combinatorial homotopy theory - richard...

Combinatorial homotopy theory

Richard Williamson

October 4, 2012

Contents

I. Introduction 3I.1. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

II. A zoo of shapes 4II.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

II.1.1. Warsaw circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4II.1.2. A moral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4II.1.3. A finite circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4II.1.4. Our moral revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 5II.1.5. Spaces for which Whitehead’s theorem holds? . . . . . . . . . . . . 5II.1.6. Lego . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6II.1.7. Towards a conceptual lego! . . . . . . . . . . . . . . . . . . . . . . 6

II.2. Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6II.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6II.2.2. Recollections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7II.2.3. Presheaves as glueings . . . . . . . . . . . . . . . . . . . . . . . . . 8

II.3. Semi-cubical sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12II.3.1. Why the topological n-cubes? . . . . . . . . . . . . . . . . . . . . . 12II.3.2. Interval in a category . . . . . . . . . . . . . . . . . . . . . . . . . 12II.3.3. Homotopy with respect to an interval . . . . . . . . . . . . . . . . 13II.3.4. Towards the category of semi-cubes . . . . . . . . . . . . . . . . . 13II.3.5. An outline of the category of semi-cubes . . . . . . . . . . . . . . . 14II.3.6. A topological picture . . . . . . . . . . . . . . . . . . . . . . . . . . 15II.3.7. A need to impose relations . . . . . . . . . . . . . . . . . . . . . . 15II.3.8. The category of semi-cubes . . . . . . . . . . . . . . . . . . . . . . 16II.3.9. A universal property of the category of semi-cubes . . . . . . . . . 17II.3.10.Semi-cubical sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

II.4. Cubical sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24II.4.1. Further structure of the topological interval? . . . . . . . . . . . . 24II.4.2. Constant homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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II.4.3. Contraction structure upon an interval . . . . . . . . . . . . . . . . 25II.4.4. Towards the category of cubes . . . . . . . . . . . . . . . . . . . . 25II.4.5. The category of cubes . . . . . . . . . . . . . . . . . . . . . . . . . 26II.4.6. A topological picture . . . . . . . . . . . . . . . . . . . . . . . . . . 28II.4.7. A universal property of the category of cubes . . . . . . . . . . . . 28II.4.8. Cubical sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

III. Kan complexes and homotopy groups 35III.1.An attempt to define homotopy groups of a cubical set . . . . . . . . . . . 35

III.1.1. Getting to know cubical sets . . . . . . . . . . . . . . . . . . . . . 35III.1.2. A moral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41III.1.3. Towards homotopy groups of a cubical set . . . . . . . . . . . . . . 41III.1.4. Combinatorially homotopic n-cubes in a cubical set . . . . . . . . 42

III.2.Greeting Kan complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46III.2.1. Horns and boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 46III.2.2. Kan complexes and the free-standing n-cubes . . . . . . . . . . . . 50

III.3.Cubical sets with connections . . . . . . . . . . . . . . . . . . . . . . . . . 54III.3.1. Connections in topology . . . . . . . . . . . . . . . . . . . . . . . . 54III.3.2. Connections in abstract homotopy theory . . . . . . . . . . . . . . 56III.3.3. Category of cubes with connections . . . . . . . . . . . . . . . . . . 60III.3.4. A universal property of the category of cubes with connections . . 63III.3.5. A different perspective . . . . . . . . . . . . . . . . . . . . . . . . . 64III.3.6. Cubical sets with connections . . . . . . . . . . . . . . . . . . . . . 65

III.4.Homotopy groups of a Kan complex . . . . . . . . . . . . . . . . . . . . . 66III.4.1. Combinatorial homotopy defines an equivalence relation for a Kan

complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66III.4.2. Group structure upon π1(X, ∗) . . . . . . . . . . . . . . . . . . . . 70III.4.3. Group structure upon πn(X, ∗) . . . . . . . . . . . . . . . . . . . . 79

A. A categorical miscellany 89A.1. Free category upon a directed graph . . . . . . . . . . . . . . . . . . . . . 89

A.1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89A.1.2. Free category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89A.1.3. A universal property of a free category . . . . . . . . . . . . . . . . 90

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I. Introduction

Lecture notes for an advanced course currently taking place at NTNU, Trondheim.

I.1. Acknowledgements

I thank Finn Faye Knudsen for his questions concerning cubical sets with connections,which led to the inclusion of §III.3.5 into these notes.

I thank Marius Thaule for his proof reading of the notes from the first lecture — thenotes have not yet been revised accordingly!

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II. A zoo of shapes

II.1. Introduction

II.1.1. Warsaw circle

Topological spaces can be very wild — at least from the point of view of ordinaryhomotopy theory! We might for instance encounter the Warsaw circle.

a∞

b∞

c d

a1

b1

a2

b2

a3

b3

a4

b4

Regarded as a subspace of the plane, in this picture we can take a∞ to be (0, 1), b∞ tobe (0, 12), c to be (0, 0), d to be (1, 0), an to be ( 1

n , 1), and bn to be ( 1n ,

12). The canonical

map from the Warsaw circle to the point gives rise to an isomorphism on all homotopygroups, but the Warsaw circle is not contractible1.

II.1.2. A moral

We may draw a moral from this example — maps from the topological interval into theWarsaw circle are not appropriate for detecting the global nature of the Warsaw circle.Intuitively, a path in the Warsaw circle cannot ‘jump’ from the vertical line from an tobn to the vertical line from a∞ to b∞, for any n, and thus the fundamental group cannotdetect any loops. With a more appropriate notion of fundamental group — for example,one cooked out of open coverings — a loop will be detected2.

II.1.3. A finite circle

Let us explore another example. Let S1fin denote the poset defined by the picture below3

4

a

b c

d

From any poset we can cook up a space4 with the same underlying set, and with Udeclared to be an open set if for any x ∈ U and x ≤ y, we have y ∈ U . Thus thetopology on S1

fin consists of exactly the following open sets: ∅, {b}, {c}, {b, c}, {a, b, c},{b, c, d}, and {a, b, c, d}. There is a map

S1 S1fin

which gives rise to an isomorphism on all homotopy groups, where S1 is the usualtopological circle5. On the other hand, every map

S1fin S1

is constant, so that S1fin cannot be homotopy equivalent to S1.

II.1.4. Our moral revisited

Though this example is of a very different nature to that of the Warsaw circle, we maydraw from it the same moral as in II.1.1 — maps from the topological interval into S1

fin

are not appropriate for detecting the global nature of S1fin. Intuitively, to build a good

homotopy theory of finite spaces, our interval should be a finite space! The Sierpinskiinterval, namely the poset defined by the following diagram, is a more suitable gadgetto look at6.

0 1

II.1.5. Spaces for which Whitehead’s theorem holds?

We thus see that if we hope for our classical theory of homotopy groups of a space X tobehave well — in particular, if we wish for Whitehead’s theorem to hold, namely thathomotopy equivalence can be detected by homotopy groups — we must work with aspace X for which maps from In to X are an appropriate setting to be working in. Howmight we arrive at the notion of such a space?

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II.1.6. Lego

As a first guess, we might begin with the topological n-cubes themselves, and try con-structing spaces out of these in a nice way. Perhaps the single most important founda-tional insight of algebraic topology is that most familiar spaces can be constructed byglueing together copies of the topological n-cubes (or other choices of building blocks)along their faces. For example, the torus can be obtained by glueing together a copy ofI2 to two copies of I, as indicated by the colours or arrowheads in the figure below.

I I I2

It turns out that Whitehead’s theorem indeed holds for spaces built up in this way.Moreover these spaces behave well in many respects, and are very general — every spacecan be shown to be weakly homotopy equivalent to one obtained by glueing togethercopies of the topological n-cubes along their faces. The key step in the proof is anapproximation theorem — variously described as cellular, cubical, simplicial, and so on,depending on the context — which goes back at least to the 1920s.

II.1.7. Towards a conceptual lego!

We have now arrived at our motivating idea: good spaces for homotopy theory can beconstructed by glueing together a small collection of building blocks. In this lecture wewill explain that, thinking about this idea from a conceptual point of view, we can movefrom topology to a purely abstract setting.

II.2. Presheaves

II.2.1. Introduction

Let us discuss a little category theory. We will show that the category of presheaves onA has a universal property — it can be viewed as asserting that SetA

opis freely built

from A by formally adding in colimits. More precisely, we will establish this universalproperty by proving that any presheaf is a colimit of representable presheaves. One cango far in category theory with only a few tools — vital, but few. This is one of them!

A colimit is a categorical notion of glueing. If we view the objects of A as buildingblocks, and the arrows of A as a prescription for how we may glue these building blocksto one another, we can — by virtue of the universal property of SetA

op— exactly view a

presheaf on A as a gadget glued out of these building blocks following this prescription.It is this intuition that leads to the idea that presheaves may model homotopy types.

If we can capture the building blocks of topology — simplexes, or cubes, or more exoticgadgets — in an abstract way as the objects of a category, and capture the recipe forglueing these shapes to one another as the arrows of this category, we might hope that

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the corresponding presheaves are a good abstract setting for homotopy theory. In therest of this course we will work towards showing that this hope can be fulfilled!

II.2.2. Recollections

II.2.2.1. Let Set denote the category of sets.

II.2.2.2. Given a category A, we denote its opposite category by Aop.

II.2.2.3. Let A0 and A1 be categories. We denote by AA01 the category of functors

from A0 to A1.

II.2.2.4. For the remainder of §II.2, let A be a category.

II.2.2.5. A presheaf on A is a functor

Aop Set.

II.2.2.6. The category of presheaves on A is the functor category SetAop

. Recall thatan arrow of SetA

opis a natural transformation.

II.2.2.7. There is a functor

A SetAop

defined by a 7→ HomA(−, a) and f 7→ HomA(−, f) for an object a of A and an arrow

a0 a1f

of A. By the Yoneda lemma, this functor is fully faithful. We say that a presheaf on A isrepresentable if it belongs to the essential image of this functor, namely if it is isomorphicin SetA

opto a presheaf HomA(−, a) for some object a of A.

II.2.2.8. As a consequence of the fact that Set has all limits and colimits, so doesSetA

op. These limits and colimits are computed levelwise7. Recall that this has the

following meaning — we will treat the case of colimits, the case of limits is entirelyanalogous. Suppose that we have a category D and a functor

D SetAop.

F

Given an object d of D, let Fd denote the presheaf F (d). Given an arrow

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d0 d1f

of D, let

Fd0 Fd1ff

denote the natural transformation

F (d0) F (d1).F (f)

For any object a of A, let

D SetFa

denote the functor defined by d 7→ Fd(a) and f 7→ Ff (a) for an object d of D and anarrow f of D. Then the colimit of F is the presheaf on A defined by a 7→ colim Fa,where colim Fa denotes the colimit of Fa.

As an alternative point of view on the functor Fa, let 1Cat denote the category withexactly one object and one (identity) arrow, and let

1Cat Aop1a

denote the functor sending the unique object of 1Cat to a. Then the following diagramin the category of categories commutes.

D SetAop

Set

F

Set1aFa

II.2.3. Presheaves as glueings

II.2.3.1. The following theorem expresses a universal property of the category SetAop

.As we mentioned earlier, the passage from A to SetA

opfreely adds in all colimits to A 8.

II.2.3.2. Theorem Let C be a category with all (small) colimits, and suppose that wehave a functor

A C.F

Then there is a unique functor

8

SetAop C

F ′

such that the following diagram in the category of categories commutes.

A SetAop

C

F ′F

II.2.3.3. The main step in the proof of Theorem II.2.3.2 is the following theorem.

II.2.3.4. Theorem Let X be a presheaf onA. Then there is a category D and a functor

D SetAopF

such that X can be equipped with the structure of a co-cone of F , and such that thisco-cone defines a colimit of F .

II.2.3.5. Proof Let

A SetAop

j

denote the functor of II.2.2.7. Given an object a of A, let Ya denote the presheafHomA(−, a). Let D denote the category 9 defined as follows.

(i) The objects of D are pairs of an object a of A and an arrow

Ya X

in SetAop

.

(ii) An arrow of D from (a0, f0) to (a1, f1), where

Ya0 Xf0

and

Ya1 Xf1

are arrows of SetAop

, is an arrow

9

a0 a1g

of A such that the following diagram in SetAop

commutes.

Ya0 Ya1

X

HomA(−, g)

f1f0

(iii) Composition of arrows of D and identity arrows of D are as in A.

The following defines a functor 10

D A.G

(i) To an (a, f) of D is associated the object a of A.

(ii) To an arrow

a0 a1g

of D is associated g itself, viewed as an arrow of A.

Let F denote the following composite functor.

D A

SetAop

G

jF

We claim that X defines a colimit of F . To be precise, the arrows

Ya Xg

of SetAop

for an object a of A assemble into a co-cone for the functor F , by definition ofthe arrows of D. We claim that X equipped with this co-cone defines a colimit of F .

To prove the claim, suppose that we have an object X ′ of SetAop

and arrows

Ya X ′Γg

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of SetAop

for every object a of A and every arrow

Ya Xg

of SetAop

, which together define a co-cone of F . We construct a morphism of presheaves

X X ′Γcan

as follows. Let a be an object of A, and let x belong to X(a). By the Yoneda lemma, xcorresponds uniquely to a morphism of presheaves

Ya X.gx

We define Γcan(x) to be the element of X ′(a) corresponding to Γgx under the Yonedalemma. That Γcan defines a morphism of presheaves can be seen as follows. Let

a0 a1f

be an arrow of A, let x belong to X(a0), and let x′ belong to X(a1). The Yoneda lemmagives us that the following diagram in SetA

opcommutes.

Ya0 Ya1

X

Hom(−, f)

gx′gx

By definition of a co-cone, we thus have that the following diagram in A commutes.

Ya0 Ya1

X ′

HomA(−, f)

Γgx′Γgx

By the Yoneda lemma once more, we deduce that f(Γcan(x)

)= Γcan(x′), as required.

Finally, let us verify that for every object

Ya Xg

of D, the following diagram in SetAop

commutes.

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Ya X

X ′

g

ΓcanΓg

Let x denote the element of X(a) corresponding to g under the Yoneda lemma, andlet x′ denote the element of X ′(a) corresponding to Γg under the Yoneda lemma. TheYoneda lemma gives us that the commutativity of the above triangle is equivalent to theassertion that Γcan(x) = x′. This holds by definition of Γcan.

II.2.3.6. We usually express Theorem II.2.3.4 as: a presheaf on A is a colimit ofrepresentable presheaves.

II.3. Semi-cubical sets

II.3.1. Why the topological n-cubes?

Let us consider the following question: from an abstract point of view, what are theproperties of the topological n-cubes I0, I1, I2, . . . that we draw upon in classicalhomotopy theory? Firstly, we have that In ' In−1 × I, for any n ≥ 1. This leads usto a simpler question: what structure does the topological interval I have that we drawupon in classical homotopy theory?

II.3.2. Interval in a category

Most fundamentally, we can observe that we rely upon the fact that I can be thoughtof as having two endpoints, 0 and 1. Reformulating this, we rely on the existence of apair of maps

I0 I,

i0

i1

where i0 sends the unique point of I0 to 0, and i1 sends the unique point of I0 to 1.Thus, as a point of depature, we might define an interval in a category A to be the

data(I0, I1, i0, i1

)of a pair of objects I0 and I1 of A together with a pair of arrows

I0 I1

i0

i1

of A.

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II.3.3. Homotopy with respect to an interval

Suppose that A is equipped with the structure (⊗, 1, λ, ρ, α)

of a monoidal category,where 1 is a unit object, λ is a natural isomorphism

−⊗ 1 −,

ρ is a natural isomorphism

1⊗− −,

and α is a natural isomorphism

−⊗ (−⊗−) (−⊗−)⊗−.

For example, if A has finite products, we may regard A as equipped with its cartesianmonoidal structure. Given an interval I =

(1, I1, i0, i1

)in A, we can define a notion of

homotopy between arrows of A with respect to I. Indeed, suppose that

a0 a1

f0

f1

are arrows of A. Then a homotopy from f0 to f1 with respect to I to be an arrow

a0 ⊗ I a1h

of A such that the following diagrams in A commute.

a0 a0 ⊗ I a0 a0 ⊗ I

a1 a1

a0 ⊗ i0

hf0

a0 ⊗ i1

hf1

We (harmlessly!) implicitly identify a0 with a0⊗1 via λ here. In the case of the categoryof spaces equipped with its cartesian monoidal structure, we recover the classical notionof homotopy.

II.3.4. Towards the category of semi-cubes

In this way, we find that from a conceptual point of view the structure(I0, I, i0, i1

)with which the topological interval I is equipped, together with the property that In 'In−1×I1 for all n ≥ 1, already furnishes us with the fundamental ingredients of classicalhomotopy theory: maps

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In X

up to homotopy, for any space X.

II.3.5. An outline of the category of semi-cubes

Thus we are led to the following idea, as our first guess at an appropriate category ofbuilding blocks for combinatorial homotopy theory: we would like to define the categoryof semi-cubes as the free strict monoidal category on an interval. Let us work towardsconstructing such a category.

We begin with the formal data of an interval, namely a directed graph with exactlytwo objects 1 and I, and two arrows

1 I,

i0

i1

and build a strict monoidal category as follows, roughly speaking. We first add in anobject I ⊗ I and four arrows

I I ⊗ I,I ⊗ i0

I ⊗ i1

and

I I ⊗ I.i0 ⊗ I

i1 ⊗ I

Next we add in an object I ⊗ I ⊗ I, and six arrows

I ⊗ I I ⊗ I ⊗ I,I ⊗ I ⊗ i0

I ⊗ I ⊗ i1

I ⊗ I I ⊗ I ⊗ I,I ⊗ i0 ⊗ I

I ⊗ i1 ⊗ I

and

I ⊗ I I ⊗ I ⊗ I.i0 ⊗ I ⊗ I

i1 ⊗ I ⊗ I

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We continue in this manner, ending up with an object I ⊗ . . .⊗ I︸︷︷︸n

for every n ≥ 0, which

we will denote by In for short, and arrows

In−1 InIi−1 ⊗ i0 ⊗ In−i

Ii−1 ⊗ i1 ⊗ In−i

for every 1 ≤ i ≤ n. It remains only to take care of composition and identities.

II.3.6. A topological picture

Before we do so, let us develop a little more feeling for our proposed category of cubes.Intuitively, we can think of In as corresponding to the topological n-cube.

I0 I1 I2 I3

We can think of the arrows


Ii−1 ⊗ i1 ⊗ In−i

for some 1 ≤ i ≤ n as a pair of faces of the topological n-cube, as illustrated below.

I ⊗ i0

I ⊗ i1

i0 ⊗ I i1 ⊗ I

We make an arbitrary choice to make when depicting the faces in this way. In the pictureabove, we could equally regard i0⊗I and i1⊗I as the horizontal faces, and regard I⊗ i0and I ⊗ i1 as the vertical faces.

II.3.7. A need to impose relations

Our first idea to take care of compositions and identities in our proposed category ofcubes might simply be to take the free category on the directed graph with the objectsand arrows we arrived at by the end of II.3.5. We might then try to equip this categorywith a strict monoidal structure in the obvious way, defining Im ⊗ In to be Im+n, andsimilarly on arrows.

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However, this will not quite work yet. In order to obtain a strict monoidal category,we must have that the following diagram commutes.

I0 I1

I1 I2

i0

i1 ⊗ Ii1

I ⊗ i0

Topologically, this corresponds to the fact that we have the following two ways of arrivingat the top right point of I2.

i1

i0I ⊗ i0

i1 ⊗ I

We must also have that three more diagrams commute, corresponding to the three othervertices of I2. Moreover we must have that further diagrams, corresponding in topologyto higher dimensional analogues of the above picture. Finally, we are led to the followingdefinition.

II.3.8. The category of semi-cubes

II.3.8.1. Definition Let Υ denote the directed graph defined as follows.

(i) We have an object for every n ≥ 0, which we denote by In.

(ii) For every 1 ≤ i ≤ n and 0 ≤ ε ≤ 1, we have an arrow

In−1 In.fni,ε

Let F(Υ) denote the free category upon Υ. The category of semi-cubes, which we willdenote by �s, is the quotient of F(Υ) by the relation ∼ defined inductively as follows.

(i) For any 1 ≤ i ≤ n and 1 ≤ j ≤ n+ 1, and any 0 ≤ δ, ε ≤ 1, we have that

fn+1j,ε ◦ f

ni,δ ∼

{fn+1i,δ ◦ f

nj−1,ε if j > i,

fn+1i+1,δ ◦ f

nj,ε if j ≤ i.

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(ii) Suppose that we have arrows

Iq Ir

g0

g′0

and

Ir Is

g1

g′1

of F(Υ) with q, r, s ≥ 0, such that g0 ∼ g1 and g′0 ∼ g′1. Then we have thatg1 ◦ g0 ∼ g′1 ◦ g′0.

II.3.8.2. Here we think of the arrow

In−1 Infni,0

as shorthand for our earlier arrow

In−1 In.Ii−1 ⊗ i0 ⊗ In−i

Similarly, we think of the arrow

In−1 Infni,1

as shorthand for our earlier arrow

In−1 In.Ii−1 ⊗ i1 ⊗ In−i

II.3.9. A universal property of the category of semi-cubes

We now work towards rigorously equipping �s with a strict monoidal structure, andproving that �s with this monoidal structure is the free strict monoidal category uponan interval, in an appropriate sense.

II.3.9.1. To help us on our way, let us recall an observation or two about a productof categories. Given categories A0 and A1, let A0 × A1 denote the category defined asfollows.

(i) The objects are pairs (a, a′) of an object a of A0 and an object a′ of A1.

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(ii) An arrow from (a0, a′0) to (a1, a

′1) is a pair (f, f ′) of an arrow

a0 a1f

of A0 and an arrow

a′0 a′1f ′

of A1.

(iii) The composition (g, g′) ◦ (f, f ′) of an arrow

(a0, a′0) (a1, a

′1)

(f, f ′)

and an arrow

(a1, a′1) (a2, a

′2)

(g, g′)

is the arrow

(a0, a′0) (a2, a

′2).

(f ′ ◦ f, g′ ◦ g)

(iv) The identity arrow from (a, a′) to itself is(id(a), id(a′)

).

To define a functor F from A0 ×A1 to a category A2, it suffices to specify F (a, a′) forany object a of A0 and any object a′ of A1, and to specify F (f, id) and F (id, f ′) for anyarrow

a0 a1f

of A0 and any arrow

a′0 a′1f ′

of A1, in such a way that the following diagram in A2 commutes.

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F (a0, a′0) F (a1, a

′0)

F (a0, a′1) F (a′0, a

′1)

F(f, id(a′0)

)F(id(a1), f

′)F(id(a0), f

′)F(f, id(a′1)

)Indeed we can then define F (f, f ′) to be the arrow of A2 obtained by taking either routethrough this diagram, we can define F (gf, id) to be F (g, id) ◦ F (f, id), and can defineF (id, g′f ′) to be F (id, g′) ◦ F (id, f ′). This observation will be useful for us below.

Before we get to this, let us observe that we have a functor

A0 ×A1 A0

pr0

given by (a, a′) 7→ a and (f, g) 7→ f . Likewise we have a functor

A0 ×A1 A1

pr1

given by (a, a′) 7→ a′ and (f, f ′) 7→ f ′. Suppose that we have a commutative diagram asfollows in the category of categories.

A0 ×A1 A0

A1 A2

pr0

F0pr1

F1

Then for any object (a, a′) of A0 ×A1 we have that F0(a) = F1(a′). Let us denote this

object of A2 by rF0,F1(a, a′). Moreover for any arrow f of A0 and any arrow f ′ of A1,we have that F0(f) = F1(f

′). Let us denote this arrow of A2 by rF0,F1(f, f ′). It is clearthat this recipe defines a functor

A0 ×A1 A2

rF0,F1

and that rF0,F1 is moreover the unique functor such that the following diagram in Acommutes.

A0 ×A1 A0

A1 A2

pr0

F0pr1

F1

rF0,F1

Thus A0×A1 together with the functors pr0 and pr1 defines a product of A0 and A1 inthe category of categories.

19

II.3.9.2. Definition Let

�s ×�s �s.−⊗−

denote the functor defined inductively by the following recipe, by virtue of our observa-tions in II.3.9.1.

(i) For m,n ≥ 0, we define Im ⊗ In to be Im+n.

(ii) For 1 ≤ i ≤ m and n ≥ 0, and 0 ≤ ε ≤ 1, we define fmi,ε ⊗ In to be fm+ni,ε .

(iii) For 1 ≤ i ≤ n and m ≥ 0, and 0 ≤ ε ≤ 1, we define Im ⊗ fni,ε to be fm+nm+i,ε.

(iv) For m,m′,m′′, n ≥ 0, and arrows

Im Im′

g0

and

Im′

Im′′

g1

of �s, we define (g1 ◦ g0)⊗ In to be (g1 ⊗ In) ◦ (g0 ⊗ In).

(v) For m,n, n′, n′′ ≥ 0, and arrows

In In′

g0

and

In′

In′′

g1

of �s, we define Im ⊗ (g1 ◦ g0) to be (Im ⊗ g1) ◦ (Im ⊗ g0).

II.3.9.3. Note that by definition of the relation ∼ in the definition of �s, the followingdiagram in A commutes for any 1 ≤ i ≤ m and 1 ≤ j ≤ n, and any 0 ≤ δ, ε ≤ 1.

Im+n−1 Im+n

Im+n Im+n+1

fm+ni,δ

fm+n+1m+j+1,εfm+n

m+j,ε

fm+n+1i,δ

20

In other words, we have that the following diagram in �s commutes.

Im+n−1 Im+n

Im+n Im+n+1

fmi,ε ⊗ In

Im ⊗ fnj,δIm ⊗ fnj,δ

fmi,ε ⊗ In

The definition of ∼ also gives us that the following diagram in �s commutes for any1 ≤ i ≤ m and 1 ≤ j ≤ n.

Im+n−1 Im+n

Im+n Im+n+1

fm+nm+j,ε

fm+n+1i,δfm+n

i,δ

fm+n+1m+j+1,ε

In other words, we have that the following diagram in �s commutes.

Im+n−1 Im+n

Im+n Im+n+1

Im ⊗ fnj,ε

fmi,δ ⊗ Infmi,ε ⊗ In

Im ⊗ fnj,δ

Thus the recipe of II.3.9.2 does allow us to cook up a functor

�s ×�s �s−⊗−

as in II.3.9.1, and it is the relation ∼, whose definition was motivated by the intuitiveconsiderations of II.3.7, which exactly ensures this.

II.3.9.4. Proposition The functor

�s ×�s �s−⊗−

equips the category �s with the structure of a strict monoidal category with unit I0.

II.3.9.5. Proof Immediate verification.

21

II.3.9.6. It is now a straightforward matter to put on firm ground our intuitive con-ception of �s as the free strict monoidal category upon an interval. Let us get down toit straight away!

II.3.9.7. Let (�s)≤1 denote the free category on the following directed graph.

0 1

ι0

ι1

We think of (�s)≤1 as the free-standing interval. If

(I0, I1, i0, i1

)defines an interval in

a category A, then we obtain a functor

(�s)≤1 A

given by 0 7→ I0, 1 7→ I1, ι0 7→ i0, and ι1 7→ i1. In particular, we obtain in this way afunctor

(�s)≤1 �s.

II.3.9.8. Proposition Let A be a category equipped with a strict monoidal structure(⊗, 1). For any functor

(�s)≤1 A,int

there is a unique strict monoidal functor

�s Acan


(�s)≤1 �s

A

canint

II.3.9.9. Proof The unique possible recipe for a strictly monoidal functor

�s Acan

fitting into the above commutative diagram is the following.

22

(i) We define can(I0) to be int(0), and for n ≥ 1 we define can(In) to be

int(1)⊗ · · · ⊗ int(1)︸︷︷︸n

.

(ii) For 1 ≤ i ≤ n, we define can(fni,0) to be the arrow

can(In−1) can(In)can(Ii−1)⊗ i0 ⊗ can(In−i)

of A.

(iii) For 1 ≤ i ≤ n, we define can(fni,1) to be the arrow


of A.

(iv) For q, r, s ≥ 0, and arrows

Iq Irg0

and

Ir Isg1

of �, we define can(g1 ◦ g0) to be can(g1) ◦ can(g0).

It remains to verify that this recipe indeed defines a functor from �s to A. Evidentlyit defines a functor from F(Υ) to A. In addition, for any 1 ≤ i < j ≤ n + 1, and0 ≤ δ, ε ≤ 1, it follows immediately from the fact that (⊗, 1) defines a strict monoidalstructure upon A that the following diagram in A commutes.

can(In−1) can(In)

can(In) can(In+1)

can(fni,δ)

can(fn+1j,ε )can(fnj−1,ε)

can(fn+1i,δ )

Moreover, for any 1 ≤ j ≤ i ≤ n, and 0 ≤ δ, ε ≤ 1, it again follows immediately from thefact that (⊗, 1) defines a strict monoidal structure upon A that the following diagramin A commutes.

23

can(In−1) can(In)

can(In) can(In+1)

can(fni,δ)

can(fn+1j,ε )can(fnj,ε)

can(fn+1i,δ )

This completes our proof.

II.3.10. Semi-cubical sets

II.3.10.1. Let us summarise our story so far. We have seen that presheaves preciselyexpress the idea of glueing from a category of building blocks. By considering abstractproperties of the topological n-cubes which we rely upon in classical homotopy theory,we have arrived at a first guess for a candidate category of building blocks for an abstracttheory of homotopy — our category �s of semi-cubes.

II.3.10.2. Definition A semi-cubical set is a presheaf on �s.

II.3.10.3. Can we construct a homotopy theory of semi-cubical sets? If so, how closelydoes it resemble the classical homotopy theory of topological spaces?

II.4. Cubical sets

II.4.1. Further structure of the topological interval?

To develop a feeling for these questions, we may pose another — from an abstract pointof view, what further structure of the topological interval do we rely upon in developingthe deeper homotopy theory of spaces? So far, we have seen that we can understandthe notion of homotopy itself in a purely conceptual way. But homotopy theory is muchmore than the definition of homotopy!

II.4.2. Constant homotopies

Let us consider the question of constructing a homotopy from a map

X Yf

between spaces to itself, in other words a constant homotopy. We cannot even do thisin the presence of the interval

(I0, I1, i0, i1

)alone! Indeed, let

I I0p

24

denote the canonical map. Then we have that the following diagrams in the category ofspaces commute.

I0 I1 I0 I1

I0 I0

i0

pid

i1

pid

Thus we see that the composite map

X × I X

Y

X × p

f

defines a homotopy from f to itself.

II.4.3. Contraction structure upon an interval

Let A be a category equipped with a monoidal structure (⊗, 1, λ, ρ, α) as in II.3.3. LetI =

(1, I, i0, i1

)be an interval inA. Motivated by II.4.2, we define a contraction structure

upon I is an arrow

I 1p

of A such that the following diagrams in A commute.

I0 I1 I0 I1

I0 I0

i0

pid

i1

pid

As in II.4.2, a contraction structure upon I allows us to construct constant homotopiesbetween arrows of A, with homotopy with respect to I being the notion we discussed inII.3.3.

II.4.4. Towards the category of cubes

We can now carry out a construction of a category of cubes entirely in analogy with ourconstruction of the category �s of semi-cubes. Rather than beginning with an intervalalone, we begin with an interval equipped with a contraction structure, and pass to

25

the free strict monoidal category upon this initial data. This will introduce a need forfurther relations.

We will shortly jump straight into the details. Before we do, we might note that theability to reverse and to compose homotopies is just as fundamental as the constructionof constant homotopies in the classical homotopy theory of spaces. Can we capture thisby means of extra structure upon an interval? Should we not build this structure intoour category of cubes, just as we are proposing to build in a contraction structure?

The answer to the first of these questions is — yes! The interested reader may consult[11] or the book [3] of Kamps and Porter11. The answer to the second question couldbe a lecture or more in itself! We may certainly build an involution structure into ourcategory of cubes, which allows us to reverse homotopies. I hope to have time to give thedetails in a future update to these notes. However, we cannot naively build a subdivisionstructure into our category of cubes which would allow us to compose homotopies — ormore precisely we could, but it would be destroyed upon passing to presheaves. However,we can build in a subdivision structure in a more sophisticated way — this leads towardshigher category theory. A paper [10] on this is in preparation.

It turns out, though, that building a contraction structure into our category of cubes issufficient to construct a homotopy theory of cubical sets which we will eventually see willbe equivalent, in an appropriate sense, to that of spaces. That this is possible revolvesaround the notion of a Kan complex, which we will introduce in the next lecture.

In fact, it is not even necessary to build in a contraction structure! In the setting ofthe simplicial analogues of semi-cubical sets, Rourke and Sanderson showed in [8] thatwe can still build a homotopy theory which is equivalent to that of spaces. However,building in a contraction structure is very useful — we will make use of it very frequently— giving rise to a richer theory, in which we have parallels of more aspects of classicalhomotopy theory.

II.4.5. The category of cubes

II.4.5.1. Definition Let Υ′ denote the directed graph defined as follows.


(ii) For every 1 ≤ i ≤ n and 0 ≤ ε ≤ 1, we have an arrow

In−1 In.fni,ε

(iii) For every 1 ≤ i ≤ n we have an arrow

In In−1.dni

26

Let F(Υ′) denote the free category upon Υ′. The category of cubes, which we will denoteby �, is the quotient of F(Υ′) by the relation ∼ defined inductively as follows.


fn+1j,ε ◦ f

ni,δ ∼

{fn+1i,δ ◦ f

nj−1,ε if j > i,

fn+1i+1,δ ◦ f

nj,ε if j ≤ i.

(ii) For any n ≥ 2, 1 ≤ i ≤ n, and 1 ≤ j ≤ n− 1, we have that

dn−1j ◦ dni ∼

{dn−1i−1 ◦ dnj if j < i,

dn−1i ◦ dnj+1 if j ≥ i.

(iii) For any n ≥ 2, 1 ≤ i, j ≤ n, and 0 ≤ ε ≤ 1, we have that

dnj ◦ fni,ε ∼

id if j = i,

fn−1i,ε ◦ dn−1j−1 if j > i,

fn−1i−1,ε ◦ dn−1j if j < i.

(iv) Suppose that we have arrows

Iq Ir

g0

g′0

and

Ir Is

g1

g′1

of F(Υ′) with q, r, s ≥ 0, such that g0 ∼ g1 and g′0 ∼ g′1. Then we have thatg1 ◦ g0 ∼ g′1 ◦ g′0.

II.4.5.2. Here we think of dni as the arrow

In In−1Ii−1 ⊗ p⊗ In−i

in our outline of the free strict monoidal category upon an interval with contraction,analogous to that of the free strict monoidal category upon an interval in II.3.5. Thismotivates the relations of (ii) as higher dimensional analogues of the defining axioms ofan interval with contraction in II.4.3. The relations of (iii) arise analogously to those of(i) — see II.3.7.

27

II.4.6. A topological picture

As in II.3.6, we think of In as the topological n-cube, and as fni,ε for 0 ≤ ε ≤ 1 as faceinclusions. We then think of dni as the projection of In onto its face fni,1 — the choiceof fni,0 or fni,1 doesn’t matter, but it is important that we make the same choice for all

i and n. For example, suppose that f21,0 and f21,1 are the inclusions onto the faces of I2

depicted in colour or by an arrow below.

Then d21 can be taken to be the following projection map.

The reader may wish to justify to themself that the relations in the definition of � reallydo hold in this topological picture, as in II.3.7.

II.4.7. A universal property of the category of cubes

II.4.7.1. Definition Let

�×� �.−⊗−





(iv) For 1 ≤ i ≤ m and n ≥ 0, we define dmi ⊗ In to be dm+ni .

(v) For 1 ≤ i ≤ n and m ≥ 0, we define Im ⊗ dni to be dm+nm+i .

(vi) For m,m′,m′′, n ≥ 0, and arrows

Im Im′

g0

and

Im′

Im′′

g1

of �, we define (g1 ◦ g0)⊗ In to be (g1 ⊗ In) ◦ (g0 ⊗ In).

28

(vii) For m,n, n′, n′′ ≥ 0, and arrows

In In′

g0

and

In′

In′′

g1


II.4.7.2. Note that by definition of the relation ∼ in the definition of �, the followingdiagram in A commutes for any 1 ≤ i ≤ m and 1 ≤ j ≤ n.

Im+n Im+n−1

Im+n−1 Im+n−2

dm+ni

dm+n−1m−1+jdm+n

m+j

dm+n−1i

In other words, we have that the following diagram in � commutes.

Im+n Im+n−1

Im+n−1 Im+n−2

dmi ⊗ In

Im−1 ⊗ dnjIm ⊗ dnj

dmi ⊗ In−1

The definition of ∼ also gives us that the following diagram in � commutes, for any1 ≤ i ≤ m and 1 ≤ j ≤ n.

Im+n Im+n−1

Im+n−1 Im+n−2

dm+nm+j

dm+n−1idm+n

i

dm+n−1m+j−1


29

Im+n Im+n−1

Im+n−1 Im+n−2

Im ⊗ dnj

dmi ⊗ In−1dmi ⊗ In

Im−1 ⊗ dnj

In addition, the definition of ∼ gives us that the following diagram in � commutes, forany 1 ≤ i ≤ m and 1 ≤ j ≤ n, and any 0 ≤ ε ≤ 1.

Im+n−1 Im+n

Im+n−2 Im+n−1

fm+ni,ε

dm+nm+jdm+n−1

m+j−1

fm+n−1i


Im+n−1 Im+n

Im+n−2 Im+n−1

fmi,ε ⊗ In

Im ⊗ dnjIm−1 ⊗ dnj

fmi,ε ⊗ In−1

Moreover, the definition of ∼ gives us that the following diagram in � commutes, forany 1 ≤ i ≤ m and 1 ≤ j ≤ n, and any 0 ≤ δ ≤ 1.

Im+n Im+n−1

Im+n+1 Im+n

dm+ni

fm+n−1m+j−1,δfm+n

m+j,δ

dm+n+1i


Im+n Im+n−1

Im+n+1 Im+n

dmi ⊗ In

Im−1 ⊗ fnj,δIm ⊗ fnj,δ

dmi ⊗ In+1

30

Together with our observations of II.3.9.3, we thus have that the recipe of II.4.7.1 doesallow us to cook up a functor

�×� �−⊗−

by virtue of II.3.9.1. As in II.3.9.3, it is the relation ∼ which exactly ensures this.

II.4.7.3. Proposition The functor

�×� �−⊗−

equips the category � with the structure of a strict monoidal category with unit I0.

II.4.7.4. Proof Immediate verification.

II.4.7.5. Let �≤1 denote the free category on the directed graph of two objects 0 and1, a pair of arrows

0 1,

ι0

ι1

and an arrow

1 0.π

We think of �≤1 as the free-standing interval equipped with a contraction structure,noting that the commutative diagrams in �≤1 other than those involving compositionwith an identity arrow are exactly the following.

0 1 0 1

0 0

ι0

πid

ι1

πid

If(I0, I1, i0, i1, p

)defines an interval equipped with a contraction structure p in a cate-

gory A, then we obtain a functor

�≤1 A

given by 0 7→ I0, 1 7→ I1, ι0 7→ i0, ι1 7→ i1, and π 7→ p. In particular, we obtain in thisway a functor

�≤1 �.

31

II.4.7.6. Proposition Let A be a category equipped with a strict monoidal structure(⊗, 1). For any functor

�≤1 A,int


� Acan


�≤1 �

A

canint

II.4.7.7. Proof The unique possible recipe for a strictly monoidal functor

� Acan

fitting into the above commutative diagram is the following.

(i) We define can(I0) to be int(0), and for n ≥ 1 we define can(In) to be

int(1)⊗ · · · ⊗ int(1)︸︷︷︸n

.

(ii) For 1 ≤ i ≤ n, we define can(fni,0) to be the arrow


of A.

(iii) For 1 ≤ i ≤ n, we define can(fni,1) to be the arrow


of A.

32

(iv) For 1 ≤ i ≤ n, we define can(dni ) to be the arrow

can(In−1) can(In)can(Ii−1)⊗ dni ⊗ can(In−i)

of A.

(v) For q, r, s ≥ 0, and arrows

Iq Irg0

and

Ir Isg1

of �s, we define can(g1 ◦ g0) to be can(g1) ◦ can(g0).

It remains to verify that this recipe indeed defines a functor from � to A. Evidently,it defines a functor from F(Υ′) to A. We now observe that, as in the proof of Propo-sition II.3.9.8, it follows immediately from the fact that (⊗, 1) defines a strict monoidalstructure upon A that the defining relations of � give rise under can(−) to commutativediagrams in A.

II.4.8. Cubical sets

II.4.8.1. Definition A cubical set is a presheaf on �.

II.4.8.2. We may now ask the same questions as in II.3.10.3. Can we construct a ho-motopy theory of cubical sets? If so, how closely does it resemble the classical homotopytheory of topological spaces? This will be the topic of the remainder of the course.

Notes

1 For example, the first Cech cohomology group of the Warsaw circle is Z, the integers.

2 There is a notion of homotopy equivalence with respect to which the Warsaw circle is homotopyequivalent to the circle, as we might intuitively expect: this is an aspect of shape theory. Very closelyrelated ideas are involved in the theory of etale homotopy types of topoi, the original examples beingthe petit etale topoi arising in algebraic geometry. A reference for shape theory, whose presentationof the Warsaw circle I have borrowed, is the book [5] of Mardesic and Segal. A reference for etalehomotopy theory is the book [1] of Artin and Mazur.

3 The space S1fin is sometimes referred to as the pseudocircle.

4 This recipe actually gives rise to one half of an equivalence of categories between finite posets andfinite spaces, and more generally between arbitrary posets and Alexandroff spaces.

5 Explicitly, thinking of S1 as a subspace of R2 in the usual way, we can take the map

33

S1 S1fin

to be given, as suggested by the picture, by

(x, y) 7→

b if x < 0,

c if x > 0,

a if (x, y) = (0, 1),

d if (x, y) = (0,−1).

6 The homotopy theory of finite spaces is very interesting — every space is weakly homotopy equivalentto a finite space, which is a priori truly remarkable! For more on the use of the Sierpinski intervalin the homotopy theory of finite spaces, see for example §4 of the paper [7] of Raptis.

7 The terminology pointwise is sometimes used, rather than levelwise.

8 In the process of freely adding colimits to our category, we may destroy any that already existed.If we wish to prescribe that a colimit of a diagram in A remains a colimit of the correspondingdiagram in SetA

op

under the Yoneda embedding, we must instead take sheaves on A with respectto an appropriate Grothendieck topology. This is a significant reason for the use of the Nisnevichtopology in Voevodsky’s approach to motivic homotopy theory, as discussed for example in the notes[9] of Weibel to a series of lectures of Voevodsky in Seattle.

9 This category is an example of a comma category.

10 If we view D as a comma category, this functor can be constructed canonically.

11 One can go far with a small rainbow of structures — in [11] it is proven that, assuming a certainstrictness hypothesis, one can cook up a model structure from an interval equipped with contraction,involution, subdivision, and connection structures. In addition to [11] and the book [3] of Kampsand Porter, the reader can find the definitions of these structures in §2 of the paper [2] of Grandis.

34

III. Kan complexes and homotopy groups

III.1. An attempt to define homotopy groups of a cubical set

III.1.1. Getting to know cubical sets

III.1.1.1. Scholium Putting together Definition II.4.5.1 and Definition II.4.8.1, we havethat a cubical set is the data of a set Xn for every n ≥ 0, a map

Xn Xn−1f i,εn

for every 1 ≤ i ≤ n and every 0 ≤ ε ≤ 1, and a map

Xn−1 Xn

din

for every 1 ≤ i ≤ n, such that the following hold.

(i) For any n ≥ 2, 1 ≤ i ≤ n, 1 ≤ j ≤ n− 1, and 0 ≤ δ, ε ≤ 1, we have that

f j,εn−1 ◦ fi,δn =

{f i−1,δn−1 ◦ f

j,εn if i > j,

f i,εn−1 ◦ fj+1,εn if i ≤ j.

(ii) For any 1 ≤ i ≤ n and 1 ≤ j ≤ n+ 1, we have that

djn+1 ◦ din =

{din+1 ◦ d

j−1n if i < j,

di+1n+1 ◦ d

j,εn if i ≥ j.

(iii) For any n ≥ 2, 1 ≤ i, j ≤ n, and 0 ≤ δ ≤, we have that

f j,δn ◦ din =

id if i = j,

dn−1i−1 ◦ fn−1j,δ if i > j,

dn−1i ◦ fn−1j−1,δ if i < j.

III.1.1.2. Definition Let X be a cubical set. We refer to an element x of Xn as ann-cube of X. We refer to

Xn Xn−1f i,εn

for any 1 ≤ i ≤ n and 0 ≤ ε ≤ 1 as a face map. Given an n-cube x of X, we refer tothe (n− 1)-cube f i,εn (x) of X as the (i, ε)th-face of x, or less precisely as a face of x. Werefer to

Xn−1 Xn

din

for any 1 ≤ i ≤ n as a degeneracy map. Given an (n− 1)-cube x of X, we refer to dni (x)as a degenerate n-cube of X. If an n-cube of X is not obtainable as din(x) for some(n− 1)-cube x of X and some 1 ≤ i ≤ n, we refer to it as non-degenerate.

35

III.1.1.3. To elaborate upon Scholium III.1.1.1, given a cubical set X, the set Xn isthe image of the object In of � under X. The face map

Xn Xn−1f i,εn

for some 1 ≤ i ≤ n and 0 ≤ ε ≤ 1 is the image of the arrow

In−1 Infni,ε

of � under X. The degeneracy map

Xn Xn−1dn

for some 1 ≤ i ≤ n is the image of the arrow

In In−1dni

of � under X. If we view � as the free strict monoidal category upon an intervalequipped with a contraction structure — Proposition II.4.7.6 — then the face map f i,0nis the image under X of the arrow


of �, and the face map f i,1n is the image under X of the arrow


of �. The degeneracy map din is the image under X of the arrow

In In−1Ii−1 ⊗ p⊗ In−i

of �.

III.1.1.4. Definition We denote by �n the presheaf on � represented by In. We referto �n as the free-standing n-cube.

III.1.1.5. Explicitly, �0 consists of a single 0-cube, corresponding to the identity ar-row

I0 I0id

36

in �, and a single degenerate n-cube for every n > 0. For example, the degenerate1-cube corresponds to the arrow

I1 I0p

of �. Although there are two arrows

I2 I1

in �, the following diagram in � commutes, so that �0 has a single degenerate 2-cube.

I2 I1

I1 I0

I1 ⊗ p

pp⊗ I1

p

III.1.1.6. In a similar way, �1 can explicitly be described as follows.

(i) There are two 0-cubes, corresponding to the arrows

I0 I1

i0

i1

of �.

(ii) There is a single non-degenerate 1-cube, corresponding to the identity arrow

I1 I1id

of �.

(iii) There are two degenerate 1-cubes, given by the following two composite arrows of�.

I1 I0 I1 I0

I1 I1

p

i0i0 ◦ p

p

i1i1 ◦ p

.

37

(iv) All n-cubes for n > 1 are degenerate. For example, there are exactly four degen-erate 2-cubes. Two of these are given by the arrows

I2 I1

I1 ⊗ p

p⊗ I1

of �. The other two are obtained by lifting each of the two degenerate 1-cubes of(iii) to a degenerate 2-cube — we obtain a single lift rather than two in each casefor exactly the same reason we discussed in III.1.1.5 when considering the 2-cubesof �0.

III.1.1.7. To give one more example, �2 can explicitly be described as follows.

(i) There are four 0-cubes, arrived at by taking either route through the followingcommutative diagrams in �.

I0 I1 I0 I1

I1 I2 I1 I2

i0

I1 ⊗ i0i0

i0 ⊗ I1

i1

I1 ⊗ i1i1

i1 ⊗ I1

I0 I1 I0 I1

I1 I2 I1 I2

i0

I1 ⊗ i1i1

i0 ⊗ I1

i1

I1 ⊗ i0i0

i1 ⊗ I1

(ii) There are four non-degenerate 1-cubes, corresponding to the arrows

I1 I2

I1 ⊗ i0

I1 ⊗ i1

and

I1 I2

i0 ⊗ I1

i1 ⊗ I1

38

of �.

(iii) There are four degenerate 1-cubes, corresponding to the four 0-cubes of �2 in thesame way as the degenerate 1-cubes of �1 corresponded in III.1.1.6 to the 0-cubesof �1.

(iv) There is a single non-degenerate 2-cube of �2, corresponding to the identity arrow

I2 I2id

of �.

(v) All n-cubes for n > 2 are degenerate.

III.1.1.8. We think of the cubical sets �n for n ≥ 0 as combinatorial analogues of thetopological n-cubes. Ignoring degenerate i-cubes, we may depict �n as follows in lowdimensions, thinking of the square in the picture of �2 as solid, and similarly thinkingof the cube and the squares in the picture of �3 as solid.

0 1

(0,1) (1,1)

(0,0) (1,0)

(0,1,0) (1,1,0)

(0,0,0) (1,0,0)

(0,1,1) (1,1,1)

(0,0,1) (1,0,1)

�0 �1 �2 �3

III.1.1.9. By a morphism from a cubical set X0 to a cubical set X1 we will mean anatural transformation from X0 to X1. By the category of cubical sets we will mean thecategory Set�

op.

III.1.1.10. Let X be a cubical set, and let x be an n-cube of X. Now that we areacquainted with the free-standing n-cubes, we can understand x, its faces, and thedegenerate (n + 1)-cubes it gives rise to, more geometrically. Indeed, by the Yonedalemma, x corresponds uniquely to a morphism

�n X

which we will also denote by x. Appealing to the Yoneda lemma once more, we havethat the following diagram in Set�

opcommutes for every 1 ≤ i ≤ n.

39

�n−1 �n

X

Hom�(−, Ii−1 ⊗ i0 ⊗ In−i)

xf i,0n (x)

Similarly we have that the following diagram in Set�op

commutes for every 1 ≤ i ≤ n.

�n−1 �n

X

Hom�(−, Ii−1 ⊗ i1 ⊗ In−i)

xf i,1n (x)

Finally we have that the following diagram in Set�op

commutes for every 1 ≤ i ≤ n.

�n �n−1

X

Hom�(−, Ii−1 ⊗ p⊗ In−i)

xdin(x)

III.1.1.11. Let X be a cubical set. Taking into account III.1.1.10, we will frequentlydepict a 1-cube x of X as

x0 x1,

if x0 is d01(x), and x1 is d11(x). Similarly, we will frequently depict a 2-cube x of X asfollows.

g0

g1g2

g3

Here g0 is f2,02 (x), g1 is f1,12 (x), g2 is f1,02 (x), and g3 is f2,12 (x). We might also depict thefaces of g1, g2, g3, and g4 as vertices of this square.

40

III.1.2. A moral

We see that cubical sets can be thought of from two points of view — as algebraic gadgetsconsisting of specific data satisfying specific axioms, or as combinatorial encodings ofgeometric gadgets. As we proceed we will typically explore an idea geometrically in lowdimensions, before working out the necessary combinatorics to express our idea in higherdimensions. In short — we shall think geometrically, and prove algebraically!

III.1.3. Towards homotopy groups of a cubical set

III.1.3.1. If we reflect upon the definition of the homotopy group πn(X,x) of a pointedtopological space (X,x) from a cubical point of view, we see that there are three ingre-dients.

(i) Maps from the topological n-cube In to X.

(ii) A condition that the boundary of In be mapped to x under a map from In to X.

(iii) Homotopies between maps from In to X satisfying the requirement of (ii).

III.1.3.2. Let X be a cubical set. In this setting, we have arrived at the analogue of(i) — a morphsm from �n to X¡ or in other words an n-cube of X, namely an elementof Xn. In particular, let us define a point of X to be a 0-cube of X.

III.1.3.3. Let us now turn to an analogue of (ii) in our abstract setting. As we havealready touched upon in III.1.1.5, the relations defining a cubical set — see III.1.1.1 —ensure that

dinn din−1

n−1 · · · d11(x)

defines the same n-cube of X for any 0-cube x of X and any choice of 1 ≤ ij ≤ j for1 ≤ j ≤ n. We will thus simply denote this n-cube by x.

III.1.3.4. Let X be a cubical set, and let ∗ be a point of X, and let x be a morphism

�n X

of cubical sets. Appealing to our observation in III.1.3.3, we can express the idea thatthe boundary of x is mapped to ∗ as the requirement that f i,εn (x) = ∗ for every 1 ≤ i ≤ nand 0 ≤ ε ≤ 1. We will later — see Definition III.2.1.2 —- that we can rigorously definethe boundary of �n as a cubical set ∂�n, which will allow to re-formulate this definitionslightly more geometrically, in III.2.1.4.

41

III.1.3.5. Let X be a cubical set, and let ∗ be a point of X. With our abstractunderstanding of (ii) in hand, let us define

Zn(X, ∗) = {x ∈ Xn | f i,εn (x) = ∗ for all 1 ≤ i ≤ n and 0 ≤ ε ≤ 1}.

When n = 0 we interpret the conditions to be vacuous, so that Z0(X, ∗) is X0. Byanalogy with (iii) above, we would like to define an equivalence relation upon Zn(X, ∗)asserting that two n-cubes

�n X

x0

x1

are homotopic, in an appropriate sense. We would then define πn(X, ∗) to be the quotientof Zn(X, ∗) by this equivalence relation, and hope to equip this quotient with a groupstructure, by further analogy with topology.

III.1.4. Combinatorially homotopic n-cubes in a cubical set

III.1.4.1. The interval in � equipped with its contraction structure gives rise to aninterval equipped with a contraction structure in Set�

op. Moreover, the monoidal struc-

ture upon � gvies rise to a monoidal structure upon Set�op

. Thus we arrive — see II.3.3— at a natural notion of homotopy in Set�

op.

III.1.4.2. We will give the details later in the course. Let (X, ∗) be a pointed cubicalset, and let x0 and x1 be n-cubes of X belonging to Zn(X, ∗) — intuitively, n-cubeswhose boundary is trivial. For now, let us take it on faith that a homotopy from themorphism

In Xx0

of cubical sets to the morphism

In Xx1

of cubical sets is exactly a morphism

In+1 Xh

of cubical sets such that the following diagram in Set�op

commute

In In+1 In In+1

X X

In ⊗ i0

hx0

In ⊗ i1

hx1

42

and moreover the following diagrams in Set�op

commute for every 1 ≤ i ≤ n.

In In+1 In In+1

X X

Ii−1 ⊗ i0 ⊗ In−i

hp

Ii−1 ⊗ i1 ⊗ In−i

h∗

III.1.4.3. Thus we are led to the following definition12 Let (X, ∗) be a pointed cubicalset, and let x0 and x1 be n-cubes of X belonging to Zn(X, ∗), for any n ≥ 0. Then wedefine x0 ∼ x1 if there is an (n+ 1)-cube h of X such that

f i,εn+1(h) =

x0 if i = n+ 1 and ε = 0,

x1 if i = n+ 1 and ε = 1,

∗ if i 6= n.

In this case we write that x0 is homotopic to x1, and that h defines a homotopy from x0to x1.

III.1.4.4. Let X be a cubical set, and let x0 and x1 be 0-cubes of X. Then x0 ∼ x1if there is a 1-cube h of X with the boundary depicted below.

x0 x1h

III.1.4.5. Let X be a cubical set, let ∗ be a 0-cube of X, and let

∗ ∗x0

x1

be 1-cubes of X belonging to Z1(X, ∗). Then x0 ∼ x1 if there is a 2-cube h of X withthe boundary depicted below — the unlabelled 1-cubes are degenerate.

∗ ∗

∗ ∗

x0

x1

Here we think of the upper horizontal face as f2,02 (h), of the lower horizontal face as

f2,12 (h), of the left vertical face as f1,02 (h), and of the right vertical face as f1,12 (h). Thereader may compare with II.3.6 — there is an arbitrary choice involved when drawingthe picture.

43

III.1.4.6. Let X be a cubical set, let ∗ be a 0-cube of X, and let x0 and x1 be 2-cubesof X belonging to Z2(X, ∗). Then x0 ∼ x1 if there is a 3-cube h of X with the followingboundary, in which the left shaded 2-cube is x0, the right shaded 2-cube is x1, and theremaining four 2-cubes are degenerate, as are all the 1-cubes.

Here we think of the left, shaded face as f3,03 (h), of the right, shaded face as f3,13 (h), of

the back face as f1,03 (h), of the front face as f1,13 (h), of the top face as f2,03 (h), and of

the bottom face as f2,13 (h).

III.1.4.7. Let X be a cubical set, and let ∗ be a point of X. In order to fulfill ourplan — see III.1.3.5 — for the definition of πn(X, ∗), we must have that ∼ defines anequivalence relation upon Zn(X, ∗). However, this is not the case for an arbitrary cubicalset!

III.1.4.8. The degeneracy maps ensure that ∼ is reflexive. Indeed, if x is an n-cubeof a cubical set X, then dn+1

n+1(x) defines a homotopy from x to itself. Recall that ourtopological motivation for introducing degeneracies was exactly in order to ensure this— see II.4.2.

III.1.4.9. However, ∼ is not symmetric. For a minimal counter-example, let X denotethe cubical set uniquely defined by the following recipe13.

(i) We have a single 0-cube ∗.

(ii) We have two non-degenerate 1-cubes

∗ ∗,x0

x1

.

(iii) We have a single non-degenerate 2-cube x with the boundary depicted below, inwhich the vertical 1-cubes are degenerate.

∗ ∗

∗ ∗

x0

x1

44

(iv) Every n-cube for n > 2 is degenerate.

Then we have that x0 ∼ x1 by virtue of the 2-cube x, but since there is no 2-cube withthe following boundary

∗ ∗

∗ ∗

x1

x0

we do not have that x1 ∼ x0.

III.1.4.10. Morevoer ∼ is not transitive. For a minimal counter-example, let X denotethe cubical set uniquely defined by the following recipe.

(i) We have a single 0-cube ∗.

(ii) We have exactly three non-degenerate 1-cubes x0, x1, and x2, all with the followingboundary.

∗ ∗

(iii) We have exactly two non-degenerate 2-cubes: one, which we will denote by x, withthe boundary depicted below

∗ ∗

∗ ∗

x0

x1

and one, which we will denote by x′, with the boundary depicted below.

∗ ∗

∗ ∗

x1

x2

(iv) Every n-cube for n > 2 is degenerate.

Then x0 ∼ x1 by virtue of the 2-cube x, and x1 ∼ x2 by virtue of the 2-cube x′, butsince there is no 2-cube with the boundary

45

∗ ∗

∗ ∗

x0

x2

we do not have that x0 ∼ x2.

III.1.4.11. If we were to work not with cubical sets but with presheaves on the freestrict monoidal category on an interval equipped with not only a contraction structurebut also an involution structure allowing us to reverse homotopies — we already pro-posed this as a possibility in II.4.4 — our homotopy relation ∼ would be symmetric.Transitivity is much more involved — we cannot ensure that it holds by equipping �with more structure, as discussed in II.4.4.

III.1.4.12. A profound insight of Kan — which goes back to the paper [4] — is thatalthough ∼ may not define an equivalence relation for arbitrary cubical sets, it willdefine an equivalence relation for cubical sets satisfying a certain extension property,now known as Kan complexes. It is to introducing these gadgets that we now turn.

III.2. Greeting Kan complexes

III.2.1. Horns and boundaries

III.2.1.1. We have become acquainted with the free-standing n-cubes �n. Let us nowdefine a couple more families of cubical sets which will be of vital importance to us.

III.2.1.2. Definition For any n ≥ 1, we denote by ∂�n the cubical set obtained asfollows, and refer to it as boundary of the free-standing n-cube. Let x denote the uniquenon-degenerate n-cube of �n.

(i) We remove x from �n.

(ii) For every m ≥ 1, we remove the (n+m)-cube

dimn+mdim−1

n+m−1 · · · di1n+1(x)

from �n, for every choice of 1 ≤ ij ≤ n+ j for 1 ≤ j ≤ m.

III.2.1.3. If we ignore keeping track of degeneracies, the boundary of �n exactlycorresponds to our topological intuition. Thinking of �1 as

0 1

as in III.1.1.8, we have that ∂�1 is the coproduct of two copies of �0, corresponding tothe 0-cubes 0 and 1 of �1. Thinking of �2 as a solid square

46

(0, 1) (1, 1)

(0, 0) (1, 0)

as in III.1.1.8, ∂�2 can be pictured in the same way, except that we no longer think ofthere being a 2-cube filling in the square. We could obtain ∂�n by glueing together fourcopies of �1.

III.2.1.4. Let X be a cubical set, and let ∗ be a 0-cube of X. We have that an n-cubex of X for any n ≥ 1 belongs to Zn(X, ∗), namely has trivial boundary, if and only ifthe following diagram in Set�

opcommutes.

∂�n �n

�0 X

x

∗

Here the morphism

∂�n �n

is the evident inclusion, and the morphism

∂�n �0

is canonical, observing that �0 is a final object of Set�op

.

III.2.1.5. Definition For any 1 ≤ i ≤ n and 0 ≤ ε ≤ 1, we denote by uni,ε the cubical

set obtained as follows, and refer to it as the (i, ε)th n-horn. Let x denote the uniquenon-degenerate n-cube of �n.

(i) We remove the (n− 1)-cube f i,εn (x) from ∂�n.

(ii) For every m ≥ 0, we remove the (n+m)-cube

dimn+mdim−1

n+m−1 · · · di0n

(f i,εn (x)

)of �n for any choice of 1 ≤ ij ≤ n+ j for 0 ≤ j ≤ m.

III.2.1.6. The 1-horns are both, up to isomorphism, �0. If we think of �1 as

0 1

as usual, we can think of u1,01 as corresponding to 1, and to u1,11 as corresponding to 0.

47

III.2.1.7. There are four 2-horns, one for each face of a square. Let us think of �2 asfollows.

(0, 1) (1, 1)

(0, 0) (1, 0)

Then the four 2-horns are as depicted below.

(0,1) (1,1)

(0,0) (1,0)

(0,1) (1,1)

(0,0) (1,0)

(0,1) (1,1)

(0,0) (1,0)

(0,1) (1,1)

(0,0) (1,0)

u1,02 u1,12

u2,02 u2,12

III.2.1.8. There are six 3-horns, one for each face of a cube. We will frequently needto construct 3-horns in which the five non-degenerate 2-cubes are labelled, and it willnot be possible for us to indicate this clearly by attempting to depict the a 3-horn asa three dimensional gadget. Instead, we will depict 3-horns as ‘nets’ of five 2-cubes,as shown below — as recipes for assembling a cube-with-one-face-missing. The coloursindicate 1-cubes which must be the same.

48

Let us think of the free-standing 3-cube in the following way.

(0,1,0) (1,1,0)

(0,0,0) (1,0,0)

(0,1,1) (1,1,1)

(0,0,1) (1,0,1)

Then we will depict the horn u2,13 in the manner shown below — the bottom face of thecube is that which is missing when we assemble our 3-horn from this net.

(0, 0, 0) (1, 0, 0)

(0, 0, 0) (0, 1, 0) (1, 1, 0) (1, 0, 0)

(0, 0, 1) (0, 1, 1) (1, 1, 1) (1, 0, 1)

(0, 0, 1) (1, 0, 1)

To give another example, we will depict the horn u3,13 as follows — the back face of thecube is that which is missing when we assemble our 3-horn from this net.

(0, 1, 1) (1, 1, 1)

(0, 1, 1) (0, 1, 0) (1, 1, 0) (1, 1, 1)

(0, 0, 1) (0, 0, 0) (1, 0, 0) (1, 0, 1)

(0, 0, 1) (1, 0, 1)

49

III.2.1.9. Definition Let X be a cubical set. A 3-horn in X is a morphism

ui,ε3 X

for some 1 ≤ i ≤ n and 0 ≤ ε ≤ 1.

III.2.1.10. Typically, when depicting a 3-horn in a cubical set X by a net, we willwish to label one or more 2-cubes. We will do this by placing a label at the centre of therelevant squares in the net. When we describe a 3-horn this way, the reader may findit helpful at first to try to draw for themself the 3-horn as a three dimensional gadget,namely a cube-with-one-face missing, and with some of the other five faces labelled.

III.2.1.11. The desperate reader, harkening back to the days of barneskole when math-ematics was more hands-on, might find it helpful to cut out the horn along its boundary,and assemble it into an actual cube-with-one-face-missing!

III.2.1.12. We can the lift the idea of presenting 3-horns by two dimensional nets toallow us to visualise a 4-horn, which is a four dimensional gadget. We will not needthis in the present lecture, but it will be helpful to us later on. For now, let us simplyobserve that a ‘net’ for a 4-horn is an arrangement of seven 3-cubes as below such thatany two 2-cubes meeting orthogonally in the boundary of this gadget are the same. Eachcube is coloured differently in the picture, in an attempt to make it easier to distinguishbetween them!

III.2.2. Kan complexes and the free-standing n-cubes

III.2.2.1. Definition Let X be a cubical set. Then X is a Kan complex if14 for every1 ≤ i ≤ n and 0 ≤ ε ≤ 1, and every morphism

uni,ε Xg

of cubical sets, there is a morphism

50

�n Xg′

of cubical sets such that the following diagram in Set�op

commutes.

uni,ε �n

X

g′g

III.2.2.2. Let X be a cubical set, and let

ui,εn Xg

be an n-horn in X. If there is an n-cube

�n Xg′

of X such that the diagram

uni,ε �n

X

g′g

in Set�op

commutes, we write that g extends to g′.

III.2.2.3. The free-standing 0-cube �0 is a Kan complex. Indeed, recalling the nxplicitdescription of �0 in III.1.1.5, there is a unique morphism

uni,ε �0g

of cubical sets for any 1 ≤ i ≤ n and 0 ≤ ε ≤ 1, and a unique morphism

�n �0g′

for any n ≥ 0. We have that the following diagram in Set�op

commutes.

uni,ε �n

�0

g′g

51

III.2.2.4. The free standing n-cube �n is not a Kan complex for any n ≥ 1. Forinstance, let us consider the possible 2-horns in �1. Let us view �1 as follows.

0 1x

Firstly, we have the following 2-horns, all of which can be extended to degenerate 2-cubesof �1. The unlabelled 1-cubes are degenerate.

0 1 0 1 0 0 0 0

0 1 0 1 1 1 0 1

x

x

x x

Secondly, we have the following 2-horns.

0 1 0 0

1 1 0 1

x

x x

x

Neither of these two 2-horns can be extended to a 2-cube of �1. Indeed, the 2-cubes of�1 are all degenerate, and degenerate 2-cubes cannot have a boundary as above.

III.2.2.5. The same reasoning on 2-horns shows that �n is not a Kan complex forn ≥ 1. It is in fact only the 2-horns of �3 that cannot be extended! For instance, thepossible 3-horns in �2 fall into two types. Let us view �2 as follows.

x

x0

x1x2

x3

Firstly, we have those 3-horns in which only copy of the non-degenerate 2-cube x appears,described by the following net — the different horns correspond to the different choicesof which face of the cube to take as x when assembling the net. All the unlabelled2-cubes and 1-cubes are degenerate.

52

x

x0

x2 x1

x3

The 3-horns of this type can be extended to degenerate 3-cubes of �2. Secondly, wehave the following 3-horns.

x x

x2 x2

x2 x2

x3 x0 x0 x3

x3 x1

x3 x1

Again, all 3-horns of this type can be extended to 3-cubes of �2. It is not possible tocook up any 3-horn in which two copies of X meet at a common face — from the pointof view of nets, the condition that two 1-cubes meeting at right angles in the boundaryof the net must be the same cannot be satisified if two copies of x are adjacent. A similarstory holds in higher dimensions.

III.2.2.6. We draw the attention of the reader acquainted with simplicial sets thatwhether or not the representable presheaves are Kan complexes exhibits a differencebetween the simplicial and cubical settings — the free-standing 1-simplex ∆1is a Kancomplex. 15.

III.2.2.7. Although �n is not a Kan complex for any n ≥ 1, we might wonder whetherwe could not simply add in ‘extra’ degeneracies to fill in the 2-horns which currentlycannot be filled in. It turns out that this is a very good idea to pursue in cubicalhomotopy theory — let us discuss it, before renewing our efforts to define the homotopygroups of a cubical set, this time for Kan complexes.

53

III.3. Cubical sets with connections

III.3.1. Connections in topology

III.3.1.1. In the classical homotopy theory of spaces, many deeper theorems rely notonly the structure of the topological interval

(I0, I1, i0, i1, p

)equipped with its contrac-

tion structure — and also further structure allowing us to reverse and compose homo-topies, for which we refer to [11] for more — but on the structure of the unit square I2.An example is Dold’s theorem on homotopy equivalences under an object, which allowsus to conclude that a trivial cofibration is a section of a strong deformation retraction.

III.3.1.2. In proving these deeper results, we typically construct certain double homo-topies, namely maps

X × I2 Y

for some spaces X and Y . Rather beautifully, it turns out that we can build all thedouble homotopies we need from only two maps.

I2 I

Γ0

Γ1

III.3.1.3. Before we describe, let us re-cast in topology the question of ‘filling in’ the2-horns which in III.2.2.4 prevented �1 from being a Kan complex. Suppose that wehave a path

I1 Xf

in a space X. Pre-composing with the map

I2 II × p

gives a map

I2 X

with the following boundary, in which the vertical sides are constant paths.

f

f

54

This corresponds to the degenerate 2-cube d22(x) for a 1-cube x of a cubical set X.Alternatively, pre-composing with the map

I2 Ip× I

gives a map

I2 X

with the following boundary, in which the horizontal sides are constant paths.

f f

This corresponds to the degenerate 2-cube d12(x) for a 1-cube x of a cubical set X.

III.3.1.4. There are two other possible boundary configurations whose faces are eitherf or a constant path, namely the following.

f

f

f

f

III.3.1.5. Can we ‘fill in’ these boundary configurations in topology? To be moreprecise, can we construct a map

I2 I

whose composition with f will yield a double homotopy with boundary as in the leftmostof the above pictures? Can we construct a map

I2 I

whose composition with f will yield a double homotopy with boundary as in the right-most of the above pictures?

III.3.1.6. The answer to both of these questions is — yes! For the picture on the left,we can take the map to be (x, y) 7→ x + y − xy. This can be taken to be the map Γ0

that we introduced in III.3.1.2. For the picture on the right, we can take the map to be(x, y) 7→ xy. This can be taken to be the map Γ1 that we introduced in III.3.1.2.

55

III.3.2. Connections in abstract homotopy theory

III.3.2.1. We refer to Γ0 and Γ1 as connections. As we have discussed, connectionsplay a crucial role from an abstract point of view in homotopy theory — we refer to [11]for more, in which connections are involved throughout. Indeed, the whole character ofclassical homotopy theory — the reason for its richness — can be seen from an abstractpoint of view to rely indispensably on the structure not only of the unit intercal I1 butthe unit square I2. Thus we can readily believe that it may be useful to expand uponour category of cubical shapes to include connections.

III.3.2.2. We can capture the fact that Γ0 has the boundary we are looking for by theobservation that the following diagrams in spaces commute.

I1 I2 I1 I2

I1 I1

I1 × i0

Γ0id

i0 × I1

Γ0id

I1 I2 I1 I2

I0 I1 I0 I1

I1 × i1

Γ0p

i1

i1 × I1

Γ0p

i1

III.3.2.3. This allows us to pass to the abstract setting — introduced in II.3.2 —of an interval I =

(1, I1, i0, i1

)in a category A equipped with a monoidal structure

(⊗, 1, λ, ρ, α). Indeed, we define an upper connection structure upon I to be an arrow

I1 ⊗ I1 I1Γ0

such that the following diagrams in A commute.

I1 I1 ⊗ I1 I1 I1 ⊗ I1

I1 I1

I1 ⊗ i0

Γ0id

i0 ⊗ I1

Γ0id

56

I1 I1 ⊗ I1 I1 I1 ⊗ I1

I0 I1 I0 I1

I1 ⊗ i1

Γ0p

i1

i1 ⊗ I1

Γ0p

i1

III.3.2.4. Similarly, we can capture the fact that Γ1 has the boundary we are lookingfor by the observation that the following diagrams in spaces commute.

I1 I2 I1 I2

I1 I1

I1 × i1

Γ1id

i1 × I1

Γ1id

I1 I2 I1 I2

I0 I1 I0 I1

I1 × i0

Γ1p

i0

i0 × I1

Γ1p

i0

III.3.2.5. This allows us to pass to the abstract setting of an interval I =(1, I1, i0, i1

)in a category A equipped with a monoidal structure (⊗, 1, λ, ρ, α). Indeed, we define anupper connection structure upon I to be an arrow

I1 ⊗ I1 I1Γ1

such that the following diagrams in A commute.

I1 I2 I1 I2

I1 I1

I1 ⊗ i1

Γ1id

i1 ⊗ I1

Γ1id

57

I1 I2 I1 I2

I0 I1 I0 I1

I1 ⊗ i0

Γ1p

i0

i0 ⊗ I1

Γ1p

i0

III.3.2.6. In spaces, we have that pre-composing a constant path with Γ0 or Γ1 givesa constant map

I2 X.

Abstractly, we can capture this for Γ0 by observing that the following diagram in thecategory of spaces commutes.

I2 I

I I

Γ0

pI × p

p

This is equivalent to the commutativity of the following diagram.

I2 I

I I

Γ0

pp× I

p

Similarly, we can capture this for Γ1 by observing that the following diagram in thecategory of spaces commutes.

I2 I

I I

Γ1

pI × p

p

Again, this is equivalent to the commutativity of the following diagram.

58

I2 I

I I

Γ1

pp× I

p

III.3.2.7. This allows us to pass to the abstract setting of an interval I =(1, I, i0, i1, p,Γ0

)equipped with a contraction structure p and an upper connection structure Γ0, in a cat-egory A equipped with a monoidal structure (⊗, 1, λ, ρ, α). Indeed, we say that Γ0 iscompatible with p if the following diagram in A commutes.

I ⊗ I I

I I

Γ0

pI ⊗ p

p

This is equivalent to requiring that the following diagram in A commutes.

I ⊗A I

I I

Γ0

pp⊗ I

p

Suppose instead that I =(1, I, i0, i1, p,Γ0

)is an interval in A equipped with a contrac-

tion structure p and a lower connection structure Γ1. We say that Γ1 is compatible withp if the following diagram in A commutes.

I ⊗ I I

I I

Γ1

pI ⊗ p

p

This is equivalent to requiring that the following diagram in A commutes.

I ⊗A I

I I

Γ1

pp⊗ I

p

59

III.3.3. Category of cubes with connections

III.3.3.1. We are now in a position to build a new category of shapes, exactly as webuilt the category of semi-cubes in II.3.5 and II.3.8, and as we built the category ofcubes in II.4.4 and II.4.5.

III.3.3.2. Definition Let Υ′′ denote the directed graph defined as follows.


(ii) For every 1 ≤ i ≤ n and every 0 ≤ ε ≤ 1, we have an arrow

In−1 In.fni,ε

(iii) For every 1 ≤ i ≤ n, we have an arrow

In In−1.dni

(iv) For every n ≥ 2, every 1 ≤ i ≤ n− 1, and every 0 ≤ ε ≤ 1, we have an arrow

In In−1.Γn−1i,ε

Let F(Υ′′) denote the free category upon Υ′′. The category of cubes with connections,which we will denote by �c, is the quotient of F(Υ′′) by the relation ∼ defined inductivelyas follows.


fn+1j,ε ◦ f

ni,δ ∼

{fn+1i,δ ◦ f

nj−1,ε if j > i,

fn+1i+1,δ ◦ f

nj,ε if j ≤ i.

(ii) For any n ≥ 2, 1 ≤ i ≤ n and 1 ≤ j ≤ n− 1, we have that

dn−1j ◦ dni ∼

{dn−1i−1 ◦ dnj if j < i,

dn−1i ◦ dnj+1 if j ≥ i.

(iii) For any 1 ≤ i, j ≤ n and 0 ≤ ε ≤ 1, we have that

dnj ◦ fni,ε ∼

id if j = i,

fn−1i,ε ◦ dn−1j−1 if j > i,

fn−1i−1,ε ◦ dn−1j if j < i.

60

(iv) For any n ≥ 2, 1 ≤ i ≤ n− 1, 1 ≤ j ≤ n, and 0 ≤ δ, ε ≤ 1, we have that

Γn−1i,ε ◦ fnj,δ ∼

id if j = i and δ = ε, or j = i+ 1 and δ = ε,

fn−1i,δ ◦ dni if j = i and δ 6= ε, or j = i+ 1 and δ 6= ε,

fn−2j,δ ◦ Γn−2i−1,ε if n ≥ 3 and j < i,

fn−2j−1,δ ◦ Γn−2i,ε if n ≥ 3 and j > i+ 1.

(v) For any n ≥ 2, 1 ≤ i ≤ n− 1, 1 ≤ j ≤ n− 1, and 0 ≤ ε ≤ 1, we have that

dn−1j ◦ Γn−1i,ε =

dn−1i ◦ dni if i = j,

Γn−2i−1,ε ◦ dnj if n ≥ 3 and j < i,

Γn−2i,ε ◦ dnj+1 if n ≥ 3 and j > i.

(vi) For any n ≥ 3, 1 ≤ i ≤ n− 1, 1 ≤ j ≤ n− 2, and 0 ≤ δ, ε ≤ 1, we impose that

Γn−2j,δ ◦ Γn−1i,ε ∼

{Γn−2i−1,ε ◦ Γn−1j,δ if j < i,

Γn−2i,ε ◦ Γn−1j+1,δ if j > i.

(vii) For any n ≥ 3, 1 ≤ i ≤ n− 2, and 0 ≤ ε ≤ 1, we have that

Γn−2i,ε ◦ Γn−1i,ε ∼ Γn−2i,ε ◦ Γn−1i+1,ε.

(viii) Suppose that we have arrows

Iq Ir

g0

g′0

and

Ir Is

g1

g′1

of F(Υ′) with q, r, s ≥ 0, such that g0 ∼ g1 and g′0 ∼ g′1. Then we have thatg1 ◦ g0 ∼ g′1 ◦ g′0.

III.3.3.3. Let (�c)≤2 denote the full subcategory of �c whose objects are I0, I1, andI2. We think of (�c)≤2 as the free-standing interval equipped with a contraction structureand connections16.

61

III.3.3.4. To avoid confusion, let us adopt the following notation for the objects andarrows of (�c)≤2.

(i) We denote I0, I1, and I2 by 0, 1, and 2 respectively.

(ii) We denote i0 and i1 by ι0 and ι1 respectively.

(iii) We denote p by π.

(iv) We denote Γ0 and Γ1 by γ0 and γ1 respectively.

Let A be a category equipped with a monoidal structure (⊗, λ, ρ, α). Suppose that(I0, I1, i0, i1, p,Γ0,Γ1

)defines an interval in A equipped with a contraction structure

p, a lower connection structure Γ0, and an upper connection structure Γ1. Supposemoreover that Γ0 is compatible with p, and that Γ1 is compatible with p. Then weobtain a functor

(�c)≤2 A

given by 0 7→ I0. 1 7→ I1, 2 7→ I2, ι0 7→ i0, ι1 7→ i1, π 7→ p, γ0 7→ Γ0, and γ1 7→ Γ1.Conversely, every functor

(�c)≤2 A

defines an interval in A equipped with a contraction structure, a lower connection struc-ture, and an upper connection structure, such that both connection structures are com-patible with the contraction structure.

III.3.3.5. Intuitively, our category of cubes with connections should be the free strictmonoidal category upon (�c)≤2. In this picture. we think of Γni,0 as the arrow

In−1 In.Ii−1 ⊗ Γ0 ⊗ In−i+1

We think of Γni,1 as the arrow

In−1 In.Ii−1 ⊗ Γ1 ⊗ In−i+1

III.3.3.6. The relations of (iv) in Definition III.3.3.2 arise by considering the definingaxioms for an upper connection structure and a lower connection structure in III.3.2.3and III.3.2.4 respectively, and formulating higher dimensional analogues. The relationsof (v) in Definition III.3.3.2 arise by considering the defining axioms in III.3.2.7 for anupper connection structure compatible with a contraction structure, and for a lowerconnection structure compatible with a contraction structure. The relations of (vi) and(vii) arise by the same considerations as those of II.3.7.

62

III.3.4. A universal property of the category of cubes with connections

III.3.4.1. It remains to capture rigorously the idea that �c should be the free strictmonoidal category upon an interval equipped with a contraction structure and connec-tion structures. We proceed exactly as in II.3.8 and II.4.5 — this time we will omit anyjustification, which is identical to that in II.3.8 and II.4.5.

III.3.4.2. Definition Let

�c ×�c �c.−⊗−





(iv) For 1 ≤ i ≤ m and n ≥ 0, we define dmi ⊗ In to be dm+ni .

(v) For 1 ≤ i ≤ n and m ≥ 0, we define Im ⊗ dni to be dm+nm+i .

(vi) For any m ≥ 2, 1 ≤ i ≤ m, n ≥ 0, and 0 ≤ ε ≤ 1, we define Γmi,ε ⊗ In to be Γm+ni,ε .

(vii) For any n ≥ 2, 1 ≤ i ≤ n, m ≥ 0, and 0 ≤ ε ≤ 1, we define Im ⊗ Γni,ε to be Γm+nm+i,ε.

(viii) For m,m′,m′′, n ≥ 0, and arrows

Im Im′

g0

and

Im′

Im′′

g1

of �, we define (g1 ◦ g0)⊗ In to be (g1 ⊗ In) ◦ (g0 ⊗ In).

(ix) For m,n, n′, n′′ ≥ 0, and arrows

In In′

g0

and

In′

In′′

g1


63

III.3.4.3. Proposition The functor

�c ×�c �c−⊗−

equips the category �c with the structure of a strict monoidal category with unit I0.

III.3.4.4. Proposition Let A be a category equipped with a strict monoidal structure(⊗, 1). For any functor

(�c)≤2 A,int


�c Acan


(�c)≤2 �c

A

canint

III.3.5. A different perspective

III.3.5.1. We may think about the category of cubes with connections in a slightlydifferent way — the reader not familiar with simplicial sets may wish to skip onto thenext section. Recall that a simplicial set is a presheaf on a category ∆ which can bedefined as the full subcategory of the category of partially ordered sets whose objectsare the partially ordered sets

0 1 n− 1 n

for n ≥ 0.

III.3.5.2. We can define � in a similar way. For any n ≥ 0, let In denote the set{0, 1}n equipped with the partial ordering defined (δ0, . . . , δn) ≤ (ε0, . . . , εn) if δi ≤ εifor all 0 ≤ i ≤ n. For any 1 ≤ i ≤ n and 0 ≤ ε ≤ 1, we denote by fni,ε the functor

In−1 In

defined by (δ0, . . . , δn) 7→ (δ0, . . . , δi−2, ε, δi−1, . . . , δn). For any 1 ≤ i ≤ n, we denote bydni the functor

64

In In−1

defined by (δ0, . . . , δn) 7→ (δ0, . . . , δi−2, δi, . . . , δn). We can then view � as the subcat-egory of the category of posets whose objects are the posets In for n ≥ 0, and whosearrows are generated by the functors fni,ε and dni for 1 ≤ i ≤ n and 0 ≤ ε ≤ 1.

III.3.5.3. However, � is not a full subcategory of the category of posets. The fullsubcategory of the category of posets whose objects are the In is exactly �c.

III.3.5.4. Thus by passing from the category of cubes to the category of cubes withconnections intuitively should bring us closer to the simplicial world.

III.3.6. Cubical sets with connections

III.3.6.1. A cubical set with connections is a presheaf upon �c.

III.3.6.2. An explicit description of a cubical set with connections X can be given indirect analogy to the explicit description of a cubical set in III.1.1.1. The new data is amap

Xn−1 Xn

for every n ≥ 2, 1 ≤ i ≤ n− 1, and 0 ≤ ε ≤ 1, which we will denote by Γi,εn . These mapssatisfy the dual relations to those of III.3.3.2.

III.3.6.3. We have put our idea of III.2.2.7 on solid ground. If we work with cubicalsets with connections, the representable presheaf �1 is a Kan complex.

III.3.6.4. It remains the case that �n is not a Kan complex for n ≥ 2 if we workwith cubical sets with connections. or example, let x denote the unique non-degenerate2-cube of �2, and let us denote its boundary faces as follows.

x0

x1x2

x3

The following defines a 2-horn in �2, but this 2-horn does not extend to a 2-cube ofcategorycubes2.

x0

x1x2

65

III.3.6.5. This gives some evidence for our feeling of III.3.5.4 — for simplicial sets itis also the case that the representable n-simplex ∆n is Kan complex for n = 0 and n = 1,and is not a Kan complex for n ≥ 2.

III.4. Homotopy groups of a Kan complex

III.4.1. Combinatorial homotopy defines an equivalence relation for a Kan complex

III.4.1.1. Let us now return to our attempt in III.1.3.5 to define the homotopy groupsof a cubical set, this time assuming that our cubical set is a Kan complex. Our stumblingblock earlier was that our homotopy relation ∼ did not define an equivalence relation —see III.1.4.11.

III.4.1.2. Proposition Let X be a Kan complex, and let ∗ be a 0-cube of X. Let x0and x1 be n-cubes of X belonging to Zn(X, ∗), namely with trivial boundary. Supposethat x0 ∼ x1. Then x1 ∼ x0.

III.4.1.3. Before proving Proposition III.4.1.2, let us explore a proof in low dimensions.Suppose that x0 and x1 are 1-cubes of X as indicated below.

∗ ∗x0

x1

Suppose that h is a 2-cube of X with the following boundary, in which the vertical1-cubes are degenerate.

∗ ∗

∗ ∗

x0

x1

The following then defines a 3-horn in X.

66

h

d22(x0)

d22(x0)

x1

x0

x0

x0

One can assemble this 3-horn by taking the front, shaded face of the 3-cube below to bethe middle face of the 3-horn, namely d22(x0).

Then the bottom face of the above 3-cube corresponds to other copy of d22(x0) in the3-horn, and the top face corresponds to the 2-cube h of the 3-horn. The back face of the3-cube is the missing face of the 3-horn. For the remainder of this lecture, all 3-hornsdepicted as nets can be assembled in this way — we shall not draw further attention tothis.

Since X is a Kan complex, this 3-horn extends to a 3-cube. The face of this 3-cubewhich is not part of the above horn has the following boundary.

∗ ∗

∗ ∗

x1

x0

Proof of Proposition III.4.1.2. Let us now give a proof in arbitrary dimensions. Let x0and x1 be n-cubes of X belonging to Zn(X, ∗), and suppose that h defines a homotopyfrom x0 to x1. The following defines an (n+ 1)-horn

un+1n,1 X

g

67

of X.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

h i = n+ 1,

dnn(x0) i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

{dnn(x0) i = n+ 1,

∗ i < n.

Since X is a Kan complex, there is an (n+ 1)-cube

�n+1 Xg′

of X such that the following diagram in Set�op

commutes.

un+1n,1 �n+1

X

g′g

Then fn+1n,1 (g′) defines a homotopy from x1 to x0.

III.4.1.4. Proposition Let X be a Kan complex, and let ∗ be a 0-cube of X. Let x0, x1,and x2 be n-cubes of X belonging to Zn(X, ∗), namely with trivial boundary. Supposethat x0 ∼ x1 and that x1 ∼ x2. Then x0 ∼ x2.

III.4.1.5. Again, let us explore a proof of Proposition III.4.1.4 in low dimensions.Suppose that x0, x1, and x2 are 1-cubes of X with the following boundary.

∗ ∗

Suppose that h0 is a 2-cube of X with the following boundary, in which the vertical1-cubes are degenerate.

∗ ∗

∗ ∗

x0

x1

Suppose that h1 is a 2-cube of X with the following boundary, in which the vertical1-cubes are degenerate.

68

∗ ∗

∗ ∗

x1

x2

The following then defines a 3-horn in X.

d22(x0)

h0

h1

x0

x0

x1

x2

Since X is a Kan complex, this 3-horn extends to a 3-cube. The face of this 3-cubewhich is not a part of the above horn has the following boundary

∗ ∗

∗ ∗

x0

x2

Proof of Proposition III.4.1.4. Let us now give a proof in arbitrary dimensions. Let x0,x1, and x2 be n-cubes of X belonging to Zn(X, ∗). Suppose that h0 defines a homotopyfrom x0 to x1, and that h1 defines a homotopy from x1 to x2. The following defines an(n+ 1)-horn

un+1n+1,1 X

g

69

of X.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

h0 i = n+ 1,

dnn(x0) i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

{h1 i = n,

∗ i < n.

Since X is a Kan complex, there is an (n+ 1)-cube

�n+1 Xg′

of X such that the following diagram in Set�op

commutes.

un+1n+1,1 �n+1

X

g′g

Then fn+1n+1,1(g

′) defines a homotopy from x0 to x2.

III.4.1.6. We have now established that if (X, ∗) is a Kan complex, then ∼ definesan equivalence relation upon Zn(X, ∗). Thus, as a set, we can define πn(X, ∗) to beZn(X, ∗)/ ∼, as we hoped to do in III.1.3.5. We now turn to equipping πn(X, ∗) with agroup structure.

III.4.2. Group structure upon π1(X, ∗)

III.4.2.1. Let x0 and x1 be 1-cubes of a pointed Kan complex (X, ∗) with the followingboundary.

∗ ∗

The following then defines a 2-horn of X.

∗ ∗

∗ ∗

x0

x1

Since X is a Kan complex, we can extend this 2-horn to a 2-cube in X. We would liketo define17 x1 · x0 to be the right vertical face of this 2-cube, as depicted below.

70

∗ ∗

∗ ∗

x1 · x0x0

x1

III.4.2.2. We must have that the following conditions are satisfied in order for this towork.

(i) If we have a 1-cube x′0 such that x0 ∼ x′0, then x1 ◦ x0 ∼ x1 ◦ x′0.

(ii) If we have a 1-cube x′1 such that x1 ∼ x′1, then x1 ◦ x0 ∼ x′1 ◦ x0.

III.4.2.3. Let us verify that (i) holds. Let h be a 2-cube of X with the followingboundary.

∗ ∗

∗ ∗

x0

x′0

Let k0 denote the 2-cube of X which extends the following 2-horn in X.

∗ ∗

∗ ∗

x0

x1


∗ ∗

∗ ∗

x′0

x1

Since X is a Kan complex, the following 3-horn in X extends to a 3-cube in X.

71

k0

h d22(x1)

k1

x1 · x0

x1

x0 x1

x′0x1

x1

x1 ◦ x′0

The face of this 3-cube which is not a part of the above 3-horn defines a homotopy withthe following boundary.

∗ ∗

∗ ∗

x1 · x0

x1 · x′0

III.4.2.4. Let us now verify that (ii) holds. Let h be a 2-cube of X with the followingboundary.

∗ ∗

∗ ∗

x1

x′1


∗ ∗

∗ ∗

x0

x1

72


∗ ∗

∗ ∗

x0

x′1


k0

d22(x0) h

k1

x1 · x0

x1

x0 x1

x0 x′1

x′1

x′1 ◦ x0

The face of this 3-cube which is missing from the above 3-horn has the following bound-ary.

∗ ∗

∗ ∗

x1 ◦ x0

x′1 ◦ x0

III.4.2.5. We have now shown that − · − gives rise to a map

π1(X, ∗)× π1(X, ∗) π1(X, ∗).

Let us prove that this map equips π1(X, ∗) with the structure of a group.

III.4.2.6. We begin with an observation. Suppose that x0 and x1 are 1-cube of Xwith the following boundary.

73

∗ ∗

Let h denote a 2-cube of X with the following boundary.

∗ ∗

∗ ∗

x1 · x0x0

x1

Suppose that we have a 2-cube k of X with the following boundary.

∗ ∗

∗ ∗

x2x0

x1

We claim that x2 ∼ x1 ·x0. To see this, note that since X is a Kan complex the following3-horn in X extends to a 3-cube.

k

d22(x0) d22(x1)

h

x2

x1

x0 x1

x0 x1

x1

x1 · x0

The face of this 3-cube corresponding to the missing face of the 3-horn has the followingboundary.

∗ ∗

∗ ∗

x2

x1 · x0

74

III.4.2.7. Let us now turn to proving that the map

π1(X, ∗)× π1(X, ∗) π1(X, ∗).

defined by − · − is associative. Suppose that x0, x1, and x2 are 1-cubes of X with thefollowing boundary.

∗ ∗

Let h0 be a 2-cube of X with the following boundary.

∗ ∗

∗ ∗

x1 · x0x0

x1


∗ ∗

∗ ∗

x2 · x1x1

x2


∗ ∗

∗ ∗

x2 · (x1 · x0)x1 · x0

x2


75

d12(x0) h0 h2

h1

x0 x0 x1 · x0 x2 · (x1 · x0)

x1 x2

x2

x2 · x1

The face of this 3-cube which is not a part of the above 3-horn has the following bound-ary.

∗ ∗

∗ ∗

x2 · (x1 · x0)x0

x2 · x1

By our observation of III.4.2.6, we deduce that x2 · (x1 · x0) ∼ (x2 · x1) · x0.

III.4.2.8. Let us prove that the degenerate 1-cube d11(∗) defines a right identity forthe map

π1(X, ∗)× π1(X, ∗) π1(X, ∗).

defined by − · −. For any 1-cube

∗ ∗x

of X, the degenerate 2-cube d12(x) of X has the following boundary.

∗ ∗

∗ ∗

xx

d11(∗)

It follows from our observation of III.4.2.6 that x · d11(∗) ∼ x.

76

III.4.2.9. If we are working with cubical sets with connection, we can also observethat d11(∗) defines a left identity for the map

π1(X, ∗)× π1(X, ∗) π1(X, ∗).

defined by − · −. Indeed the 2-cube Γ1,12 (x) of X has the following boundary.

∗ ∗

∗ ∗

xd11(∗)

x

It would then follow from our observation of III.4.2.6 once more that d11(∗) · x ∼ x.

III.4.2.10. If we do not assume that our cubical sets have connections, the author isnot aware18 of any straightforward geometric argument to show that d11(∗) defines a leftidentity for the map

π1(X, ∗)× π1(X, ∗) π1(X, ∗).

defined by − · −. This can nevertheless be proven to be the case —indeed it suffices19

to prove that every 1-cube of X admits a right inverse up to homotopy with respect to− · −.

III.4.2.11. Let us turn to this. Let

∗ ∗x

be a 1-cube of X. Since X is a Kan complex, the following 2-horn in X extends to a2-cube h.

∗ ∗

∗ ∗

x

Let us denote the face of this 2-cube which is not part of the above 2-horn by x−1, sothat the boundary of h is as shown below.

∗ ∗

∗ ∗

x

x−1

It follows immediately from our observation of III.4.2.6 that x · x−1 ∼ d11(∗).

77

III.4.2.12. If we are working with cubical sets with connection, we can also observethat x−1 defines a left inverse of x up to homotopy. Indeed, let k be a 2-cube of X withthe following boundary, for some 1-cube x′ of X.

∗ ∗

∗ ∗

x′x−1

x

Since X is a Kan complex, the following 3-horn in X extends to a 3-cube of X.

d12(x)

Γ1,02 (x) h d22(x)

k

x x

x x

x

x−1 x

x

x′

The face of this 3-cube which is not part of the above 3-horn has the following boundary.

∗ ∗

∗ ∗x

It follows from our observation of III.4.2.6 that x−1 · x ∼ d11(∗).

III.4.2.13. As in III.4.2.10, if we do not assume that our cubical sets have connections,then the author is not aware20 of any straightforward geometric argument to show thatx−1 defines a left inverse of x up to homotopy. Nevertheless, we have shown that d11(∗)equips

π1(X, ∗)× π1(X, ∗) π1(X, ∗).

defined by − · − with a right identity, and that every 1-cube x of X has a right inversex−1 up to homotopy. As we mentioned in III.4.2.10, it follows formally from these two

78

observations that d11(∗) moreover defines a left identity, and that x−1 moreover is a leftinverse of x.

III.4.2.14. This concludes our proof that the map

π1(X, ∗)× π1(X, ∗) π1(X, ∗).

defined by − · − equips π1(X, ∗) with the structure of a group.

III.4.3. Group structure upon πn(X, ∗)

III.4.3.1. Our construction of a group structure upon π1(X, ∗) for any pointed Kancomplex (X, ∗) can be carried out in any dimension. We now give the details.

III.4.3.2. Definition Let X be a Kan complex, and let ∗ be a 0-cube of X. Let x0 andx1 be n-cubes of X belonging to Zn(X, ∗), for some n ≥ 1. The following defines an(n+ 1)-horn

un+1i,1 X

g

in X.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

{x1 i = n+ 1,

∗ i < n.

Let

�n Xg′

denote the21 extension of g to an (n + 1)-cube of X. We denote by x1 · x0 the n-cubefn,1n+1(g) of X.

III.4.3.3. Proposition Let X be a Kan complex, and let ∗ be a 0-cube of X. Let x0,x′0, and x1 be n-cubes of X, and suppose that x0 ∼ x′0. Then x1 · x0 ∼ x1 · x′0.

Proof. Let h be an (n + 1)-cube of X which defines a homotopy from x0 to x′0. Let k0be an (n+ 1)-cube of X with the following boundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x0 i = n,

∗ i < n.

79

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x1 i = n+ 1,

x1 · x0 i = n,

∗ i < n.

Let k1 be an (n+ 1)-cube of X with the following boundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x′0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x1 i = n+ 1,

x1 · x′0 i = n,

∗ i < n.

The following defines an (n+ 2)-horn

un+2,1n+2 X

g′

in X.

Ii−1 ⊗ i0 ⊗ In+2−i 7→

h i = n+ 2,

∗ i = n+ 1,

k0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

dn+1n+1(x1) i = n+ 1,

k1 i = n,

∗ i < n.

Let g′ denote the extension of g to an (n+2)-cube of X. Then the (n+1)-cube fn+2,1n+2 (g)

defines a homotopy from x1 · x0 to x1 · x′0.

III.4.3.4. Proposition Let X be a Kan complex, and let ∗ be a 0-cube of X. Let x0,x1, and x′1 be n-cubes of X, and suppose that x1 ∼ x′1. Then x1 · x0 ∼ x′1 · x0.

Proof. Let h be an (n + 1)-cube of X which defines a homotopy from x1 to x′1. Let k0be an (n+ 1)-cube of X with the following boundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x1 i = n+ 1,

x1 · x0 i = n,

∗ i < n.

80

Let k1 be an (n+ 1)-cube of X with the following boundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x′1 i = n+ 1,

x′1 · x0 i = n,

∗ i < n.


un+2,1n+2 X

g′

in X.

Ii−1 ⊗ i0 ⊗ In+2−i 7→

d22(x0) i = n+ 2,

∗ i = n+ 1,

k0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

h i = n+ 1,

k1 i = n,

∗ i < n.

Let g′ denote the extension of g to an (n+2)-cube of X. Then the (n+1)-cube fn+2,1n+2 (g)

defines a homotopy from x1 · x0 to x′1 · x0.

III.4.3.5. Lemma Let X be a Kan complex, and let ∗ be a 0-cube of X. Let x0 andx1 be n-cubes of X belonging to Zn(X, ∗), for some n ≥ 1. Suppose that h0 is an(n + 1)-cube of X with the following boundary, for some n-cube x2 of X belonging toZn(X, ∗).

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x1 i = n+ 1,

x2 i = n,

∗ i < n.

81

Suppose that h1 is an (n+ 1)-cube of X with the following boundary, for an n-cube x′2of X belonging to Zn(X, ∗).

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x1 i = n+ 1,

x′2 i = n,

∗ i < n.

Then x2 ∼ x′2.

Proof. The following defines an (n+ 2)-horn

un+2,1n+2 X

g

in X.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

dn+1n+1(x0) i = n+ 2,

∗ i = n+ 1,

h0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

d22(x1) i = n+ 1,

h1 i = n,

∗ i < n.

Since X is a Kan complex, g extends to an (n + 2)-cube g′ in X. The (n + 1)-cubefn+2,1n+2 (g′) defines a homotopy from x2 to x′2.

III.4.3.6. Proposition Let X be a Kan complex, and let ∗ be a 0-cube of X. The map

πn(X, ∗)× πn(X, ∗) πn(X, ∗)

defined by (x, y) 7→ x · y equips πn(X, ∗) with the structure of a group.

Proof. Firstly, let us show that this map is associative. Let x0, x1, and x2 be n-cubes ofX belonging to Zn(X, ∗). Let h0 be an (n+ 1)-cube of X with the following boundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x0 i = n,

∗ i < n.

82

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x1 i = n+ 1,

x1 · x0 i = n,

∗ i < n.

Let h1 be an (n+ 1)-cube of X with the following boundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x1 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x2 i = n+ 1,

x2 · x1 i = n,

∗ i < n.

Let h2 be an (n+ 1)-cube of X with the following boundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x1 · x0 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x2 i = n+ 1,

x2 · (x1 · x0) i = n,

∗ i < n.


un+2,1n+2 X

g

in X.

Ii−1 ⊗ i0 ⊗ In+2−i 7→

h0 i = n+ 2,

d12(x0) i = n+ 1,

∗ i ≤ n.

Ii−1 ⊗ i1 ⊗ In+2−i 7→

h2 i = n+ 1,

h1 i = n,

∗ i < n.

Since X is a Kan complex, g extends to an (n + 2)-cube g′ in X. The (n + 1)-cubefn+2,1n+2 (g′) of X has the following boundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x0 i = n,

∗ i < n.

83

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x2 · x1 i = n+ 1,

x2 · (x1 · x0) i = n,

∗ i < n.

Appealing to Lemma III.4.3.5, we deduce that x2 · (x1 · x0) ∼ (x2 · x1) · x0, as required.Secondly, let us prove that ∗, regarded as a degenerate n-cube in the unique possible

way as usual, defines a right identity for the map of the proposition when viewed asbelonging to πn(X, ∗). Let x be an n-cube of X belonging to Zn(X, ∗). The (n+1)-cubednn+1(x) of X has the following boundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

∗ i = n+ 1,

x i = n,

∗ i < n.

Appealing to Lemma III.4.3.5, we deduce that x · ∗ ∼ x.To complete the proof, it suffices22 to show that for any n-cube x of X belonging to

Zn(X, ∗), there is an n-cube x−1 of X belonging to Zn(X, ∗) such that x · x−1 ∼ ∗. Tosee this, note that the following defines an (n+ 1)-horn

un+1,1n+1 X

g

in X.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→{∗ i ≤ n.

Since X is a Kan complex, g extends to an (n + 1)-cube g′ in X. Let x−1 denote then-cube fn+1,1

n+1 (g′) of X. By Lemma III.4.3.5, x · x−1 ∼ ∗.

III.4.3.7. If we assume in Proposition III.4.3.6 thatX is a cubical set with connections,we can see directly that ∗ defines a left identity up to homotopy for the map

πn(X, ∗)× πn(X, ∗) πn(X, ∗)

defined by (x, y) 7→ x · y. Indeed the (n + 1)-cube Γn,1n+1(x) of X has the followingboundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

{∗ i = n+ 1,

∗ i ≤ n.

84

Ii−1 ⊗ i1 ⊗ In+1−i 7→

x i = n+ 1,

x i = n,

∗ i < n.

Appealing to Lemma III.4.3.5 once more, we deduce that ∗ · x ∼ x.

III.4.3.8. Let us again assume that X is a cubical set with connections in PropositionIII.4.3.6, and let x be an n-cube of X belonging to Zn(X, ∗). We can see directly thatthe n-cube x−1 constructed in the proof of Proposition III.4.3.6 defines a left inverse upto homotopy for the map

πn(X, ∗)× πn(X, ∗) πn(X, ∗)

defined by (x, y) 7→ x · y. Indeed, let h denote an (n + 1)-cube of X with the followingboundary.

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

{x−1i = n+ 1,

∗ i ≤ n.

Let k denote an (n + 1)-cube of X with the following boundary, for some n-cube x′ ofX belonging to Zn(X, ∗).

Ii−1 ⊗ i0 ⊗ In+1−i 7→

∗ i = n+ 1,

x−1 i = n,

∗ i < n.

Ii−1 ⊗ i1 ⊗ In+1−i 7→

{xi = n+ 1,

x′ i ≤ n.


un+2,1n+2 X

g

in X.

Ii−1 ⊗ i0 ⊗ In+2−i 7→

h i = n+ 2,

dnn+1(x) i = n+ 1,

Γn,0n+1(x) i = n,

∗ i < n.

85

Ii−1 ⊗ i1 ⊗ In+2−i 7→

k i = n+ 1,

dn+1n+1(x) i = n,

∗ i < n.

Since X is a Kan complex, g extends to an (n + 2)-cube g′ of X. The (n + 1)-cubefn+2,1n+2 (g′) of X defines a homotopy ∗ ∼ x′. Appealing to Lemma III.4.3.5, we deduce

that x−1 · x ∼ ∗.

III.4.3.9. In a later lecture, we will prove that for any pointed Kan complex (X, ∗),we have that πn(X, ∗) is abelian for n ≥ 2.

Notes

12 We will see later that the requirement that f i,εn+1(h) be degenerate for i 6= n arises naturally fromthe conceptual definition of homotopy we touched upon in III.1.4.1 — for n = 1, it is akin to thecommutativity requirement which a natural transformation must satisfy in category theory. Cubicalsets are an algebraic notion, we should expect that ‘everything must be specified’ ! Nevertheless, thereader may be puzzled by the fact that this requirement might not seem to be present in topology.In fact, however, there exists a homotopy between two paths in the usual sense in topology if andonly if there exists a homotopy between these paths satisfying the extra condition we impose in thecombinatorial setting. Indeed, let

I2 I2l

denote the map depicted below.

In words, l is the identity on the diamond, whilst the rest of the square retracts onto the boundaryof the diamond. Then given a map

I2 Xf

with the boundary

f0

f1f2

f3

we have that the composite map

86

I2 I2

X

l

ff ◦ l

has the following boundary, where the vertical faces are constant maps.

f1 ◦ f0

f3 ◦ f2

An analogous story holds in higher dimensions. In abstract homotopy theory, this idea has beenpursued by Grandis, in [2] for example.

13 We will later explain formally how to cook up a cubical set from a recipe of this kind. Roughlyspeaking, one adds any degeneracies which one needs in order to obtain a cubical set, and nothingelse!

14 We shall interpret the definition of a Kan complex constructively, though the reader happy withnon-constructive foundations for mathematics may harmlessly ignore this if it is not to their taste!To be precise, we will think of a Kan complex as a cubical set equipped for every 1 ≤ i ≤ n and0 ≤ ε ≤ 1 with a map Σni,ε from the set HomSet�

op (uni,ε, X) to HomSet�op (�n, X) such that diagram

uni,ε �n

X

gΣni,ε(g)

in Set�op

commutes for every g. A Kan complex in this constructive sense is referred to as analgebraic Kan complex in [6]. The reader who prefers not to implicitly work constructively in thisway must appeal to the axiom of choice at various places later in this work, for example when wecome to define the homotopy groups of a Kan complex.

15 Indeed, the only possible 2-horns in ∆1 are the following, which may be extended to degenerate2-simplexes. The free-standing n-simplex ∆n is not a Kan complex for n ≥ 2 — the following 3-hornin ∆2 cannot be extended to a 3-simplex, for example.

16 To be more precise, we think of (�c)≤2 as the free standing interval equipped with a contractionstructure, an upper connection structure, and a lower connection structure, such that both connectionstructures are both compatible with contraction.

17 In the simplicial setting, the analogous definition of the group structure on π1(X, ∗) is perhapsmore ‘minimal’, and more immediately clearly related to composition in category theory: given1-simplexes

∗ ∗x0

x1

of a pointed simplicial Kan complex (X, ∗) the following defines a 2-horn in X.

87

∗

∗ ∗

x0

x1

Since X is a Kan complex, this 2-horn extends to a 2-simplex. We define x1 · x0 to be the diagonalface of this 2-simplex.

∗

∗ ∗

x0

x1

x1 · x0

In higher dimensions, the (n + 2)-horn giving rise to the group structure on πn(X, ∗) will containdegenerate (n + 1)-simplexes, just as in the cubical setting — of course there are slightly fewer ofthem!Note that the idea behind our notion of composition-up-to-homotopy in a Kan complex X — cubicalor simplicial — more generally allows us to the composite of any pair of 1-cubes

x0

x1 x2

f0

f1

in X to be the right vertical face of the 2-cube obtained by extending the following 2-horn in X.

x0 x0

x1 x2

f1 · f0

f0

This composite will be well-defined up to homotopy — this notion of homotopy is a slight general-isation of that we have seen so far, but is again a special case of the notion of the general notionof homotopy in Set�

op

mentioned in III.1.4.1, which we will come to in the next lecture. Moregenerally, the notion of composition-up-to-homotopy between n-cubes of X belonging to Zn(X, ∗)which we will give later extends to a notion of composition-up-to-homotopy of n-cubes of X witharbitrary compatible faces. These considerations are a route towards higher category theory.

18 In addition to being clear for cubical sets with connections, it is clear for simplicial sets.

19 It is an elementary algebraic observation that a set equipped with an associative binary relation is agroup if and only if this binary relation admits a right identity, and if every element admits a rightinverse. The same holds if we replace right by left in both cases (but not if we mix the two).

20 Again, it is straightforward to prove for simplicial sets.

21 Recall — see 14 — that we are thinking of our definition of a Kan complex as constructive, so thatour n-cube g′ is part of the structure of a Kan complex. Traditionally, one would only know that ann-cube extending g exists, and would then need the axiom of choice to make a simultaneous choicefor every 2-horn, as mentioned in 14.

22 That this indeed suffices is for the same formal reasons as in 19.

88

A. A categorical miscellany

A.1. Free category upon a directed graph

A.1.1. Introduction

In the course, we will frequently construct a category as the free category upon a directedgraph. We recall the details — it is possible to cook up the free category functor veryquickly by a formal argument which we will see later in the course, but we will insteadconstruct it explicitly here.

A.1.2. Free category

A.1.2.1. Definition Let Υ be a directed graph. The free category on Υ, which we willdenote by F(Υ) is defined as follows.

(i) The objects of F(Υ) are the objects of Υ.

(ii) An arrow of F(Υ) is a morphism from a directed graph

• • • •

n

to Υ for some n ≥ 0. The source of this arrow is the object of Υ to which theleftmost object of the directed graph is sent. The target of this arrow is the objectof Υ to which the rightmost object of the directed graph is sent.

(iii) Composition is concatenation.

(iv) The identity arrow corresponding to an object of Υ is the morphism of directedgraphs from • to Υ defined by this object.

A.1.2.2. Example The free category on the directed graph

0 1 2

is the directed graph

0 1

2

equipped with the unique possible structure of a category. In particular, we have that(1 → 2) ◦ (0 → 1) = (0 → 2). This example captures the essence of the free categoryconstruction — we formally add in composites.

89

A.1.2.3. Definition Let

Υ0 Υ1F

be a morphism of directed graphs. We denote by

F(Υ0) F(Υ1)F(F )

the functor defined as follows.

(i) To an object x of F(Υ0), we associate the object F (x) of F(Υ1).

(ii) To an arrow

x0 x1 xn−1 xnf1 fn

of F(Υ0), we associate the arrow

F (x0) F (x1) F (xn−1) F (xn)F (f1) F (fn)

of F(Υ1).

A.1.2.4. Associating to a directed graph Υ the category F(Υ), and associating to amorphism of directed graphs

Υ0 Υ1F

the functor

F(Υ0) F(Υ1)F(F )

defines a functor from the category of directed graphs to the category of directed graphs.We refer to it as the free category functor.

A.1.3. A universal property of a free category

A.1.3.1. Proposition The free category functor F is left adjoint to the forgetful functorU from categories to directed graphs.

90

A.1.3.2. Proof This follows from the following observations.

(1) Let A be a category, and let Υ be a directed graph. Suppose that

Υ U(A)F

defines a morphism of directed graphs. Then the following defines a functor

F(Υ) A.L(F )

(i) To an object x of F(Υ) we associate the object G(a) of A.

(ii) To an arrow

x0 x1 xn−1 xnf1 fn

of F(Υ) we associate the — unique, by associativity! — composite of thefollowing arrows in A.

F (x0) F (x1) F (xn−1) F (xn)F (f1) F (fn)

(2) Let A be a category, and let Υ be a directed graph. Suppose that

F(Υ) AF

defines a functor. Then the following defines a morphism

Υ U(A)R(F )

of directed graphs.

(i) To an object x of Υ, we associate the object F (x) of U(A).

(ii) To an arrow

x0 x1f

of Υ, we associate the arrow

91

F (x0) F (x1)F (f)

of A.

(3) Let A be a category, and let Υ be a directed graph. It is immediately verifiedthat L(−) defines a natural transformation from the functor HomCat

(F(−),−

)to

the functor HomdGraphs

(−,U(−)

), and that R(−) defines a natural transformation

in the other direction. Here Cat denotes the category of categories, and dGraphsdenotes the category of directed graphs.

(4) It is clear that L(−) and R(−) are inverse to one another.

92

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zbmath/search/?q=an%3A0182.26001.

[2] M. Grandis, Categorically algebraic foundations for homotopical algebra, Appl.Categ. Structures, 5 (1997), pp. 363–413. Preprint version: http://www.

dima.unige.it/~grandis/HX1.pdf. Review: http://www.zentralblatt-math.

org/zbmath/search/?q=an%3A0906.55015.

[3] K. H. Kamps and T. Porter, Abstract homotopy and simple homotopy theory,World Scientific, 1997. Review: http://www.zentralblatt-math.org/zbmath/

search/?q=an%3A0890.55014.

[4] D. M. Kan, Abstract homotopy. I, Proc. Nat. Acad. Sci. U.S.A., 41 (1955),pp. 1092–1096. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528202/.Review: http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0065.

38601.

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www.zentralblatt-math.org/zbmath/search/?q=an%3A0495.55001.

[6] T. Nikolaus, Algebraic models for higher categories, Indag. Math. (N.S.), 21(2011), pp. 52–75. Preprint version: http://arxiv.org/abs/1003.1342. Review:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1236.55024.

[7] G. Raptis, Homotopy theory of posets, Homology, Homotopy Appl., 12 (2010),pp. 211–230. Review: http://www.zentralblatt-math.org/zbmath/search/?q=

an%3A1215.18017.

[8] C. P. Rourke and B. J. Sanderson, ∆-sets, I. Homotopy theory, Quart.J. Math. Oxford Ser. (2), 22 (1971), pp. 321–338. Review: http://www.

zentralblatt-math.org/zbmath/search/?q=an%3A0226.55019.

[9] C. Weibel, Voevodsky’s Seattle lectures: K-theory and motivic cohomology, in Al-gebraic K-theory (Seattle, WA, 1997), vol. 67 of Proc. Sympos. Pure Math., 1999,pp. 283–303. Preprint: http://www.math.uiuc.edu/K-theory/0316/.

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http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0182.26001


http://www.dima.unige.it/~grandis/HX1.pdf

http://www.dima.unige.it/~grandis/HX1.pdf





http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528202/





http://arxiv.org/abs/1003.1342






http://www.math.uiuc.edu/K-theory/0316/

combinatorial homotopy theory - richard...

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