like H-space structure on
$\mathbb{S}^{n}\mathrel{{\cdot}{\to}}\mathbb{S}^{n}$, such that the degree map
becomes a group isomorphism. Compared to the previous approach based on
$\pi_{n}$, where we had difficulty writing down a generator of $\pi_{n}\mskip
0.0mu\lparen\mskip 0.0mu\mathbb{S}^{n}\mskip 0.0mu\rparen\mskip 0.0mu$, a
generator here is simply the identity function on $\mathbb{S}^{n}$.

A _co-H-space_ is the notion dual to that of an H-space. This is a natural
notion in homotopy theory, see [arkowitz] for a survey in the classical
setting. A _co-H-space_ is a pointed type $A$ together with a
_comultiplication_ map $\sigma:A\mathrel{{\cdot}{\to}}A\vee A$, and witnesses
for the left and right _counit_ laws, stating that $\pi_{1}\circ\sigma$ and
$\pi_{2}\circ\sigma$ are homotopic as pointed maps to
$\mathrm{id}:A\mathrel{{\cdot}{\to}}A$, where $\pi_{i}:A\vee A\to A$ denotes
the projection on the left or right summand, for $i=1,2$ respectively. This
definition ensures that the type of pointed maps from a co-H-space to any
pointed type is an H-space. A _cogroup_ is a (homotopy) associative co-H-space
together with a left inverse map $\lambda:A\mathrel{{\cdot}{\to}}A$ such that
$\nabla\circ(\lambda\vee\mathrm{id})\circ\sigma$ is homotopic to the constant
pointed map, where $\nabla:A\vee A\mathrel{{\cdot}{\to}}A$ is the folding map,
which is the identity on each summand. Our main interest in cogroups stems
from the following fact.

######  Proposition 2.

Any suspension $\mathrm{susp}(A)$ has the structure of a cogroup with
comultiplication given by the pinch map
$\sigma:\mathrm{susp}(A)\to\mathrm{susp}(A)\vee\mathrm{susp}(A)$.

The proof is easy and we omit it here. We note that there are cogroups that
are not suspensions, for example the two-cell complex corresponding to an
element of order $3$ in $\pi_{34}(\mathbb{S}^{5})$, see [berstein-harper].

The next fact is the main result we need.

######  Proposition 3.

If $X$ is a cogroup and $Y$ is a pointed type, then the set $\mskip
0.0mu\lVert\mskip 1.0muX\mathrel{{\cdot}{\to}}Y\mskip 1.0mu\rVert\mskip
0.0mu_{0}$ has a natural group structure.

For example, the Moore space $M(\mathbb{Z}/q\mathbb{Z},n)$ is a suspension
when $n\geq 2$, allowing us to define for a pointed type $X$ the _$n$ th
homotopy group of $X$ with coefficients in $\mathbb{Z}/q\mathbb{Z}$_ as
$\pi_{n}(X;\mathbb{Z}/q\mathbb{Z})\mathrel{\vcentcolon\equiv}\mskip
0.0mu\lVert\mskip
1.0muM(\mathbb{Z}/q\mathbb{Z},n)\mathrel{{\cdot}{\to}}X\mskip
1.0mu\rVert\mskip 0.0mu_{0}$. (We shall not need these groups for our present
purposes.)

Having thus equipped each type
$\mathbb{S}^{n}\mathrel{{\cdot}{\to}}\mathbb{S}^{n}$ for $n\geq 1$ with the
structure of a group up to homotopy, it remains to actually define the degree
functions.

For $n=1$ we must choose the winding number per property (4). Since we want
property (3) to hold, we use the fact that the action of suspension on pointed
maps,
$(X\mathrel{{\cdot}{\to}}Y)\to(\mathrm{susp}(X)\mathrel{{\cdot}{\to}}\mathrm{susp}(Y))$
is a group homomorphism whenever $X$ is a cogroup. So it remains to verify
that this map is an group isomorphism in the case of endo-arrows on
$\mathbb{S}^{n}$, for $n\geq 1$. This follows from the Freudenthal suspension
theorem [hott-as:book, Theorem 8.6.4], in the same way that we prove that
$\pi_{n}(\mathbb{S}^{n})\simeq\mathbb{Z}$. In fact, the loop-suspension
adjunction refines to a group isomorphism
$(\mathrm{susp}(X)\mathrel{{\cdot}{\to}}Y)\simeq(X\mathrel{{\cdot}{\to}}\Omega
Y)$, using the cogroup structure on $\mathrm{susp}(X)$ and the (H-)group
structure on $\Omega Y$.

##  6\. Eilenberg–Steenrod Cohomology

Unlike the above explicit construction, there is also an _axiomatic_ framework
for cohomology. Participants to the special year at Institute for Advanced
Study have brought into HoTT the standard abstract framework for cohomology
theories—Eilenberg–Steenrod axioms [cohomology-in:utt-by:mike, mayer.vietoris-
in:utt, eilenberg.steenrod.cohomology.axioms]. An (ordinary)444A cohomology
theory is _ordinary_ if it satisfies the last Eilenberg–Steenrod axiom. See
below. reduced cohomology theory in HoTT may be defined as a contravariant
functor $h$ from pointed types to sequences of abelian groups (indexed by
integers $\mathbb{Z}$) satisfying the axioms below.

Before presenting these axioms, however, we need to define what it means to
_satisfy set-level axiom of choice_ , a condition stating that $\prod$
quantifiers and $0$-truncation commute. This will be used in one of the
cohomology axioms shown later. [set-level axiom of choice] A type $A$
_satisfies the set-level axiom of choice_ if, for any family of types $B$
indexed by $A$, the canonical function from $\mskip
0.0mu\mathopen{\Big{\lVert}}\mskip 1.0mu\prod_{a:A}B(a)\mskip
1.0mu\mathclose{\Big{\rVert}}\mskip 0.0mu_{0}$ to $\prod_{a:A}\mskip
0.0mu\mathopen{\big{\lVert}}\mskip 1.0muB(a)\mskip
1.0mu\mathclose{\big{\rVert}}\mskip 0.0mu_{0}$ is an equivalence. See [hott-
as:book] for more discussion about the axiom of choice and [cohomology-in:utt-
by:mike] for its role in cohomology theory in HoTT. Essentially, one could
present the Eilenberg–Steenrod axioms _without_ the axiom of choice, but it
would be difficult for pointed arrows whose codomains are Eilenberg–Mac Lane
spaces, an important example of cohomology theories, to satisfy these axioms
_within_ HoTT. In any case, because we only deal with cell complexes with
finite cell sets, we do not have to worry about the axiom of choice, for it is
provable in HoTT for finite sets.

Anyway, here are the axioms we use. We write $h^{n}(X)$ to denote the $n$th
group in the sequence for a pointed type $X$.

Suspension:

There is an isomorphism between $h^{n+1}(\mathrm{susp}(X))$ and $h^{n}(X)$,
and the choice of isomorphisms is natural in $X$.

Exactness:

For any pointed arrow $f:X\mathrel{{\cdot}{\to}}Y$, the following sequence is
exact, which means the kernel of $h^{n}(f)$ is exactly the image of
$h^{n}(\mathrm{cfcod})$.

${h^{n}(\mathrm{cofiber}(f))}$${h^{n}(Y)}$${h^{n}(X)}$$\scriptstyle{h^{n}(\mathrm{cfcod})}$$\scriptstyle{h^{n}(f)}$

Wedge:

Let $I$ be a type satisfying the set-level axiom of choice. For any family of
pointed types $X$ indexed by $I$, the group morphism

$\iota^{*}:h^{n}\mskip 0.0mu\mathopen{\Bigg{\lparen}}\mskip
0.0mu\bigvee_{i:I}X(i)\mskip 0.0mu\mathclose{\Bigg{\rparen}}\mskip
0.0mu\to\prod_{i:I}h^{n}(X_{i})$

induced by inclusions $X(i)\to\bigvee_{i:I}X(i)$ is a group isomorphism.

Dimension:

For any integer $n\neq 0$, the group $h^{n}\mskip 0.0mu\lparen\mskip
0.0mu\mathbb{2}\mskip 0.0mu\rparen\mskip 0.0mu$ is trivial.

The word _ordinary_ refers to satisfying the

######  Axiom 4.

dimension axiom. Interesting examples violating this axiom (but satisfying the
rest), such as $K$-theories or complex cobordism, were discovered after the
introduction of the framework, and are called _extra_ ordinary cohomology
theories. Our result only handles ordinary ones.

It will be shown that these axioms uniquely identify cohomology groups for
finite CW complexes. To begin with, we can calculate all the groups of the
spheres directly from the axioms:

######  Lemma 5.

For any $m,n:\mathbb{Z}$ such that $n\geq 1$, $h^{m}\mskip 0.0mu\lparen\mskip
0.0mu\mathbb{S}^{n}\mskip 0.0mu\rparen\mskip 0.0mu$ is isomorphic to
$h^{0}\mskip 0.0mu\lparen\mskip 0.0mu\mathbb{2}\mskip 0.0mu\rparen\mskip
0.0mu$ if $m=n$ and trivial otherwise.

######  Proof 6.1.

Because the spheres are iterated suspensions of $\mathbb{2}$, one can apply
the

And similarly the groups for the bouquets, or wedges of spheres. They play an
important role in our calculation of the groups of the CW complexes.

######  Lemma 8.

For any $m,n:\mathbb{Z}$ such that $n\geq 0$ and any finite set $A$,
$h^{m}\mskip 0.0mu\mathopen{\big{\lparen}}\mskip
0.0mu\bigvee_{\underline{\hskip 3.5pt}:A}\mathbb{S}^{n}\mskip
0.0mu\mathclose{\big{\rparen}}\mskip 0.0mu$ is isomorphic to
$\prod_{\underline{\hskip 3.5pt}:A}h^{0}\mskip 0.0mu\lparen\mskip
0.0mu\mathbb{2}\mskip 0.0mu\rparen\mskip 0.0mu$ if $m=n$ and trivial
otherwise.

There is also an important consequence from the


#1     #1#1#1     #1     #1     #1     #1#1          #1#1

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§ MULTI-SCALE NODE REPRESENTATIONS
These techniques encode location and node attribute information coming from increasing orders of proximity in the graph with separate blocks of representations for each node. We summarized how these techniques differ from each other in Table <ref> based on these desirable characteristics:

* Supervised: Node level embeddings can be trained to be optimal with respect to a downstream machine learning task.

* Implicit: The higher-order proximity matrix (e.g. random walk based pointwise mutual information) of nodes is not calculated explicitly by the model.

* Attributed: Vertex representations are learned by the distribution of generic attributes in the neighbourhoods not just pairwise node proximity.

* Multimodal: The attributes of vertices which are described by various data modalities (e.g. image, sound, text) can be incorporated by the model.

* Inductive: The representations of new nodes can be inferred without updating the model weights or learning a representation for the new node.

Based on our understanding MOOMIN is the first multi-scale node representation learning algorithm which incorporates multimodal data.