The Dendroidal Dold-Kan Correspondence

MSc Mathematics Master Thesis

Author Examiner Marco Pievani dr. Hessel Posthuma

Supervisor Second examiner dr. Gijs Heuts dr. Raf Bockland ii If you have a garden and a library, you have everything you need.

Cicero, 106 BCE - 43 BCE

The aim of theory really is, to a great extent, that of systematically organizing past experience in such a way that the next generation, our students and their students and so on, will be able to absorb the essential aspects in as painless a way as possible, and this is the only way in which you can go on cumulatively building up any kind of scientiﬁc activity without eventually coming to a dead end.

Sir Michael F. Atiyah, 1929 - 2019

Abstract We introduce simplicial sets and review the Dold-Kan correspondence, an equivalence between the category of chain complexes and the category of simplicial abelian groups. Then, we present the theory of DK-triples, which allows to prove a more general kind of equivalences of Dold-Kan type. After familiarizing with trees, operads and dendroidal sets, we generalize the previous ideas to equip the category of trees with an opportune DK-triple. Finally, this leads to the dendroidal Dold-Kan correspondence, an equivalence between the category of dendroidal chain complexes and the category of dendroidal abelian groups.

Acknowledgments I want to thank my parents for giving me the amazing opportunity of studying my Master’s abroad. During this time I have always been aware of how lucky I was, pursuing my dream of studying what I truly love, you made it possible and I owe you this. I want to thank my supervisor Gijs for making me passionate about algebraic topology in the course he taught, but especially for his patience in supervising my work on this thesis. Last but not least, my biggest thanks must go to the fellow students I met during these two years: Pirates from KdVI, class- mates from Utrecht, Leiden and Barcelona. I have learnt more from you than from any textbook ever. Thank you for sharing with me the beauty of mathematics, thank you for always being there to help me, and thank you for the uncountably many hours spent together consuming chalk on blackboards. I will never forget this time of pure joy together.

iii iv Contents

Introduction vii

1 The Simplicial Dold-Kan Correspondence 1 1.1 The simplex category ∆ ...... 1 1.1.1 Combinatorial properties of ∆ ...... 2 1.2 Simplicial objects ...... 4 1.2.1 Chain complexes ...... 5 1.2.2 Connection with algebraic topology ...... 6 1.3 Three different chain complexes ...... 7 1.3.1 Semi-simplicial abelian groups ...... 11 1.4 The correspondence ...... 11 1.4.1 Γ quasi-inverse of N ...... 12 1.4.2 Proof of the correspondence ...... 13

2 DK-triples 17 2.1 Categorical preliminaries ...... 17 2.1.1 Free pointed categories ...... 17 2.1.2 Quotient categories ...... 19 2.2 DK-triples ...... 20 2.3 The main theorem ...... 22 2.4 Revisiting the simplicial Dold-Kan correspondence ...... 24

3 Trees 29 3.1 Basics on trees ...... 29 3.2 Operads ...... 32 3.3 The category of trees Ω ...... 34 3.3.1 Elementary faces and degeneracies ...... 36 3.3.2 Combinatorial properties of Ω ...... 37 3.3.3 Linear faces ...... 40 3.4 Dendroidal abelian groups & dendroidal chain complexes ...... 41 3.4.1 Dendroidal chain complexes ...... 42 3.4.2 A description via reduced trees ...... 43

4 The Dendroidal Dold-Kan Correspondence 47 4.1 The planar case ...... 47 4.2 DK-triple on Ω ...... 50 4.3 The correspondence ...... 54 4.4 Final comments ...... 55

Popular Summary 59

Bibliography 59

v vi CONTENTS Introduction

Chain complexes are a very central object in pure mathematics, arising unilaterally in every area from algebra to geometry and topology. On the other hand, simplicial sets, i.e. presheaves on the simplex category ∆, form a quite simple but elaborated structure which can serve as a model in algebraic topology and category theory, especially in connection with higher categories. The Dold-Kan correspondence is an equivalence of categories between the category of simplicial abelian groups sAb, i.e. abelian presheaves on ∆, and the category of (non-negatively graded) chain complexes Ch≥0. It is named after Albrecht Dold and Daniel Kan who independently worked on it around 1957. The standard proof of this equivalence is given via the construction of the normalization functor N : sAb → Ch≥0 which expresses a concrete ‘simpliﬁcation’ of the simplicial structure.

Generalizing the category of linear orders ∆ to the category of planar rooted trees Ωp, of which the former is in fact a full subcategory, it is possible to generalize the theory of simplicial sets to the theory of planar dendroidal sets, in parallel with the generalization of ordinary categories to non-symmetric operads. In [GLW11] it was proved a generalization of the Dold-Kan correspondence, renamed planar dendroidal Dold-Kan correspondence. This is an equivalence of categories op between the category of planar dendroidal abelian groups, i.e. abelian presheaves on Ωp , and the brand-new category of dendroidal chain complexes. By constructing a variation of the normalization functor, their proof mimics closely the one from the simplicial correspondence.

In [Wal19] the process of generalizing the Dold-Kan correspondence is brought to a further level. By equipping a category C with the combinatorial structure of a so-called DK-triple, it is possible to ‘simplify’ the shape of the category to obtain the normalized pointed category NC. This simpliﬁcation is inspired by the construction of the previously mentioned normalization functors, but it is operated via quotienting C modulo some subclass of arrows, and it ultimately leads to the equivalence of categories

Fun(C, A) =∼ Fun0(NC, A), for any weakly idempotent complete category A. This last result generalizes more abstractly the previous ones, embedding them into the broader context of equivalences of Dold-Kan type.

The inspiration of this thesis is trying to combine the latter two ideas to push forward the correspondence of [GLW11], but instead working in the more ﬂexible category of trees Ω, i.e. without assuming the choice of a planar structure, within the frame work of dendroidal sets and symmetric operads. By equipping Ω with the adequate DK-triple, it is indeed possible to provide a simpliﬁcation, which translates into a simpler version of dendroidal abelian presheaves called dendroidal chain complex. As ultimate goal of this work, we hence obtain the equivalence between the category of dendroidal abelian groups and the category of dendroidal chain complexes

dAb =∼ dCh, that we will name dendroidal Dold-Kan correspondence.

vii viii INTRODUCTION

Chapter 1 presents the standard proof of the Dold-Kan correspondence, as found for example in [GJ99]. Originality is hard to achieve in this kind of review, but certainly details are much more spelled out, making the proof easier to grasp. In particular, an introduction from scratch to the theory of simplicial sets is presented, with attention to the connections with algebraic topology. Chapter 2 synthesizes the main ideas of the paper [Wal19] introducing some categorical tools, such as pointed categories and quotient categories, necessary to state and understand the main theorems involved. At the end, we revisit the original Dold-Kan correspondence showing how it can fit in this more general class of equivalences. Chapter 3 provides an accessible introduction to the category of trees and dendroidal theory. This is not actually complicated, but it takes some time to get used to working in this category, and grasping its structure is the key point to get to our final result. We finally introduce the concept of dendroidal chain complexes giving some intuition about its relation with dendroidal abelian groups. Chapter 4 starts with a review of the proof of the planar correspondence as presented in [GLW11], which will serve as motivation to understand the following section. Then, we move on to showing that Ω admits a DK-triple structure and we finish by applying Walde’s result to finally get the desired dendroidal Dold-Kan correspondence. Chapter 1

The Simplicial Dold-Kan Correspondence

In this chapter we are going to prove the original Dold-Kan correspondence, spelling out all the details from the classical proof and introducing the key concepts involved. In particular, we will discuss the fundamental concept of simplicial sets, more speciﬁcally in the variation of simplicial abelian groups, which provide a purely combinatorial model for describing the singular homology functor. As a consequence of the correspondence, we ﬁnd that considering the simplicial abelian group of singular chains is equivalent to considering the singular homology chain complex, i.e. we essentially lose no information by passing from one to the other.

1.1 The simplex category ∆

Recall that a partially ordered set or poset is a set P equipped with an order relation “≤”, i.e. a relation which is a reﬂexive, antisymmetric and transitive. A poset P is called totally ordered or linear order if for any a, b ∈ P then either a ≤ b or b ≤ a.

Definition 1.1. We define the simplex category ∆ to be the category whose objects are finite linear orders and morphisms are order-preserving maps between them.

Concretely, objects in ∆ are dented by [n] for every n ∈ N and are of the form

[n] = {0 ≤ 1 ≤ · · · ≤ n − 1 ≤ n}.

A morphism α : [m] → [n] in ∆ is a weakly monotone function between linear orders, that is a map α : {0 ≤ · · · ≤ m} → {0 ≤ · · · ≤ n} such that i ≤ j implies α(i) ≤ α(j) for any i, j ∈ [m].

Observation 1.2. We can view a poset P as a small category P with set of objects Ob(P) = P and hom-sets defined by ( {∗} if a ≤ b HomP (a, b) = ∅ otherwise In other words, there is exactly one morphism i → j if and only if i ≤ j. Indeed, reflexivity provides the identity morphisms and transitivity gives a well-defined composition. Any category originated from a poset in this way will be referred to as a poset category. Furthermore, a functor between poset categories is exactly the same as an order preserving function between posets.

1 2 CHAPTER 1. THE SIMPLICIAL DOLD-KAN CORRESPONDENCE

Since objects in ∆ are totally ordered sets, we can view them as poset categories deﬁning a functor from the simplex to the category of small categories Cat

i : ∆ → Cat.

Thanks to the previous observation, this functor is full and faithful, hence we can view ∆ as a full subcategory of Cat. Thus we can identify each object [n] with the small category represented by the string 0 → 1 → · · · → n − 1 → n.

1.1.1 Combinatorial properties of ∆ The category ∆ has a rather interesting combinatorial structure which relies mainly in the pres- ence of two particular classes of morphisms: elementary faces and elementary degeneracies. ? Faces. For each 0 ≤ i ≤ n there is the injective function ( k if k < i δi : [n − 1] → [n] δi(k) = k + 1 if k ≥ i that skips the element i. These maps are called elementary face maps. ? Degeneracies. For each 0 ≤ j ≤ n − 1 there is the surjective function ( k if k ≤ j σj : [n] → [n − 1] σj(k) = k − 1 if k > j that hits twice the element j, with j and j + 1, and once all the others. These maps are called elementary degeneracies.

[0] [1] [2] [3] ...

Figure 1.1: Elementary faces and degeneracies.

Remark 1.3. Many authors, for instance [GJ99] and [Rie11], call these maps cofaces and code- generacies to stress that they go in the “opposite” direction and call faces, degeneracies the ones induced by them applying a contravariant functor on ∆. We follow this convention here, as done in [HM18], because it will not be confusing and also will agree with the generalization to the context of trees. Elementary faces and degeneracies behave in a particularly nice way under composition. Ba- sically, they compose according to some ﬁxed rules called cosimplicial identities. Their proof is nothing more than a tedious veriﬁcation which, as in every serious textbook, is left as an exercise.

Lemma 1.4 (Cosimplicial identities). Elementary face maps and degeneracies satisfy the following identities: δjδi = δiδj−1 for i < j

σiσj = σj−1σi for i < j

 δ − σ if i < j − 1  j 1 i σiδj = id if i = j − 1 or i = j  δjσi−1 if i > j 1.1. THE SIMPLEX CATEGORY ∆ 3

In particular, from the last one we see that any elementary degeneracy has exactly two sections given by elementary face maps. This also shows that elementary face maps and degeneracies are respectively split monomorphisms and split epimorphisms in ∆. The other key property about elementary faces and degeneracies consists in the fact that any other map in ∆ can be written as a composition of these.

Lemma 1.5 (Epi-Mono factorization). Any morphism f : [m] → [n] in ∆ can be factored as f = h ◦ g where g is a composition of elementary degeneracies and h is a composition of elementary face maps.

Proof. We start by showing that an injective map h : [m] ,→ [n] is the composition of elementary face maps. First, we must have m ≤ n otherwise h cannot be injective; moreover, if m = n then we have only the map id[m] which can be seen as an empty composition of elementary face maps. For m < n we can write n = m + i with i ≥ 1 and proceed by induction on i. For i = 1 we only have the elementary face maps [m] ,→ [m + 1]. Suppose the inductive hypothesis holds true for i = k − 1 and consider the case i = k. Let h : [m] ,→ [n] be an injective map with n = m + k, then we have exactly k elements of [n] not in the image of h: pick one, say j ∈ [n], and consider the face map δj : [n − 1] → [n]. Since im(h) ⊆ im(δj) we can invert δj and deﬁne

0 −1 h := δj ◦ h : [m] → [n − 1].

By deﬁnition h0 is injective and by inductive hypothesis it is a composition of elementary face 0 maps. Finally, we clearly have h = δj ◦ h so we conclude that h is a composition of elementary face maps.

We now show that a surjective map g : [m] [n] is the composition of elementary degeneracies. Again, we immediately notice that m ≥ n otherwise surjectivity is impossible; then if m = n we only have the map id[m] which can be seen as an empty composition of elementary degeneracies. For m > n we can write m = n + i with i ≥ 1 and proceed by induction on i. For i = 1 we only have the elementary degeneracies [m] [m − 1]. Assume the inductive hypothesis for i = k − 1 and consider the case i = k. Let g : [m] [n] be a surjective map with m = n + k, then there will be a j ∈ [n] such that |g−1(j)| ≥ 2. Let y, y + 1 ∈ [m] be two consecutive elements mapped to j by g, and consider the elementary degeneracy σj : [n + 1] → [n]. Then deﬁne a map g0 : [m] → [n + 1] by sending ( g(x) if x ≤ y x 7→ g(x) + 1 if x > y

Then g0 is surjective and hence by inductive hypothesis is a composition of elementary degen- 0 eracies, moreover g = σj ◦ g so we conclude that g is a compositon of elementary degeneracies. Lastly, we observe that any map f : [m] → [n] factors as a surjection followed by an injection:

g h f : [m] −−− [k] ,−−−→ [n].

Indeed, let k = |im( f )| − 1 so that we have im( f ) = {i0 ≤ · · · ≤ ik} ⊆ [n]. Then we can deﬁne the surjective map g and the injective map h:

−1 g : [n] [k] mapping f (ij) 7→ j

h : [k] ,→ [m] mapping j 7→ ij. Taking their composition we clearly have f = h ◦ g and we are done because g is a composition of elementary degeneracies and h is a composition of elementary face maps. 4 CHAPTER 1. THE SIMPLICIAL DOLD-KAN CORRESPONDENCE

These two lemmas depict completely the combinatorial structure of the simplex category: any injective map is a composition of elementary face maps and hence a split monomorphism, and dually any surjective map is a composition of elementary degeneracies and hence a split epimorphism.

Remark 1.6. There is a slight asymmetry between the two classes of structural maps: there are more face maps than degeneracies! For example, between [n − 1] and [n] we have n + 1 face maps, but only n degeneracies in the other direction. At the end of the day, we could phrase this mild difference by saying that “pairing faces and degeneracies into section-retraction pairs, one face map is left out”. This subtlety might look innocuous but is essentially the core of the Dold-Kan correspondence. Furthermore, as we will see in the next chapter, we could generalize this correspondence to other combinatorial settings where the same phenomenon of asymmetry occurs. This will be the investigated in depth with the tools of DK-triples.

We close this section by presenting a classical connection with the world of topology and a signiﬁcant ingredient for the deﬁnition of singular homology that we will encounter in the next section.

Example 1.7. Recall that for every n ≥ 0 the standard n-simplex ∆n is deﬁned as:

n n+1 ∆ := {(t0,..., tn) ∈ R | t0 + ··· + tn = 1 , ti ≥ 0 ∀i}

m n For any map α : [m] → [n] in ∆ we can deﬁne the map α∗ : ∆ → ∆ by

α∗(t0,..., tm) = (s0,..., sn) with sj = ∑ ti i∈α−1(j)

Roughly speaking, in the jth entry we sum the coordinates with indices mapped to j by α. In particular, for an elementary face map δi : [n − 1] → [n], the maps

n−1 n (δi)∗ : ∆ → ∆ embed ∆n−1 into ∆n as face opposite to the ith vertex. For an elementary degeneracy σj : [n] → [n − 1] the induced map

n n−1 (σj)∗ : ∆ → ∆ collapses ∆n onto ∆n−1 by projecting parallely to the line connecting the jth vertex to the j + 1th. In other words, we have just described a covariant functor ∆• : ∆ → Top sending [n] 7→ ∆n.

1.2 Simplicial objects

Deﬁnition 1.8. A simplicial object in a category C is a contravariant functor ∆op → C.

As in every functor category, natural transformations play the role of morphisms, so we obtain the category of simplicial objects in C indicated by sC = Fun(∆op, C).

Unpacking the deﬁnition, a simplicial object X in C can be concretely described as a sequence of ∗ objects in C Xn := X([n]) indexed by n ≥ 0 and equipped with morphisms α : Xn → Xm for every map α : [m] → [n] in ∆. In particular, functoriality implies

∗ (id[n]) = id : Xn → Xn

∗ ∗ ∗ β α (αβ) = β α : Xn → Xk for [k] −→ [m] −→ [n]. 1.2. SIMPLICIAL OBJECTS 5

A morphism ϕ between two simplicial objects X and Y in C is then a sequence of morphisms ∗ ϕn : Xn → Yn in C (n ≥ 0) compatible with every map α for α : [m] → [n], that is the following square commutes for any choice of such maps: α∗ Xn Xm

ϕn ϕm α∗ Yn Ym

By the Epi-Mono factorization (Lemma 1.5), we might as well focus only on the structural maps of ∆ that generate all the other morphisms, namely elementary faces and degeneracies. Hence we deﬁne the following maps, called respectively face maps and degeneracy maps:

∗ di := (δi) : Xn → Xn−1 for 0 ≤ i ≤ n ∗ sj := (σj) : Xn−1 → Xn for 0 ≤ j ≤ n − 1

These two classes of maps satisfy some ﬁxed composition rules called simplicial identities, which are obtained simply by dualizing the cosimplicial identities holding in ∆. Their proof follows immediately by (contravariant) functoriality applied to the relations from Lemma 1.4.

Lemma 1.9 (Simplicial identities). Face maps and degeneracy maps satisfy the following identities:

didj = dj−1di for i < j

sjsi = sisj−1 for i < j  s d − if i < j − 1  i j 1 djsi = id if i = j − 1 or i = j  si−1dj if i > j As a consequence of these, we recognize that face maps and degeneracy maps form section- retraction pairs, but swapping their role compared to what happened in ∆: face maps are now split epimorphisms and degeneracy maps are split monomorphisms.

Example 1.10. Two of the most important cases of simplicial objects arise when considering the category of sets Set and the category of abelian groups Ab. These are actually the only two cases that we will encounter in this thesis. We then deﬁne the category of simplicial sets sSet and the category of simplicial abelian groups sAb. These two categories come naturally into play in the ﬁeld of algebraic topology, it is not surprising that when dealing with a simplicial set (Xn)n≥0 the element of the set Xn are called n-simplices.

1.2.1 Chain complexes After having introduced simplicial abelian groups from the more general point of view of simplicial objects, we now recall some basic notions about chain complexes, the other main character involved in the Dold-Kan correspondence.

Deﬁnition 1.11. A (non-negatively graded) chain complex (C•, ∂•) is a sequence of abelian groups

∂n ∂1 · · · → Cn −→ Cn−1 → · · · → C1 −→ C0 connected by group homomorphisms, called differentials or boundary maps, satisfying ∂n ◦ ∂n+1 = 0 for every n ≥ 0. 6 CHAPTER 1. THE SIMPLICIAL DOLD-KAN CORRESPONDENCE

The condition ∂n ◦ ∂n+1 = 0 is equivalent to im(∂n+1) ⊆ ker(∂n), so we can deﬁne the quotient

Hn(C•) := ker(∂n)/im(∂n+1) called the n-th homology group of C•. This is perhaps the most important tool for studying chain complexes as it measures how far a complex is from being exact, which means satisfying the condition im(∂n+1) = ker(∂n).

A morphism of chain complexes f : C• → D•, referred to as a chain map, is a sequence of level-wise homomorphisms fn : Cn → Dn for every n ≥ 0 compatible with the differentials, in the sense that the following diagram commutes:

C C ∂n+1 ∂n ... Cn+1 Cn Cn−1 ...

fn+1 fn fn−1 D D ∂n+1 ∂n ... Dn+1 Dn Dn−1 ...

In particular, the commutation of the previous diagram implies that:

C D C D • ∂n (x) = 0 ⇒ ∂n ( fn(x)) = fn−1(∂n (x)) = 0 ⇒ fn(x) ∈ ker(∂n ) C C D D • y = ∂n+1(x) ⇒ fn(y) = fn(∂n+1(x)) = ∂n+1( fn+1(x)) ⇒ fn(y) ∈ im(∂n+1) C D C D In other words, fn(ker(∂n )) ⊆ ker(∂n ) and fn(im(∂n+1) ⊆ im(∂n+1), consequently every chain map induces a well-defined map between homology groups in every degree. It is interesting to notice how the definition of chain map resembles pretty much the definition of natural transformation and in particular, the one of morphism between simplicial objects. We define the category of (non-negatively graded) chain complexes Ch≥0 to have chain complexes as objects and chain maps as morphisms.

Remark 1.12. Chain complexes are often indexed over the integers Z, but in this thesis we will only consider non-negatively graded chain complexes, i.e. indexed over the natural numbers N. Therefore, from now on, we might as well assume this convention and omit the speciﬁcation.

1.2.2 Connection with algebraic topology We now discuss the probably most important and motivating example for the study of simplicial structures: the construction of singular homology. Recall that ∆n indicates the standard topological th n-simplex and δi∗ is the map embedding into it the i face of dimension n − 1. Let X be a topological space and let n ≥ 0, a singular n-simplex in X is a continuous map σ : ∆n → X, and n S(X)n := {σ : ∆ → X | σ is continuous } denotes the set of singular n-simplices in X. For any 0 ≤ i ≤ n we can restrict a singular n-simplex th n n−1 n to the i face of ∆ by precomposing with the inclusion δi∗ : ∆ ,→ ∆ . We hence get a map of sets n di : S(X)n → S(X)n−1 (σ : ∆ → X) 7→ σ ◦ δi∗, which acts by restricting a singular simplex to the ith face. Similarly, we can deﬁne the precom- n n−1 position with each σj∗ : ∆ ∆ to get maps of sets ( ) → ( ) ( n → ) 7→ ◦ sj : S X n−1 S X n σ : ∆ X σ σj∗. Let now A be an abelian group, for all n ≥ 0 we deﬁne the group

Cn(X; A) := A[Sn(X)] 1.3. THREE DIFFERENT CHAIN COMPLEXES 7 of singular n-chains by taking the A-linearization of S(X)n, which means linear combinations of singular n-simplices with coefﬁcients in A. The set-theoretic maps di that we have just deﬁned extend linearly to homomorphisms between these abelian groups.

Then we deﬁne the singular boundary operator by taking the alternating sum of the di’s

n i ∂ = ∂n := ∑(−1) di : Cn(X; A) → Cn−1(X; A) i=0

One can check (we will do it in the next section) that these homomorphisms satisfy ∂n ◦ ∂n+1 = 0 for every n ≥ 0, so the sequence Cn(X; A) forms a chain complex called the singular chain complex. Finally, we can take the n-homology groups of the singular chain complex to get the nth singular homology group of X Hn(X; A) := Hn(Cn(X; A)).

n Chain X ⇒ S(X)n = {σ : ∆ → X} ⇒ Cn = {n-chains} ⇒ Complex ⇒ Hn

Figure 1.2: Summary of the construction of singular homology.

It should now be evident where the simplicial structure is ’hidden’ in this procedure, but let us point it out. The construction of the set of singular n-simplices in X produced a contravariant functor over the simplex category ∆, i.e. it forms a simplicial set by assigning

[n] 7→ S(X)n and δi 7→ di σj 7→ sj.

Moreover, this assignment is functorial over the category of topological spaces Top in the sense that for any continuous map f : X → Y there is a well-defined morphism between simplicial sets, i.e. a natural transformation. In fact, by postcomposition we can define the induced morphism for all n ≥ 0 S(X)n → S(Y)n σ 7→ f ◦ σ and this commutes with all the faces and degeneracies because they act by precomposition. Also, identities on spaces induce identity morphisms and compositions of maps induce compositions of natural transformations. We can briefly rephrase this final observation by saying that there is a functor called the total singular complex

Sing : Top → sSet X 7→ (S(X)n)n≥0 f 7→ f ◦ −.

After all this machinery is set up, the rest is relatively easy. We transform the simplicial set S(X)• into the simplicial abelian group C(X; A)• by applying the universal construction of the A-linearization functor. Then we construct a very natural chain complex out of the simplicial abelian and ﬁnally we apply the homology functor to our complex.

i Sing A[−] ∑(−1) di Hn Hn(−; A) : Top −−−→ sSet −−−−−→ sAb −−−−−→ Ch≥0 −−−−→ Ab

Figure 1.3: Singular homology as a composition of functors.

1.3 Three different chain complexes

From the previous example, we have already seen a way to go from simplicial abelian groups to chain complexes. We will now study in detail that construction and two others closely related 8 CHAPTER 1. THE SIMPLICIAL DOLD-KAN CORRESPONDENCE which will be central in the proof of the Dold-Kan correspondence. Throughout this section we will consider a simplicial abelian group A = (An) ∈ sAb.

First, we deﬁne the Moore complex CA•, the one that is used to build singular homology:

n C i CAn := An ∂n := ∑(−1) di. i=0

It is immediate to observe that this is well-deﬁned since taking the alternating sum of the face maps di : An → An−1 gives again a map An → An−1. Then, we must check that this is actually a chain complex, namely ∂2 = 0. Let us do the calculations:

n n+1 ! n n+1 C C i j i+j ∂n ◦ ∂n+1 = ∑(−1) di ∑ (−1) dj = ∑ ∑ (−1) di ◦ dj i=0 j=0 i=0 j=0 n i n n+1 i+j i+j = ∑ ∑(−1) di ◦ dj + ∑ ∑ (−1) di ◦ dj i=0 j=0 i=0 j=i+1 n i n n+1 i+j i+j = ∑ ∑(−1) di ◦ dj + ∑ ∑ (−1) dj−1 ◦ di i=0 j=0 i=0 j=i+1 n i n n i+j i+j0 = ∑ ∑(−1) di ◦ dj − ∑ ∑ (−1) dj0 ◦ di = 0. i=0 j=0 i=0 j0=i

We used the simplicial identity didj = dj−1di for i < j to get the third line, then we substituted the indexed j0 = j − 1 to obtain the last one. Finally, one realizes that the two last terms cancel out because one is a summation over 0 ≤ j ≤ i ≤ n and the other over 0 ≤ i ≤ j0 ≤ n.

Next, we deﬁne the normalized complex NA• by taking the intersection of the kernels of all the th face maps di : An → An−1, except the n :

n−1 \ N n NAn := ker(di) ⊆ An ∂n := (−1) dn. i=0

For n = 0 we have an empty intersection, so we set NA0 := A0. We must check that this is N well-deﬁned chain complex, namely that ∂n (NAn) ⊆ NAn−1. Let x ∈ NAn, then for i < n − 1 we have N n n di(∂n (x)) = (−1) di ◦ dn(x) = (−1) dn−1 ◦ di(x) = 0 N by the same simplicial identity as before, so we see that ∂n (x) ∈ NAn−1. Now we should also 2 check that ∂ = 0. Let x ∈ NAn+1, then we have:

N N n+n+1 ∂n ◦ ∂n+1(x) = (−1) dn ◦ dn+1(x) = −dn ◦ dn(x) = −dn(0) = 0 by using again the same simplicial identity and the fact that x ∈ ker(dn). Notice that the sign n N C (−1) is actually irrelevant in this calculation, although it is needed to have ∂ = ∂ |NA• . In particular, the latter fact could have also been used to deduce ∂2, but more importantly we have NA• ⊆ CA•, i.e. the normalized complex is a subcomplex of the Moore complex.

Last, we deﬁne the degenerate complex DA• generated by the elements in the image of the degeneracy maps sj : An−1 → An:

n−1 = ( ) ⊆ D = C| DAn : ∑ im sj An ∂n : ∂n DA• . j=0 1.3. THREE DIFFERENT CHAIN COMPLEXES 9

For n = 0 there are no degeneracies, so we set DA0 := {0}. Again, we have to check that this is D well-deﬁned, i.e. ∂n (DAn) ⊆ DAn−1. Let x = sj(y) ∈ An for some y ∈ An−1 and 0 ≤ i ≤ n, then we have: n D i ∂n (x) = ∑(−1) disj(y) i=0 j−1 n i j j+1 i = ∑(−1) disj(y) + (−1) djsj(y) + (−1) dj+1sj(y) + ∑ (−1) disj(y) i=0 i=j+1 j−1 n i j i = ∑(−1) sidj−1(y) + (−1) (¡y − y¡) + ∑ (−1) si−1dj(y). i=0 i=j+1

The middle terms in the last line, obtained from the simplicial identity djsj = id = dj+1sj, cancel out. The remaining terms, obtained by swapping sj and di thanks to the previously D used simplicial identity, show that ∂n (x) ∈ DAn−1. We can extend this result to the whole D 2 group generated by the degeneracies to conclude ∂n (DAn) ⊆ DAn−1. The property ∂ = 0 follows from the fact that this holds for ∂C, of which ∂D is just a restriction. Thence, as for the normalized complex, we have that DA• ⊆ CA•, meaning that the degenerate complex is a subcomplex of the Moore complex.

Last but not least, we would like to ensure that all these three constructions provide functors sAb → Ch≥0. In practice we must verify that a morphism in sAb naturally deﬁnes a morphism in Ch≥0. To see this, we just have to realize that a natural transformation, i.e. a morphism between simplicial abelian groups, is a family of maps commuting with any structural morphism (faces and degeneracies). In particular, this family of maps will commute with the differential of the Moore complex, which means it will form a chain map. Furthermore, we clearly have that the identity maps will induce the identity maps, and compositions of maps will induce compositions of induced maps. For the normalized and degenerate complex this holds as well since they are subcomplexes. One may ask why constructing three different chain complexes: is one not enough? The answer is that these three are intimately linked together, as shown in the following lemma. Moreover, as a little spoiler, we can anticipate that the normalization functor N : sAb → Ch≥0, i.e. the construction of the normalized complex from a simplicial abelian group, will provide the equivalence of categories proving the Dold-Kan correspondence.

=∼ Lemma 1.13. The morphism NAn ⊕ DAn −→ An = CAn induced by the inclusions is an isomorphism. Proof. We are going to show the isomorphism level-wise for each n ≥ 0.

We start with the case n = 0: by definition we have DA0 = {0} and NA0 = A0, so this ∼ tautologically shows DA0 ⊕ NA0 = A0. Let us now consider the general case for a fixed n ≥ 1. We need to define partial versions of the normalized and degenerate complexes in degree n: for k < n define

k k Nk An := ∩i=0 ker(di) Dk An := ∑ im(si). i=0 Notice that we have the following chain of inclusions:

N0 An ⊇ N1 An ⊇ · · · ⊇ Nn−1 An = NAn D0 An ⊆ D1 An ⊆ · · · ⊆ Dn−1 An = DAn. ∼ We will prove by induction on k that Nk An ⊕ Dk An = An for every 0 ≤ k < n and for k = n − 1 we will get the desired result. 10 CHAPTER 1. THE SIMPLICIAL DOLD-KAN CORRESPONDENCE

We start the induction with k = 0. In this case we have N0 An = ker(d0) and D0 An = im(s0). ◦ = Using the simplicial identity d0 s0 idAn−1 , the result follows from these two facts:

• N0 An ∩ D0 An = {0}: let x = s0(x0) for some x0 ∈ An−1 and x ∈ ker(d0). Then 0 = d0(x) = d0s0(x0) = x0 so we conclude x = s0(0) = 0.

0 0 • N0 An + D0 An = An: let x ∈ An, deﬁne x := s0(d0(x)) so that d0(x ) = d0(x) meaning 0 0 0 0 0 x − x ∈ ker(d0), thus we have x = (x − x ) + x with (x − x ) ∈ N0 An and x ∈ D0 An.

Now, let 0 < k < n and assume the inductive hypothesis is true for k − 1, meaning we already ∼ ∼ have a splitting Nk−1 An ⊕ Dk−1 An = An. We want to show that Nk An ⊕ Dk An = An.

k k−1 • Nk An ∩ Dk An = {0}: let x = ∑i=0 si(xi) = ∑i=0 si(xi) + sk(xk) for some xi ∈ An−1 and di(x) = 0 for i ≤ k. By the inductive hypothesis on An−1 we have the splitting ∼ An−1 = Nk−1 An−1 ⊕ Dk−1 An−1

so we can write xk = α + β with α ∈ Nk−1 An−1 and β ∈ Dk−1 An−1, i.e. di(α) = 0 for every k−1 i ≤ k − 1 and β = ∑i=0 si(bi) for some bi ∈ An−2. Then applying sk we get

k−1 k−1 sk(xk) = sk(α) + sk(β) = sk(α) + ∑ sksi(bi) = sk(α) + ∑ sisk−1(bi) i=0 i=0

using the simplicial identity sjsi = sisj−1 for i < j; in particular, we have sk(β) ∈ Dk−1 An. Now we substitute in the original expression of x gathering together all the elements in the image of each si:

k−1 k−1 x = ∑ si(xi + sk−1(bi)) + sk(α) = γ + sk(α) with γ := ∑ si(xi + sk−1(bi)) ∈ Dk−1 An. i=0 i=0

Moreover, from the simplicial identity disj = sj−1di if i < j, we have

di(sk(α)) = sk−1(di(α)) = 0 for i < k

since α ∈ Nk−1 An−1. Hence, we have sk(α) ∈ Nk−1 An and x ∈ Nk An ⊆ Nk−1 An, so in particular γ ∈ Nk−1 An too, hence γ = 0 by inductive hypothesis. Finally we have x = sk(α), but applying dk we have 0 = dk(x) = dk(sk(α)) = α so we conclude x = 0.

• Nk An + Dk An = An: let x ∈ An and write it as x = α + β with α ∈ Nk−1 An and β ∈ Dk−1 An 0 0 by inductive hypothesis. Deﬁning α := sk(dk(α)) ∈ Dk An, then we have dk(α ) = dk(α) 0 and so α − α ∈ ker dk. Using simplicial identities, for i < k we get

0 di(α ) = di(sk(dk(α))) = sk−1(di(dk(α))) = sk−1(dk−1(di(α))) = sk−1(di(0)) = 0,

0 0 0 so α ∈ ker(di) for every i < k. Ultimately, we are able to write x = (α − α ) + (β + α ) 0 0 where (α − α ) ∈ Nk An and (β + α ) ∈ Dk An as desired.

Roughly speaking, we could rephrase the result from this lemma by saying that “the normalized complex NA• keeps track of non-degenerate information of CA•”. Moreover, one can prove (see for example [GJ99]) that the inclusion NA• ,→ CA• is a chain homotopy equivalence. Even though we will not need this last fact, it tells us that somehow NA• loses no homological information of CA•. 1.4. THE CORRESPONDENCE 11

1.3.1 Semi-simplicial abelian groups We discuss now an intermediate structure between simplicial abelian groups and a chain complexes, and its connections with these two.

Let ∆in be the wide subcategory of ∆ whose morphisms are only the injective maps. In other words, we preserve the face maps and identites of ∆ and forget about the degeneracies. op Deﬁnition 1.14. A semi-simplicial abelian group is a functor ∆in → Ab. We indicate with ssAb the category of semi-simplicial abelian groups.

We have the natural inclusion ∆in ,→ ∆ and precomposing with it we get the forgetful functor F : sAb → ssAb which restricts a simplicial abelian group to ∆in, forgetting about all the degeneracy maps. We can observe that the normalization complex could be deﬁned for any semi-simplicial abelian group since it does not involve the degenercy maps in the deﬁnition. Put in another way, the normalization functor N : sAb → Ch≥0 factors through the forgetful functor F as pictured in the diagram below.

N sAb Ch≥0

F N ssAb

Last but not least, we can construct a rather special semi-simplicial abelian group (Cn)n≥0 from a chain complex (C•, ∂•). Not surprisingly, we set C([n]) := Cn and for 0 ≤ i < n we assign the elementary face maps ∗ di = (δi) := 0 : Cn → Cn−1 to be the zero morphism, while setting

∗ n dn = (δn) := (−1) ∂n : Cn → Cn−1.

One can check that this is indeed a well-defined semi-simplicial abelian group, i.e. cosimplicial 2 identities are satisfied. In particular, from ∂ = 0 we find dn ◦ dn+1 = 0 as an instance of n the simplicial identity dn ◦ dn+1 = dn ◦ dn. The sign (−1) is just a formality to compensate the minus sign in the definition of the normalized complex. This construction gives rise to a well-defined functor G : Ch≥0 → ssAb, which is the right inverse of the normalization functor N : ssAb → Ch≥0 since we obviously have ◦ = N G idCh≥0 . In particular, G is a full and faithful functor which allows us to see chain complexes as a rather special case of semi-simplicial abelian groups.

1.4 The correspondence

Recall that a functor F : C → D is an equivalence of categories if there exists a functor G : D → C ' ' and two natural isomorphisms of functors η : G ◦ F −→ idC and e : F ◦ G −→ idD. We have now all the ingredients to state the Dold-Kan correspondence.

Theorem 1.15 (Dold-Kan). The normalization functor N : sAb → Ch≥0 is an equivalence of categories.

To prove the result, we will construct another functor Γ : Ch≥0 → sAb that will be the quasi- inverse of the normalization functor N. 12 CHAPTER 1. THE SIMPLICIAL DOLD-KAN CORRESPONDENCE

1.4.1 Γ quasi-inverse of N

Given a chain complex C = (C•, ∂•) ∈ Ch≥0 we aim to construct a simplicial abelian group, that is a functor ∆op → Ab, thus we must specify its action on objects and morphisms of ∆.

• Objects: in degree n, i.e. as image of [n] ∈ ∆, we deﬁne M ΓC([n]) = Γn(C) := Ck [n][k]

meaning that we take the direct sum of as many copies of Ck as the number of surjections [n] [k] in ∆; obviously, in this indexing it is considered k ≤ n simply because there cannot be surjective maps [n] → [m] with m > n. Let us make some explicit computations to clarify the construction:

– Γ0(C) = C0 corresponding to id[0]

– Γ1(C) = C1 ⊕ C0 corresponding respectively to id[1] and to [1] [0]

– Γ2(C) = C2 ⊕ C1 ⊕ C1 ⊕ C0 corresponding respectively to id[2], to the two degeneracies σ0, σ1 : [2] [1] and to the unique surjection [2] [0].

• Morphisms: let ν : [m] → [n], and let τ : [n] [k] be a surjection indexing the component σ ι Ck in degree n. Using the Epi-Mono factorization we can express τν as [m] −− [j] ,−→ [k] as shown in the following commutative square:

[m] ν [n]

σ τ [j] ι [k]

Now we deﬁne a homomorphism Ck → Cj induced by ι as follows:  id if j = k  Ck k (Ck → Cj) = (−1) ∂k if j = k − 1 and ι = δk   0 otherwise.

In other words, this map is the zero morphism except for the cases in which ι is an identity or the top-degree elementary face map. Next, composing with the inclusion of the component indexed by σ into the direct sum, we get a map Ck → Cj ,→ Γm(C). Specifying a map for every component, i.e. such an indexing surjection τ, by universal property we obtain a map ∗ ν : Γn(C) → Γm(C), which we deﬁne to be the one contravariantly induced by ν.

To ensure that this construction produces a simplicial abelian group, we need to verify that we actually obtain a functor ΓC : ∆op → Ab.

First of all, it is not hard to see that id[n] induces the identity map Γn(C) → Γn(C): in this case the previous commutative square in ∆ has identity maps horizontally, so by the way we have → deﬁned it we are considering identity maps idCk : Ck Ck in each component, which of course sum up to an identity in degree n. µ ν Then, consider a composition [l] −→ [m] −→ [n] in ∆, we want to see that (νµ)∗ = µ∗ν∗. As before, we build the following commutative diagram in order to deﬁne µ∗, ν∗ and (νµ)∗: 1.4. THE CORRESPONDENCE 13

µ [l] [m] ν [n]

[i] θ [j] ι [k]

∗ ∗ The bottom horizontal maps induce the componentwise morphisms θ : Cj → Ci, ι : Ck → Cj ∗ ∗ ∗ ∗ ∗ ∗ and (ιθ) : Ck → Ci. Now we reason by cases to realize that (ιθ) = θ ι : if θ and ι are identities, so it is going to be (ιθ)∗; similarly, if one among θ∗ and ι∗ is a zero map, so it is going to be (ιθ)∗. The only case we must be careful about is when both θ∗ and ι∗ are the chain complex ∗ 2 differential (with sign): in this case (ιθ) = 0 = ∂ since (C•, ∂•) is a chain complex. Ensured this ’functoriality’ on each summand, it naturally extends to the direct sum and we ﬁnd that the ∗ ∗ ∗ two morphisms (νµ) , µ ν : Γn(C) → Γl(C) coincide.

Last but not least, we now want to check that the construction of Γ actually provides a functor

Γ : Ch≥0 → sAb, that is a chain map f• : C• → D• deﬁnes a natural transformation Γ( f ) : ΓC → ΓD. In fact, we have componentwise maps fn : Cn → Dn, i.e. maps between each summands of ΓC and ΓD, which commute with the chain complexes differentials; therefore, by universal property, we will get maps Γ( f )n : Γn(C) → Γn(C) for each n ≥ 0 and they commute with structural maps of the simplicial abelian groups from the way we deﬁned the structure of Γ.

Observation 1.16. Let A = (An)n≥0 be a simplicial abelian group, consider the composition of functors Γ ◦ N : sAb → sAb. For the simplicial abelian group Γ(NA•) in degree n we have:

M Γn(NA•) = NAk = NAn ⊕ NAn−1 ⊕ · · · [n][k]

For every surjection σ : [n] [k] we have a term NAk associated to σ in the direct sum and a ∗ structural map σ : Ak → An being this a simplicial abelian group. Moreover, NAk naturally includes into Ak by deﬁnition, so combining these together we can consider the composition

σ∗ NAk ,→ Ak −→ An for any such term NAk indexed by a surjection σ : [n] [k]. Then, by universal property, this family of morphisms from each summand uniquely induces the sum map:

M ∗ Ψn : Γn(NA•) = NAk → An mapping (xσ)σ 7→ ∑ σ (x). [n][k] σ Furthermore, considering this construction for every n ≥ 0 leads to a natural transformation Ψ = (Ψn)n≥0 given by Ψ : Γ ◦ N → idsAb.

1.4.2 Proof of the correspondence

We will break the proof of the main theorem into the two following propositions each of which shows the desired natural isomorphism.

Proposition 1.17. The morphism Ψn : Γn(NA•) → An is an isomorphism for every n ≥ 0. In =∼ particular, the family of morphisms Ψ = (Ψn)n≥0 provides a natural isomorphism Γ ◦ N −→ idsAb. 14 CHAPTER 1. THE SIMPLICIAL DOLD-KAN CORRESPONDENCE

Proof. Let A ∈ sAb be a simplicial abelian group, we are going to prove that the natural map constructed in the previous observation Ψ = (Ψn) is an isomorphism by induction on n. As a base case we consider n = 0, for which we only have the surjection id[0] : [0] → [0] which gives Γ0(NA•) = NA0 = A0, so the statement is tautologically satisﬁed. As inductive hypothesis, let us assume that Ψm is proven to be an isomorphism for every m < n; we now want to show that Ψn is again an isomorphism.

• Ψn is surjective: When restricting Ψn to the component indexed by id[n] we have ∗ id[n] NAn ,→ An −−→ An

which is just the inclusion NAn ,→ An, hence we get NAn ⊆ imΨn. By the inductive hypothesis Ψn−1 is surjective onto An−1, hence for any degeneracy sj(x) ∈ An there is a y ∈ Γn−1(NA) such that Ψn−1(y) = x. Therefore, by the naturality of Ψ, we have sj(x) = sj(Ψn−1(y)) = Ψn(sj(y)), which implies DAn ⊆ imΨn. From the canonical splitting ∼ An = NAn ⊕ DAn, we conclude that An ⊆ imΨn since both DAn and NAn lie in imΨn. ∗ • Ψn is injective: suppose that Ψn maps (xσ) ∈ Γn(NA) to zero, i.e. ∑ σ (x) = 0. For σ:[n][k] k < n, a surjection σ : [n] [k] has a canonical maximal section

νσ : [k] ,→ [n] νσ(i) := max{j ∈ [n] | σ(j) = i}. 0 Given two surjections σ, σ : [n] [k] we say 0 σ ≤ σ ⇔ νσ(i) ≤ νσ0 (i) for all i ∈ [k]. 0 0 0 In particular, σ νσ = id[k] ⇒ σ ≤ σ since νσ0 is the maximal section of σ , hence νσ ≤ νσ0 . If there exists a surjection τ : [n] [k] such that the component xτ 6= 0, choose a maximal such τ (with respect to the ordering just deﬁned). Then consider the morphism ∗ ντ : Γn(NA) → Γk(NA) which on the τ component is deﬁned by the diagram

ν [k] τ [n]

id[k] τ id[k] [k] [k]

The bottom arrow induces the identity map id : NAk → NAk, which then is composed with the inclusion NAk ,→ Γk(NA) of the component indexed by id[k]. In particular, by the ∗ simplicial structure of Γ we see that the component of ντ (xσ) ∈ Γk(NA) indexed by id[k] is precisely xτ ∈ NAk. But then by commutativity we have ∗ (xσ) ∈ ker Ψn ⇒ ντ (xσ) ∈ ker Ψk ⇒ xτ = 0

where Ψk is considered to be injective by inductive hypothesis. So we must have xσ = 0 for all surjections σ 6= id[n]. The only remaining case is indeed σ = id[n], but Ψn on the NA NA ,→ A x = component n indexed by id[n] is just the inclusion n n, so we conclude id[n] 0 too, and hence (xσ) = 0 meaning Ψn is injective.

Now, let C = (C•, ∂•) be a chain complex, we are going to examine the composition of functors N ◦ Γ : Ch≥0 → Ch≥0. We can compare this with the Moore complex composed with Γ: from the canonical inclusion of normalized complex into the Moore complex for every n ≥ 0 we get the maps: M Φn : N(ΓC)n ,→ C(ΓC)n = Γn(C) = Ck = Cn ⊕ · · · [n][k] 1.4. THE CORRESPONDENCE 15

Proposition 1.18. The natural inclusions Φn : N(ΓC)n ,→ C(ΓC)n have image the factor Cn indexed ∼ = ( ) ◦ −→= by id[n]. In particular, the family of morphisms Φ Φn n≥0 is a natural isomorphism N Γ idCh≥0 . M Proof. We need to compute N(ΓC) in degree n. It consists of the elements of Γn(C) = Ck [n][k] that are killed by the maps di for i < n. We claim that this consists exactly of the elements of Cn considered as summand indexed by id[n].

• Cn ⊆ N(ΓC)n: by the simplicial structure of Γ, for i < n the map δi : [n − 1] → [n] induces the zero map di : Γn(C) → Γn−1(C), therefore Cn ⊆ ker(di) for every i < n and so Cn ⊆ N(ΓC)n.

• N(ΓC)n ⊆ Cn: note that for an indexing surjection σ : [n] [k] with k < n we have a factorization of the form σi [n] −− [n − 1] [k],

so the factor Ck of Γ(C)n indexed by σ lies in the image of the degeneracies D(ΓC)n. By the canonical splitting we have ∼ M N(ΓC)n ⊕ D(ΓC)n = Γ(C)n = Ck = Cn ⊕ · · · [n][k]

from which we deduce N(ΓC)n ⊆ Cn.

Finally, the combination of these two facts gives us N(ΓC)n = Cn, hence we are able to conclude ◦ = N Γ idCh≥0 . 16 CHAPTER 1. THE SIMPLICIAL DOLD-KAN CORRESPONDENCE Chapter 2

DK-triples

In this chapter we introduce the concept of DK-triples, which allows to state a broad and very general variety of theorems called of Dold-Kan type. We will provide the necessary machinery from category theory and then get speciﬁcally into key constructions involved in the main theorem. Finally, we will see how to adapt this theory to re-obtain the classical Dold-Kan correspondence and observe how this more abstract point of view can give more insight into a new reinterpretation of the original result.

2.1 Categorical preliminaries

Deﬁnition 2.1. A category C is called pointed if it has a zero object, i.e. if there exists an object 0 ∈ C which is both initial and terminal in C. A functor between pointed categories is called pointed if it sends the zero object to the zero object.

As for any universal property in category theory, an object is initial or ﬁnal up to unique isomorphism, so we are consciously allowed to call a zero object just the zero object, as already done in the previous deﬁnition.

We denote by Cat0 the category of (small) pointed categories and pointed functors between them. This comes with a canonical forgetful functor

U : Cat0 → Cat.

Given two pointed categories C, C0 we denote Fun0(C, C0) ⊂ Fun(C, C0) the full subcategory spanned by the pointed functors.

2.1.1 Free pointed categories

Construction 2.2. (Free pointed category) Let C be a category, we deﬁne a pointed category C+ by freely adjoining a zero object to C. Freely means that we are formally adding one object which we impose to behave according to the properties we want it to have; explicitly this means the following:

— The objects of C+ are the same as the objects of C with an additional object, indicated with 0, that obviously will be the zero object.

— For every object x ∈ C+ we set

HomC+ (x, 0) = {0} and HomC+ (0, x) = {0}

17 18 CHAPTER 2. DK-TRIPLES

Given two objects x, y ∈ C we set

HomC+ (x, y) := HomC (x, y) ∪˙ {0}

where 0 denotes the unique zero morphism x → 0 → y.

— The composition in C+ is induced by the composition in C.

i The free pointed category C+ comes equipped with the canonical inclusion functor C −→C+ which is obviously not full.

We can rephrase construction 2.2 into a functor

0 + : Cat → Cat acting as follows:

• Objects: a category C is sent to the free pointed category C+ deﬁned as above.

• Morphisms: for a functor F : C → P with P a pointed category, we can deﬁne the pointed functor F+ : C+ → P which we call the free pointed extension of F. F+ coincides with F when restricted to the original category C, plus it sends 0 ∈ C+ to 0 ∈ P. In particular, for any two objects x, y ∈ C we have

F+ (x → 0 → y) 7−−→ (Fx → 0 → Fy)

so we observe that F+ maps the zero morphisms in C+ to the zero morphisms in P. More generally, given a functor F : C → D, then + maps F to the free pointed extension of F i the composition C −→D −→D+, i.e. we get the pointed functor (F ◦ i)+ : C+ → D+.

Notice that even for a pointed category P we have P 6=∼ P+. This happens because we are adding formally the new zero object 0 to the category changing all the previous hom-sets: in particular, an old zero object x cannot be ﬁnal nor initial anymore since it has now two endomorphisms, namely the identity idx and the new zero morphism x → 0 → x. In the next lemma we should ﬁnally see why construction 2.2 is reasonably called free.

Lemma 2.3 (Universal Property of the FPC). The ’free pointed category’ functor

0 + : Cat → Cat is left adjoint to the forgetful functor U : Cat0 → Cat.

Proof. To prove the adjunction we must show that, for any category C and for any pointed category P, we have a natural bijection

0 Fun (C+, P) =∼ Fun(C, UP) = Fun(C, P)

From the LHS to the RHS we have the canonical restriction of any pointed functor along the i canonical inclusion C −→C+

i (F : C+ → P) 7→ (F|C : C −→C+ → P)

From the RHS to the LHS we have the free pointed extension taking advantage of the fact that P is already pointed (G : C → P) 7→ (G+ : C+ → P) 2.1. CATEGORICAL PRELIMINARIES 19 where G+ is deﬁned to agree with G on C and is forced to map 0 ∈ C+ to 0 ∈ P.

These two maps are inverses of each other: for G ∈ Fun(C, P) we have (G+)|C = G since 0 by definition G+ and G agree on the original category C, while for F ∈ Fun (C+, P) we have (F|C )+ = F because a pointed functor is completely characterized by its action on the original category C, since the pointed condition is independent and satisfied by definition. Therefore, taking the free pointed extension of F|C we find a pointed functor that agrees with F when restricted to C and is imposed to map 0 ∈ C+ 7→ 0 ∈ P, i.e. we get back F itself.

2.1.2 Quotient categories Let N be the poset category over the natural numbers as in Chapter 1, a (non-negatively graded) chain complex in a pointed category P is a functor Nop → P, which we can visualize as

• ←−−−−•d ←−−−−•d ←−−−−•d ←−−−−·d · · satisfying the condition that any composition of more than one arrow d equals the zero morphism in P. Recalling that chain maps are just natural transformations between such functors, we can see the category of chain complexes in P as a full subcategory of Fun(Nop, P). We would like to make this deﬁnition more precise and clean, in order to eventually see chain complexes simply as functors over a certain category where the trivialization condition ∂2 = 0 is already encoded. To do so, we now want to specify what it means to quotient over a particular set of morphisms within a category.

Construction 2.4. Let C be a category, a two-sided ideal S ⊆ C is a set of arrows which is closed under the action of Ar(C) on the right by pre-composition and on the left by post-composition, in other words C ◦ S ◦ C ⊆ S. C In this setup we deﬁne the quotient category S to be the following pointed category: C — The objects of S are the objects of C plus an additional zero object 0. C — Morphisms in S are determined by setting

HomC (x, y) Hom C (x, y) := = { f ∈ HomC (x, y) | f ∈/ S} ∪˙ {0} S S for any pair of objects x, y ∈ C with the composition induced by the one in C.

C Similarly to the situation with quotient of algebraic structures, the category S comes canonically C equipped with a projection functor p : C → S which acts as identity on all objects and morphisms not in S, while it sends all and only the arrows in S to the zero map. Not surprisingly, this C construction has the universal property expressing the fact that S is the biggest pointed category trivializing the arrows in S, as stated in the following lemma.

Lemma 2.5 (Universal Property of the Quotient). Given a functor F : C → P, where P is pointed, such that all the morphisms in the two sided ideal S ⊂ C are sent to the zero morphism by F, there exists a unique pointed functor F : C/S → P such that F = F ◦ p.

C F P

p F C/S

Proof. We can give a quite obvious recipe to cook up the functor F: 20 CHAPTER 2. DK-TRIPLES

– Objects: for any object x ∈ C/S, F(x) = Fx and clearly F(0) = 0 ∈ P.

– Morphisms: for any arrow f ∈ Hom C (x, y) we set F f = F f ∈ HomP (Fx, Fy) and without S much choice the zero morphisms must be mapped to the zero morphisms in P.

Immediately we can see that the factorization F = F ◦ p holds. Furthermore we can observe that, since p acts identically on everything but S, these assignments were in fact the only ones which could allow such a factorization. The uniqueness follows from the fact that another functor G : C/S → P satisfying F = G ◦ p would need to satisfy the same assignments we have given for F, so it would necessarily agree with it.

Using the quotient construction, we can now deﬁne an auxiliary category taking quotients over the poset category N. We will consider the ideal

(→→) := { f : m → n | n − m ≥ 2} for any m, n ∈ N. In other words, this is the ideal generated by any composition of two or more arrows.

Deﬁnition 2.6. Let P be a pointed category, a (non-negatively graded) chain complex in P is a N op pointed functor (→→) → P. So we deﬁne the category

N op Ch (P) := Fun0 , P ≥0 (→→) of (non-negatively graded) chain complexes in P.

2.2 DK-triples

Let us consider a category B equipped with two wide subcategories E, E∨ ⊂ B, i.e. such that both of them contain all isomorphisms in B. In particular, this implies that they both contain every object of B, so what we are really going to focus on are the morphisms in these two subcategories. We will call Epis the arrows in E and dual Epis the arrows in E∨, and we will graphically indicate them respectively with two-headed arrows and with tailed arrows ; this notation is not random at all, indeed it should recall the usual convention for surjective and injective maps and, as we will see, this will be the case in the examples. For an object b ∈ B we will consider the co-slice category E(b) of Epis under b and the slice category E∨(b) of dual Epis over b: in these categories the objects are arrows of E∨, E and morphisms are commutative diagrams.

• • •

Figure 2.1: Dual Epis over b and Epis under b.

We still need some more terminology before getting to the real deﬁnition of DK-triples.

∨ f g ∨ • We will consider the right ideal Sing := E6' ◦ B = {−→−→| f ∈ ArB, g ∈ E not invertible}, the B-action is on the right by pre-composition. Arrows in B are called singular if they lie in Sing and regular if they lie in Reg := B \ Sing. 2.2. DK-TRIPLES 21

h f • We will consider the left ideal B ◦ E6' = {−→−→| f ∈ ArB, g ∈ E not invertible}, the B-action is on the left by post-composition. We then deﬁne M := B \ (B ◦ E6') and call an arrow Mono if it lies in M.

Finally, for each object b ∈ B we have a pairing − ◦ − : E(b) × E(b)∨ → ArB given by composition: g f ( f , g) 7→ f ◦ g = • b •. This induces a pairing on the isomorphism classes of the categories, denoted by:

∨ h−, −ib : π0E(b) × π0E(b) → π0ArB.

Deﬁnition 2.7. The data B := (B, E, E∨) is called a DK-triple if it satisﬁes the following axioms:

T1. Every morphism f in B can be written uniquely (up to unique isomorphism) as a composition of the form f = e∨ ◦ f¯ ◦ e0 where e0 ∈ E, e∨ ∈ E∨ and f¯ ∈ M ∩ Reg

0 e f¯ e∨ • • −→• •.

T2. For every object b ∈ B the pairing h−, −ib can be represented as a ﬁnite square matrix with invertible arrows on the diagonal and non-invertible arrows below it. More concretely, there exists a ∼ ∼ ∨ number n ≥ 1 and bijections π0E(b) = {1, . . . , n} = π0E (b), and the pairing h−, −ib induces th an n × n matrix (aij)1≤i,j≤n, where aij = i ◦ j with j standing for the j dual Epi over b and i for the ith E-pi under b (both according to the opportune linear order provided by the above bijections); the obtained matrix has the form

' ?...?   .. .. .  6' . . .     . . .   . .. .. ?  6' · · · 6' '

with values in π0ArB: it has invertible morphisms on the diagonal and non-invertible morphisms below it (no conditions are required on the morphisms above the diagonal).

f g ∨ ∨ T3. The set E ◦ E = { | f ∈ E, g ∈ E } is closed under composition. T4. Composing two regular Monos gives a Mono (not necessarily regular), i.e. (M ∩ Reg) ◦ (M ∩ Reg) ⊂ M.

T5. The singular arrows form a left module over M with left action given by post-composition, i.e. M ◦ Sing ⊂ Sing.

Moreover, a DK-triple is called diagonalizable if the pairing matrix can be made diagonal modulo non- isomorphisms and reduced if B = E∨ ◦ E, i.e. if every morphism in B can be factored through Epis and dual Epis.

From the deﬁnition, we can already see that a DK-triple is a combinatorial structure that encodes a symmetry in the factorization system. Epis and dual Epis balance each other cancelling out, so this symmetry will allow to simplify the data of a functor on such a category focusing only on what is left out. For instance, as we will see, this symmetry is the one that exists in ∆ between degeneracies and all but one face maps. 22 CHAPTER 2. DK-TRIPLES

2.3 The main theorem

We introduce now the most important construction induced by a DK-triple, which will be a simplified version of the category, cancelling out the symmetry. Given a DK-triple B we construct a pointed category N0(B) called the normalized pointed category of B which is defined as the quotient M N := 0 M ∩ Sing of M by the two-sided ideal M ∩ Sing. Observe that by definition we have Reg := B \ Sing, i.e. B = Sing ∪˙ Reg, hence M = M ∩ B = (M ∩ Sing) ∪˙ (M ∩ Reg) and therefore only the regular Monos will ’survive’ in the quotient.

Explicitly we can describe N0 as follows:

• The pointed category N0 has a zero object 0 and for each object b ∈ B an object b¯ ∈ N.

• For every pair of objects b¯, b¯0 ∈ N, we have the hom-set

0 0 0 0 N0(b¯, b¯ ) := M(b, b )/(M ∩ Sing) = (M ∩ Reg)(b , b) ∪ {b¯ → 0 → b¯ }.

• Composition in N0 is induced by composition in B and it is well-deﬁned because of axioms T4 and T5.

We are now ready to state the main theorem from [Wal19], which is an abstract and very general version of the Dold-Kan correspondence.

Theorem 2.8 (Correspondences of Dold-Kan type). Let B = (B, E, E∨) be a DK-triple with associated normalized pointed category N0 = N0(B), then for every weakly idempotent complete additive category A we have an equivalence of categories

0 ' 0 Fun (B+, A) ←−−−→ Fun (N0, A).

The proof of this theorem goes far beyond the scope of this Master’s thesis, but it might be interesting to spend a few words to brieﬂy explain how the equivalence is constructed. First, we construct the auxiliary pointed category V = V(B) ! N R M Sing \ B V := 0 0 := Sing 0 B+ 0 B+ associated to the N0-B+-bimodule R0 := Sing \ B. Explicitely, the category V consists of the following:

• The objects of V are given by objects n ∈ N, the objects b ∈ B and a zero object 0, that is Ob(V) = Ob(N0) t{0} Ob(B+).

• Morphisms come in three different kinds of hom-sets:

– the hom-sets in V between two objects both belonging to N0 or to B+ is inherited from N0 and B+ respectively.

– The zero morphism is the only arrow in V from an object in N0 to an object in B+. – The hom-set in V between an object b ∈ B and an object n ∈ N is deﬁned to be

V(b, n) := R0(b, n) := Sing\B(b, [n]) = Reg(b, [n])∪{˙ b → 0 → n}. 2.3. THE MAIN THEOREM 23

• Composition in V is induced by the composition in N0 and B+; the composition 0 0 0 0 N0(n, n ) × R0(b, n) × B+(b , b) −→ R0(b , n )

is well-deﬁned because M ◦ Sing ◦ B ⊆ Sing from axiom T5.

This construction is made in such a way that both B+ and N0 embed fully faithfully into it

B+ ,→ V ←- N0.

Then we can consider the restriction functors along these two embeddings

0 0 0 0 ResB+ : Fun (V, A) → Fun (B+, A) and ResN0 : Fun (V, A) → Fun (N0, A). The remarkable thing is now that these two functors admit respectively, a left adjoint given by left Kan extension and a right adjoint given by right Kan extension. The composite adjunction

Res LKE N0 0 0 0 Fun (B+, A) ⊥ Fun (V, A) ⊥ Fun (N0, A) ResB+ RKE is then proven to be an adjoint equivalence of categories and gives the result.

Remark 2.9. The original theorem, as stated and proved in [Wal19], is expressed in the language of ∞-categories, but here we propose this special case since we are only interested in ordinary categories. Also, the theorem holds for the very weak hypothesis of A being just an additive and weakly idempotent complete category. For our purposes, much less generality will be needed since we will focus on the case A = Ab, the category of abelian groups, or better its dual Abop. Speciﬁcally, Ab is an abelian category so it is clearly additive, it is bicomplete having all products/coproducts and equalizers/coequalizers, and hence it is idempotent complete, i.e. any idempotent splits as retraction-section composition; so, in particular Abop is additive and weakly idempotent complete. This ensures that we can apply the previous theorem to our cases of interest.

We can notice that in the theorem we only deal with pointed categories, while the Dold-Kan correspondence arises on the category ∆ which is not pointed. The missing step is ﬁlled in the next corollary.

Corollary 2.10 (Abstract Dold-Kan Correspondence). Each DK-triple B = (B, E, E∨) induces a natural equivalence of categories

op ' 0 op Fun(B , Ab) ←−−−→ Fun (N0(B) , Ab)

Proof. By the remark, we know that Abop satisﬁes the hypothesis of Theorem 2.8, so we have

0 op ' 0 op Fun (B+, Ab ) ←−−−→ Fun (N0(B), Ab ).

On the other hand, we know from the adjunction of Lemma 2.2 that

op op 0 op Fun(B, Ab ) = Fun(B, UAb ) =∼ Fun (B+, Ab ) and composing these two equivalences we get

op ' 0 op Fun(B, Ab ) ←−−−→ Fun (N0(B), Ab ).

Finally, dualizing the source categories instead of the target ones, we get the result. 24 CHAPTER 2. DK-TRIPLES

2.4 Revisiting the simplicial Dold-Kan correspondence

We will now show how to equip the simplex category ∆ with a DK-triple in order to apply Corollary 2.10 and obtain another proof of Theorem 1.15. The idea we have to keep in mind is that the simpliﬁcation done by the normalization functor must be translated into the simpliﬁcation done with the normalized pointed category construction.

In the classical Dold-Kan correspondence, given a simplicial abelian group (Xn)n≥0, the normalization functor forgets about all the maps of the simplicial structure and keeps track only of the nth face maps which become the differentials of the chain complex:

dn : Xn → Xn−1 ⇒ ∂n : NXn → NXn−1.

Actually, there is also the sign (−1)n, but it is only relevant to express the normalized complex as a subcomplex of the Moore complex, indeed we could omit it and still get a chain complex since ∂2 = 0 occurs for another reason, as explained in Chapter 1. So only the nth face maps are surviving and all the rest is ‘forgotten’; from the DK-triple perspective, this translates into the fact that ‘the rest’ is what encodes the symmetry, and the dn’s are what is left out from it.

With this motivation in mind, it is not surprising that the Epis will be the surjective maps, i.e. the morphisms generated by the degeneracies, and the dual Epis will be the injective maps not involving dn’s, i.e. the morphisms generated by the face maps δk : [n − 1] → [n] for 0 ≤ k < n. So, let us set E ⊂ ∆ to be the wide subcategory of surjective maps and let E∨ ⊂ ∆ be the wide subcategory of those injective maps that preserve the maximal element, i.e. injections [m] ,→ [n] mapping m 7→ n. From the factorization lemma, such an injection is a composition th of elementary face maps, but none of them can be of the n face map δn : [n − 1] → [n], otherwise the composition would not preserve the maximal element. On the other hand, a composition of elementary face maps not of top-degree gives an injective map which preserves the maximal element. These two observations combined show that the dual Epis are precisely those morphisms spanned by all but the top-degree elementary faces, as anticipated above.

Remark 2.11. This set up might look a bit asymmetric at first: why on one hand do we take all surjections, while on the other hand we consider only the maximal preserving injections? First of all, we should notice that surjections already preserve the maximal elements by definition, so this extra specification would be useless. Moreover, we should keep in mind Remark 1.6 and remember that in ∆ there are more injective maps than surjective. For example, between [n] and [n − 1] we have n elementary degeneracies and n + 1 elementary faces in the other directions, of which precisely n preserve the maximal elements. Recalling that elementary faces and degeneracies can be combined into section-retraction pairs, specifying the condition maximal- preserving for injections allows us to select the right number of sections we need. In fact, we leave out only one unpaired injection, which in our setting is δn : [n − 1] → [n], namely the only elementary face which does not preserve the maximal element.

Let us prove that the data ∆ = (∆, E, E∨) forms a DK-triple. First we have to identify the special classes of morphisms.

∨ • Sing := E6' ◦ B = {· · · →} is the right ideal of morphisms whose factorization ends with a dual Epi, i.e. a maximal-preserving injection which is not an isomorphism. Since the only isomorphisms in ∆ are the identities id[n], using the unique factorization, we have that a map belongs to Sing if and only if it can be expressed as a composition ending with an elementary face map δk : [n − 1] → [n] for 0 ≤ k < n, which will skip k ∈ [n]. Rephrasing this, a map [m] → [n] is singular if and only if there is a k < n which is not in its image. A morphism without this property is then regular, by deﬁnition of Reg = B \ Sing. 2.4. REVISITING THE SIMPLICIAL DOLD-KAN CORRESPONDENCE 25

• B ◦ E6' = {→ · · · } is the left ideal of morphisms whose factorization starts with an Epi, i.e. a surjection, which is not an isomorphism. With the same reasoning as before, we have that a map belongs to this ideal if and only if it can be expressed as a composition starting with an elementary degeneracy, and this is equivalent to not being injective. So we have that M = Mono = B \ ◦E6' consists exactly of all the injective maps.

The set M ∩ Reg of regular Monos consists of those injective maps which are not singular. This th amounts only to the identities and the n face maps δn : [n − 1] → [n]. It is important to remark that this means exactly these maps and not the morphisms generated by them! Indeed M ∩ Reg is not closed under composition: in fact, composing two such face maps, using cosimplicial identities, we ﬁnd δn ◦ δn−1 = δn−1 ◦ δn−1 ∈ Sing.

We are now going to check that the axioms T1-T5 are satisﬁed. T1. Unique factorization: Any map f : [m] → [n] in ∆ admits a factorization of the form E∨ ◦ (M ∩ Reg) ◦ E: indeed, from the Epi-Mono factorization of Lemma 1.5 we can write f as

g h [m] −−−− im( f ) ' [k] ,−−−→ [n] where g is surjective, hence belongs to E, and h is injective. Now we just need to show that h ∈ E∨ ◦ (M ∩ Reg). We distinguish two cases: n ∈ im( f ) or n ∈/ im( f ). In the ﬁrst case, f preserves the maximal element, so also h must do it since g preserves the maximal element by ∨ surjectivity; this means h ∈ E hence we can write h = h ◦ id[k] and get the result. In the second case, f does not preserve the maximal element, so clearly also h does not; by writing the latter as = ◦ · · · ◦ = − composition of elementary face maps h δil δi1 (with l n k) at least one of these must be a top-degree elementary face map. Pick the ﬁrst such map in the order of composition, say

δiα : [iα − 1] → [iα] skipping the maximal element iα, then by assumption all the previous maps in the composition preserve the maximal element, so in particular, they have a strictly lower index. Using the simplicial identity

δj ◦ δi = δi ◦ δj−1 for i < j we can swap δiα with all the preceding ones, and in this way we lower its degree till ﬁnally getting δk+1 : [k] → [k + 1] as ﬁrst map in the composition. If there are other top-degree elementary faces in this new factorization, we can repeat this process with each of them in order to place all of them at the beginning of the composition, leaving the others coming after to be maps preserving the maximal element. Finally, we can use again the previous simplicial identity to operate the following trick on these top-degree faces placed at the beginning of the composition

· · · ◦ δk+3 ◦ δk+2 ◦ δk+1 = · · · ◦ δk+2 ◦ δk+2 ◦ δk+1 = · · · ◦ δk+2 ◦ δk+1 ◦ δk+1.

In this way we ensure that we actually have only one top-degree elementary face map in the factorization. In particular, we see that we can factor h as δk+1 ∈ M ∩ Reg followed by a composition of maximal preserving at the end, i.e. a map belonging to E∨. T2. Pairing matrix: We are going to consider now the pairing induced on the isomorphism classes of ∆, which correspond exactly to the objects since the identities are the only isomorphisms. Let m, n ∈ N with m ≤ n and let α, β : [m] [n] be dual Epis over [n]. We say that α ≤ β ⇔ α(i) ≤ β(i) for all i ∈ [a]. In particular, this order relation gives a linear order on E∨([m], [n]) for each [m], [n] ∈ ∆. Now deﬁne a map

(−)∨ : E([n], [m]) −→ E∨([m], [n])

∨ σ : [n] [m] 7→ σ : [m] [n] 26 CHAPTER 2. DK-TRIPLES sending a surjection σ to its maximal section σ∨ deﬁned by the formula

σ∨(i) := max σ−1{i}.

∨ Indeed, we immediately see that σ ◦ σ = id[m], and by definition it is maximal among the sections of σ with respect to the linear order we defined. Moreover, the map (−)∨ is well-defined because any surjection maps n 7→ m, and the formula imposes the section to send m 7→ n, so σ∨ is in fact a dual Epi. We claim that (−)∨ is a bijection.

∨ ∨ • Injective: given two surjections σ, τ : [n] [m] suppose that σ 6= τ , i.e. there exists i ∈ [m] such that σ∨(i) 6= τ∨(i), meaning

j := max σ−1{i} 6= max τ−1{i} := j0.

Without loss of generality, assume j < j0, then

τ(j0) = i = σ(j) < σ(j0)

by deﬁnition of σ∨, so we conclude σ 6= τ.

• Surjective: let δ : [m] [n] be an injection mapping m 7→ n, then we can deﬁne a surjection σδ : [n] [m] mapping {0 ≤ · · · ≤ δ(0)} 7→ 0 and for any 1 ≤ i ≤ m {δ(i − 1) + 1 ≤ · · · ≤ δ(i)} 7→ i. ∨ ∨ Then σδ ∈ E([n], [m]) and clearly (σδ) = δ. We basically deﬁned the inverse of (−) .

Now, observing that we have standard decompositions [ [ E([n]) = E([n], [m]) E∨([n]) = E∨([m], [n]) m≤n m≤n and considering simultaneously all the bijective maps E([n], [m]) → E∨([m], [n]) for all m ≤ n we can combine them to obtain a bijection

' (−)∨ : E([n]) −−−−→ E∨([n]).

Now we want to order these two sets to ﬁll a table with the dual Epis in the top row and the Epis in the leftmost column, matching the entries to complete the pairing matrix with their composition. We order E∨([n]) by ordering the subsets of the decomposition with increasing source (even though this order turns out to be irrelevant)

E∨([0], [n]) , E∨([1], [n]) ,..., E∨([n − 1], [n]) , E∨([n], [n]) and ordering the maps in each component decreasing according to the linear order deﬁned at the beginning of this paragraph. Using the bijection (−)∨ we get a rule to give a linear order to E([n]) as well, so that we ﬁll the leftmost column. An example of such a table for [2] is pictured below.

[0] [2] δ0 : [1] [2] δ1 : [1] [2] id[2]

[2] [0] id[0] [1] [0][1] [0][2] [0] σ0 : [2] [1] 0 7→ 1 id[1] id[1] σ0 σ1 : [2] [1] 0 7→ 1 {0, 1} 7→ 1 id[1] σ1 id[2] 0 7→ 2 δ0 δ1 id[2] 2.4. REVISITING THE SIMPLICIAL DOLD-KAN CORRESPONDENCE 27

We clearly have isomorphisms (identities) on the diagonal, since we match section-retraction pairs. Considering a matching of a dual Epi with an Epi having different source and target, then their composition cannot be an isomorphism. So we are left to consider the matching of dual Epi E∨([m], [n]) with Epis E([n], [m]) which correspond to square submatrices appearing as blocks aligned on the diagonal. Let us consider an entry below the diagonal in such a submatrix: supposing we have σ ◦ δ = id in this entry, this means δ is a section for σ, but on the other hand, by the way we ordered the dual Epis, we have σ∨ < δ contradicting the maximality of σ∨. So we conclude that the pairing matrix has isomorphisms on the diagonal and non-isomorphisms below it. T3. E∨ ◦ E closed under composition: from the factorization E∨ ◦ (M ∩ Reg) ◦ E shown for T1, we see that E∨ ◦ E consists precisely of maps preserving the maximal element, corresponding to the case in which the factor in M ∩ Reg is just an identity. We conclude by observing that the composition of two maps preserving the maximal element is still a map which preserves the maximal element. T4. (M ∩ Reg) ◦ (M ∩ Reg) ⊂ M: M consists of injective maps and the composition of two injective maps is still injective. T5. M ◦ Sing ⊂ Sing: let f : [m] → [n] be singular, which means equivalently that there exists n 6= i ∈ [n] such that i ∈/ im( f ), and let g : [n] → [p] be a Mono, i.e. simply an injective map. Then we have g(i) 6= p, otherwise g would not be injective since it should map also i + 1, ··· , n to p being monotone. Hence g(i) ∈/ im(g ◦ f ) which means that g ◦ f is singular.

Now that we have veriﬁed that the ∆ = (∆, E, E∨) provides an actual DK-triple on the simplex category ∆, we can apply Corollary 2.10 that gives us the equivalence of categories

op ' 0 op Fun(∆ , Ab) ←−−−→ Fun (N0(∆) , Ab).

The only thing left to do is to understand how the normalized pointed category N0(∆) looks like, as explained in the following proposition.

∨ Proposition 2.12. The normalized pointed category N0(∆) induced by the DK-triple ∆ = (∆, E, E ) is N isomorphic to the category (→→) .

N Proof. We start by constructing a pointed functor ϕ : (→→) → N0(∆); as always, we specify its action on objects and morphisms.

• Objects: for any n ∈ N we set ϕ(n) = [n] and ϕ(0) = 0, i.e. the zero object is mapped to the zero object.

th • Morphisms: the arrow n − 1 → n is mapped to the n face map δn : [n − 1] → [n].

ϕ is functorial and well-deﬁned because of the simplicial identity

δn ◦ δn−1 = δn−1 ◦ δn−1 the composition of two maximal face maps is singular, and hence equivalent to the zero morphism in the quotient modulo M ∩ Sing. Roughly speaking, in both categories ∂2 = 0 holds. Finally, this is an isomorphism because we can easily exhibit the inverse ϕ−1, since all the assignments that we gave to deﬁne ϕ are clearly bijective. 28 CHAPTER 2. DK-TRIPLES

With this last isomorphism, we can now update our previous equivalence of categories to

op ' N Fun(∆op, Ab) ←−−−→ Fun0 , Ab (→→) and renaming each side properly we obtain the original Dold-Kan correspondence

' sAb ←−−−→ Ch≥0(Ab). Chapter 3

Trees

In this chapter we introduce the category of trees Ω, a direct generalization of the simplex category ∆ which embeds into it. We will also need to discuss the notion of operad which generalizes the concept of category allowing arrows having different arities. In this generalization flow, simplicial objects, i.e. presheaves on ∆, generalize to dendroidal objects, that are presheaves on Ω. Similarly, one can ask how to generalize chain complexes to the dendroidal case, simpler objects encoding almost the same information but with less structure, posing the question of what should be the best definition. The motivation will be to make them fit into an equivalence of categories in the flavour of the Dold-Kan correspondence.

3.1 Basics on trees

Recall that a graph is a pair (V, E) consisting of a set V of vertices and a set E of edges which are two-elements subsets of V, since an edge can be identiﬁed by the set of two vertices which connects. A graph with half-edges is a graph in which edges are also allowed to be attached to only one vertex at one end, so in this case E is a collection of one- or two-elements subsets of V.

Deﬁnition 3.1. A tree is a ﬁnite graph with half-edges in which there is precisely one path connecting any two edges. An edge is called inner if it connects two vertices and outer if it is attached only to one vertex. A rooted tree is a tree with a distinguished outer edge, called the root, whereas the remaining outer edges are called leaves.

Rephrasing the deﬁnition, a tree is a ﬁnite graph (with half-edges) which is connected and has no loops. For us all trees will always be rooted, so from now on we might omit the adjective. For a tree T we write E(T) for its set of edges and V(T) for its set of vertices. The following is a typical picture of a tree T with V(T) = {t, u, v, w} and E(T) = {a, b, c, d, e, f , g, h, i}.

w d c e h i u f v b g t a

The choice of a root turns a tree into a directed graph, with the direction intended as ’from the leaves to the root’. With this in mind, for any vertex v in a tree we can speak of the set input edges in(v) and the unique output edge out(v). For instance, in the previous example we have that b, f , g are the input edges of t and a is its output, while w has no input edges at all and

29 30 CHAPTER 3. TREES output i. We define the valence of a vertex to be the number of its input edges and call a vertex unary if its valence is 1 and nullary if its valence is 0. We say that a tree is open if it has no nullary vertices and it is closed if it has no leaves. The direction ’leaves → root’ can also be used to define a partial order on E and V: given two edges e, f we set e ≤ f if and only if the unique directed path going from f to the root passes through e, and similarly for vertices. We call two edges e and f incomparable if they are not related in the order, and denote this by e ⊥ f . In this partial order setup, not completely conventional, the root is the unique minimal element and the leaves are the maximal elements. Moreover, one immediately verifies that

e ≤ e0, f ≤ f 0, e⊥ f ⇒ e0⊥ f 0

In the previous example we have for instance a ≤ b ≤ c, a ≤ g ≤ i and b⊥g, therefore c⊥i.

Remark 3.2. By considering this poset structure on the set of edges of a tree, we can already get a glimpse of how the concept of tree allows to generalize totally ordered sets of ∆ to posets where not every pair of elements is comparable. To be precise, not every poset P can be the underlying set of edges of a tree: we must require that there exists a minimal element r ∈ P, representing the root, and that P satisﬁes the property that for every e ∈ P the set

P≤e := {x ∈ P | x ≤ e} is totally ordered, representing the fact that there is only one path from each edge to the root. In fact, given such poset P and considering the subset L ⊆ P of maximal elements, then there exists a tree T unique up to isomorphism with set of edges E(T) = P and leaves L(T) = L.

Deﬁnition 3.3. A planar structure on a tree T is a linear ordering on the set of input edges of every vertex of T.

Unfortunately, any picture of a tree automatically endows it with a planar structure, but we should stress that such a structure is not part of the data of the tree itself. Thus, for instance the following is a picture of the same tree as before, but corresponding to a different planar structure.

w e i h c d v f u g b t a

Example 3.4. Let us present now some particularly simple trees.

(a) The simplest tree is the one with only one edge and no vertices at all. We denote it by η.

(b) A tree is called linear if all its vertices are unary. The linear tree with n vertices and n + 1 edges, running from the root 0 to the leaf n, is denoted by Ln. There is an evident similarity 3.1. BASICS ON TREES 31

between Ln and [n] in ∆, represented on the right as a poset category.

n n

. . . .

1 1

0 0

...

C0 C1 C2 C3

Even though we will properly deﬁne morphisms of trees in the next section, we can already introduce a special case of maps called embeddings, which allow us to also talk about subtrees of a given tree and to recognize when two trees are isomorphic.

Deﬁnition 3.5. An embedding of trees ϕ : S → T consists of injective maps ϕE : E(S) → E(T) and ϕV : V(S) → V(T) such that for any vertex v of S the map ϕE gives a bijection between in(v) and in(ϕv) and ϕ(out(v)) = out(ϕv). An isomorphism of trees is an embedding which is bijective both on edges and vertices. If ϕ : S → T is an embedding and the map ϕ is just given by inclusion of subsets E(S) ⊆ E(T) and V(S) ⊆ V(T), then S is called a subtree of T.

In other words, an embedding S → T is the same as an isomorphism of S onto a subtree of T. If S is a subtree of T, then S can be obtained from T by successively pruning away external vertices and the outer edges attached to them. Note that any edge e of T determines an embedding denoted by e : η → T.

b c b c v v a a

We should observe now that some trees can have non-trivial automorphisms, i.e. isomorphisms from a tree to itself. For instance, for any permutations σ ∈ Σn on n elements, there is an isomorphism Cn → Cn swapping the leaves of the n-corolla accordingly. Moreover, it is not hard ∼ to realize that any automorphism must be of this kind so we can conclude Aut(Cn) = Σn. In the following picture we see an example of a non-trivial automorphism: it shufﬂes the leaves at the vertex v by sending a 7→ c, b 7→ a, c 7→ b and it swaps the ones at w. Actually, this kind of permutations generates the only possible automorphisms for this tree, in fact notice that v cannot be mapped to w because the former has three input edges, while the latter has only two. We must underline that planar structure in this picture is not part of the data and it is simply used to visually suggest how the map is acting. 32 CHAPTER 3. TREES

a b c d e b c a e d

v w v w

Remark 3.6. There is also a more subtle problem regarding how to distinguish an automorphism from another isomorphism. For example, consider a tree T and a tree T0 which is exactly the same except that we consider formally different sets for the edges and the vertices. Clearly T and T0 are isomorphic, and the isomorphism is essentially an identity map, so the question is: is this actually relevant? A similar issue arises when considering the category of finite sets: there are infinitely many sets with n elements and they are all canonically isomorphic; what is actually interesting are the permutations over each finite set, i.e. the isomorphisms which are non-identical automorphisms. For the same reason, we will sometimes talk generally about isomorphisms of trees, but we will keep in mind that the only relevant ones are the automorphisms.

3.2 Operads

In order to define a category whose objects are rooted trees we will need to specify what a morphism of trees is. Even though at the end of the day the description of such maps will turn out to be combinatorial, the precise definition relies on the concept of operad, that we will introduce now. The purpose of this is twofold: on one hand morphisms of trees will in fact be defined as morphisms between the associated operads, while on the other hand operads can be intuitively understood via trees. We start by giving the definition of operad, sometimes also referred to as multicategory, suggest- ing that it is a sort of category but with arrows allowing multiple inputs. We will see that it is indeed a generalization of the concept of category, just replacing the set of objects with the set of colours and considering sets of operations between them instead of sets of morphisms. Definition 3.7. An operad P consists of a set of colours cl(P) and a set of operations

P(c1,..., cn; c) for every n ≥ 0 and any sequence of colours c1,..., cn, c, thought of as taking the n inputs c1,..., cn to the output c. Moreover, we have the following additional structures:

- for each colour c ∈ cl(P) there is a unit operation 1c ∈ P(c; c), resembling the identity morphisms of ordinary categories, ∗ - for each permutation σ ∈ Σn there is a map σ : P(c1,..., cn; c) → P(cσ(1),..., cσ(n); c), usually written σ∗ p = p ◦ σ,

i i - for any sequence of colours c ,..., cn, c, and any n-tuple of sequences d ,..., d for i = 1, . . . , n, a 1 1 ki composition of operations

n γ : P(c ,..., c ; c) × P(di , ..., di ; c ) → P(d1,..., d1 , d2,..., dn ; c) 1 n ∏ 1 ki i 1 k1 1 kn i=1

usually written γ(p, q1,..., qn) = p ◦ (q1,..., qn) or just p(q1,..., qn). These structural maps satisfy some compatibility axioms expressing that composition is unital and asso- ciative, and that the symmetric group induces a right action which is compatible with the composition. In detail, they are: 3.2. OPERADS 33

- for an operation p ∈ P(c1,..., cn; c), we have γ(1c, p) = p and γ(p, 1c1 , . . . , 1cn ) = p, i i i - for operations p, q ,..., qn as above and another sequence of operations r ,..., r , where r has 1 1 ki j i output dj we have (p(q ,..., q ))(r1,..., rn ) = p(q (r1,..., r1 ),..., q (rn,..., rn )), 1 n 1 kn 1 1 k1 n 1 kn ∗ ∗ ∗ ∗ ∗ - for permutations σ, τ ∈ Σn and associated operations σ , τ as above, we have (στ) = τ σ , ∈ ∈ - for operations p, q1,..., qn as before and permutations σ Σn, τi Σki , we have ∗ ∗ ∗ ∗ σ p(τσ(1)qσ(1),..., τσ(n)qσ(n)) = (σ ◦ (τ1,..., τn)) (p(q1,..., qn)), n where σ ◦ (τ1,..., τn) is the element of the symmetric group on Σi=1ki letters formed by letting τi th permute the letters within the i block of length ki and letting σ permute then different blocks.

Let P be an operad with operations p ∈ P(c1,..., cn; c) and q ∈ P(d1,..., dk; ci) for some 1 ≤ i ≤ n. We will sometimes use the following notation:

p ◦i q = p ◦ci q := p(1c1 ,..., q . . . , 1cn ) th That is, p ◦i q is the composition of p with q in the i variable. We usually visualize an operation with n inputs as an n-corolla. In the picture below p represents an operation with inputs the colours c1, c2, c3, c4 and output c.

c2 c3 c1 c4 p c

Since we are not assuming the planar structure, we should beware that this picture does not strictly represent only the operation p itself, but also the operations σ∗ p for any permutation σ ∈ Σ4, corresponding to all the possible shufﬂes of the inputs. This visualization gives a tangible intuition about the link between trees and operads: the idea is that trees parametrize the operations of an operad and the ways in which they can be composed.

Given two operads P and Q, a morphism of operads φ : P → Q consists of a map between the corresponding sets of colours φ : cl(P) → cl(Q) and for each sequence of colours c1,..., cn, c in P a map between the corresponding sets of operations

φc1,...,cn,c : P(c1,..., cn; c) → Q(φ(c1),..., φ(cn); φ(c)) such that it respects units and composition of operations and it is compatible with the Σn-actions. We define category of operads, denoted by Oper, to be the category whose objects are operads and whose arrows are the morphisms of operads we have just defined. Remark 3.8. Operads generalize small categories since any small category can be seen as an operad with only unary operations: in fact, given a small category C we can define an operad C by assigning clr(C) := Ob(C) and assigning to any pair of colours c and d a set of unary operations C(c, d) := HomC (c, d). Moreover, this standard construction defines a full and faithful embedding j : Cat → Oper from the category of small categories to the category of operads. Indeed, functors between small categories, i.e. morphisms in Cat, are the same thing as morphisms in Oper between the corresponding unary operads. 34 CHAPTER 3. TREES

We will now show how to construct an operad from a tree. Construction 3.9. Given a tree T, we deﬁne the associated operad Ω(T) as follows: – the set of colours is the set of edges E(T),

– the operations are freely generated by the vertices of T, that is if v ∈ V(T) is a vertex with input edges c1,..., cn and output edge c, then Ω(T)(c1,..., cn; c) = {v}.

In other words, an operation p ∈ Ω(T)(c1,..., cn; c) is a subtree of T with leaves c1,..., cn and root c; so vertices represent the most elementary operations, i.e. the simplest building blocks to obtain all the others via composition. Note that for any sequence of colours c1,..., cn, c there is at most one such subtree, therefore the sets of operations of Ω(T) are either empty or singletons. Moreover, for different ordering of such a sequence of inputs, we get different operations according to the Σn-action. Example 3.10. Consider the rooted tree T pictured below.

c2 c3 e1 c1 e2 u v b d t a

For each vertex there is a generating operation:

u ∈ Ω(T)(c1, c2, c3; b) , v ∈ Ω(T)(e1, e2; d) and t ∈ Ω(T)(b, d; a).

Moreover, we have the following composed operations:

t ◦b u ∈ Ω(T)(c1, c2, c3, d; a) , t ◦d v ∈ Ω(T)(b, e1, e2; a) and t(u, v) ∈ Ω(T)(c1, c2, c3, e1, e2; a).

Also, applying a permutation to the inputs we get another operation corresponding to same subtree: for instance, the operation τ∗(t) ∈ Ω(T)(d, b; a) corresponds to the exact same subtree of the operation t, the only difference being that we have listed its leaves in a different order.

The operad Ω(T) is an example of a free operad since its operations are freely generated by the vertices of T. This means that for any operad P, any morphism of operads φ : Ω(T) → P is uniquely determined by its effect on the colours E(T) and on the elementary operations V(T). In fact, given such a morphism, then for every vertex v of T with inputs c1,..., cn and output c, we assign the operation φ(v) ∈ P(φ(c1),..., φ(cn); φ(c)). Conversely, any assignment of a colour of P to each edge of T and a compatible assignment of an operation of P to each vertex of T extends uniquely to a map of operads φ. Finally, notice also that any planar structure on T ﬁxes an order for the input edges of each vertex, hence it ﬁxes a set of generating operations, the others being obtained just by Σn-action.

3.3 The category of trees Ω

Deﬁnition 3.11. The category of trees Ω is the category whose objects are rooted trees and whose morphisms S → T are maps between the associated operads Ω(S) → Ω(T). Observation 3.12. By deﬁnition we have that Ω is a full subcategory of the category of operads, so we have a full and faithful embedding

Ω → Oper. 3.3. THE CATEGORY OF TREES Ω 35

Recall also that we have a full and faithful embedding

∆ → Cat by viewing a totally ordered set as a poset category and realizing that monotone maps are just functors between posets. Moreover, we have that ∆ embeds naturally into Ω by considering linear trees: we can deﬁne a functor i : ∆ → Ω by assigning [n] 7→ Ln. The corresponding operad Ω(Ln) is nothing more than just the poset category n → · · · → 1 → 0. Hence a morphisms [m] → [n] in ∆, that is a functor between the corresponding poset categories, corresponds bijectively to morphisms of linear trees Lm → Ln. In other words, via the embedding i we are allowed to identify ∆ with the full subcategory of linear trees in Ω. These embeddings are compatible with the previously deﬁned full and faithul embedding j : Cat → Oper, in the sense that they form the following commutative diagram (up to isomorphism).

∆ Cat i j Ω Oper

We can informally rephrase this square by saying that trees generalize totally ordered sets in the same ways as operads generalize small categories. Let us consider now the morphisms in Ω more closely. By the deﬁnition of an associated operad, a morphism ϕ : S → T sends edges to edges and each vertex v ∈ V(S) to subtrees ϕ(v) of T. If e1,..., en are the input edges of v and e the output, then ϕ(e1),..., ϕ(en) are the leaves of ϕ(v), whereas ϕ(e) is the root. Since the operad Ω(T) has at most one operation from given sequence of colours to another colour, it is clear that the morphism ϕ is uniquely determined by its effect on colours. In particular, notice that a morphism of trees always provides a well-deﬁned map between the sets of edges ϕE : E(S) → E(T) but it does not give in general a map between the sets of vertices. We will often call a morphism of tree injective or surjective if the induced map between the set of edges is respectively injective or surjective.

We observe also that the underlying map on edges ϕE is a map of posets, i.e.

e ≤ f ⇒ ϕ(e) ≤ ϕ( f ) and preserves independence of edges,

e ⊥ f ⇒ ϕ(e) ⊥ ϕ( f ).

To see the latter fact, observe that the paths from e and f to the root of S first meet in a vertex v. This vertex has two leaves e0 and f 0 such that e0 ≤ e and f 0 ≤ f . These two leaves are distinct and therefore incomparable; moreover, the edges ϕ(e0) and ϕ( f 0) are distinct leaves of the subtree ϕ(v), and therefore also incomparable. Since ϕ(e0) ≤ ϕ(e) and ϕ( f 0) ≤ ϕ( f ) it follows that ϕ(e) and ϕ( f ) are incomparable as well. In conclusion, let us recognize that each embedding of trees S → T as defined in the previous section defines an actual morphism in Ω. In fact, this morphism is defined through a map between the sets of vertices, so in operadic terms it sends generating operations to generating operations, or equivalently vertices to subtrees with one vertex. 36 CHAPTER 3. TREES

3.3.1 Elementary faces and degeneracies

For our purposes, once we set up all the formal deﬁnition of Ω, we need to understand better its combinatorial structure in order to play with it. In Chapter 1 we explained in what sense the simplex category is generated by the face maps δi : [n − 1] → [n] and the degeneracy maps σj : [n] → [n − 1]. Analogously, we will now discuss similar sets of generating morphisms for the category Ω, called again faces and degeneracies and which generalize their homonyms in ∆. There is one important difference, however, namely that the automorphism groups of objects of ∆ are all trivial while in Ω they need not be. ? Outer faces. Consider a tree T and an external vertex v ∈ V(T), i.e. a vertex with exactly one inner edge attached to it. Denote by ∂vT the tree obtained from T by pruning away v and all the outer edges attached to it. This vertex v can be any leaf vertex or it can be the vertex attached to the root of T, provided it is external. We write

δv ∂vT −−−−→ T for the inclusion of this tree and refer to it as an outer face of T. If v is a leaf vertex we will sometimes speciﬁcally speak of a leaf face and similarly for a root face. In the picture below we see an example of a leaf face δv and a root face δw.

w δw δv v v w

There is another situation which requires particular attention. For most trees T the notions of leaf vertex and root vertex are distinct and the preceding discussion sufﬁces; however, this is not the case for corollas since they possess only one vertex. There are n + 1 different maps

η → Cn corresponding to the inclusion of each of the n leaves and the root: all of these are by deﬁnition outer faces as well. ? Inner faces. Let T be a tree and e ∈ E(T) an inner edge connecting the vertices v and w, and assume v ≤ w. Denote by ∂eT the tree obtained from T by deleting the edge e and identifying the two vertices v and w. We will usually say ∂eT has been obtained from T by contracting e. This gives an inclusion of trees denoted

δe ∂eT −−−−→ T which sends the new vertex obtained by contraction e to the subtree of T obtained by grafting the two corollas with vertices v and w, i.e. the smallest subtree of T containing these vertices. In terms of the associated operads, δe sends the new vertex in ∂eT to the composed operation w ◦e v in Ω(T). A morphism of this kind will be referred to as an inner face of T.

f δ δf e e 3.3. THE CATEGORY OF TREES Ω 37

? Degeneracies. Let T be a tree and e ∈ E(T) any edge. Denote by σeT the tree obtained from T by splitting this edge into two edges e1 and e2 by placing a unary vertex v in the middle of e. Then there is a morphism σe σeT −−−−→ T e in Ω which sends both edges e1 and e2 to e and the new unary vertex v to the subtree η −→ T of T. In operadic terms, it sends the new vertex v to the identity operation of the colour e in the operad Ω(T). A morphism of this kind is called a degeneracy of T.

e1 σe e

Observation 3.13. We can immediately note that face maps are injective on edges and increase the number of vertices by one, while degeneracy maps are surjective on edges and decrease both the number of vertices and the number of edges by one. It is not necessarily true that a face map increases the number of edges: just think of an outer face corresponding to adding a nullary vertex, that is for example the inclusion η → C0. We will sometimes refer to the face and degeneracy maps discussed above as elementary faces and elementary degeneracies. We will also use the general terms face and degeneracy to indicate any non-empty composition of respectively elementary faces and elementary degeneracies. Lastly, notice that a composition of elementary outer faces is precisely the same as the embedding of a subtree.

Example 3.14. Let us examine for a moment these elementary maps in the case of linear trees. Via the full and faithful functor i : ∆ → Ω we know that they correspond precisely to elementary maps in ∆. So we have n + 1 elementary face maps Ln−1 → Ln, each of them skips exactly one edge of Ln coinciding with δi : [n − 1] → [n]. In particular, δ0 and δn are embeddings since they are outer faces, respectively root face and leaf face, while the others are all inner faces. Analogously, we have n elementary degeneracies Ln → Ln−1 coinciding with the degeneracies σi : [n] → [n − 1].

3.3.2 Combinatorial properties of Ω Similar to the simplex category, these elementary maps compose according to some ﬁxed rules, called codendroidal identities. They are necessarily more complicated than the cosimplicial ones and we list them in the next lemma, omitting the proof which is just a routine check.

Lemma 3.15 (Codendroidal identities). Let T be a tree, e, f ∈ E(T) and v, w ∈ V(T). Elementary faces and degeneracies satisfy the following relations:

(1) δvδw = δwδv if v and w are distinct outer vertices.

(2) δeδf = δf δe if e and f are distinct inner edges.

(3) δeδv = δvδe if e is an inner edge and v an external vertex not attached to e.

(4) δeδx = δvδw if v is an external vertex and e is an inner edge connecting v to another vertex w, provided w is an external vertex of the tree ∂vT. Here x is the new vertex of ∂eT arising as the composition of v and w along e. 38 CHAPTER 3. TREES

(5) σeσf = σf σe if e and f are distinct edges.

(6) σeσe1 = σeσe2 if e1 and e2 are the two new edges in the tree σeT.

(7) σeδei = id for i = 1, 2 if ei is an internal edge.

(8) σeδx = id if e is an outer edge and x is the new vertex connecting e1 and e2.

(9) σeδf = δf σe if e and f are distinct and f is an inner edge.

(10) σeδv = δvσe if v is an external vertex and e is not an external edge attached to v.

To have a little bit more of understanding about these relations, let us consider the ﬁrst one. First of all, the equation makes sense because w is still an external vertex of ∂vT and v is an external vertex of ∂wT. Then the equality is equivalent to the commutativity of the following diagram.

∂w(∂vT) = ∂v(∂wT)

δv δw

∂vT ∂wT

δv δw T One can recognize in these identities a generalization of the cosimplicial ones: indeed when we restrict to the full subcategory of linear trees we obtain exactly those, although more complicat- edly because of the distinction between inner and outer face maps.

Remark 3.16. From (7) and (8) we see that we have section-retraction pairs formed by degeneracies and two kinds of face maps. In particular, this shows that these maps are respectively split epimorphisms and split monomorphisms.

Analogously to the category ∆, the other key feature of elementary faces and elementary degeneracies is that they generate every morphism in Ω, in the sense that any map can be decomposed into these elementary pieces. However, there is an important difference: in Ω objects may have non trivial automorphisms, so the factorization is going to be Epi-Iso-Mono instead of just Epi- Mono. We give the precise statement and proof of this property in the next two lemmas.

Lemma 3.17 (Epi-Iso-Mono factorization). Any morphism ϕ : S → T in Ω can be factored as

σ α δ S −→ S0 −→ T0 −→ T

where σ is a degeneracy, α an isomorphism and δ a face map.

Proof. We start by considering the effect of ϕ on the set of edges, i.e. by looking at induced function ϕE : E(S) → E(T). We identify the edges in S that have the same image by deﬁning the equivalence relation 0 0 e ∼ e ⇔ ϕE(e) = ϕE(e ). Let us denote by A := E(S)/∼ the set of equivalence classes. It follows that we can factor ϕE as a surjection ϕs, the projection on the quotient, followed by an injection ϕi, sending each equivalence class to the corresponding common image:

ϕs ϕ E(S) −→ A −→i E(T).

Since any map of trees preserve the independence of edges, e ∼ e0 implies that they cannot be independent in S, so let us assume e ≤ e0. Then ϕ sends all the edges on the path from e0 to e to 3.3. THE CATEGORY OF TREES Ω 39 the same element in E(T), so in particular, all the vertices on this path must be unary. Denoting 0 by Se,e0 the tree obtained by collapsing to a single edge the segment from e to e in S, we have a degeneracy S → Se,e0 . Applying this reasoning to every equivalence class of edges in A, we get a degeneracy σ : S → S0 such that E(S0) = A. Moreover, by the universal property of the quotient, we have that ϕ factors as i ◦ σ for some injective map i : S0 → T corresponding to the function on edges ϕi. This map i may be factored as

α δ S0 −→ T0 −→ T where α is an isomorphism onto the image of i. Finally, the map δ is an injective map of trees which on the level of edges acts simply as an inclusion of subsets E(T0) ⊆ E(T). Any such map must be a face map: we can consider the edges not in the image of ϕ and inductively decompose the last inclusion δ as composition of maps ’skipping’ these edges, i.e. a composition of elementary face maps.

In the next lemma we show how we can decompose a face map into elementary face map respecting a precise order.

Lemma 3.18. Any face map ϕ : S → T in Ω can be factored as

δ ε S −→ S0 −→ T in the following two ways: (1) δ is a composition of inner faces and ε a composition of outer faces (2) δ is a composition of outer faces and ε a composition of inner faces,

0 Proof. (1) Let l1,..., ln be the leaves of S and r its root. Deﬁne S to be the maximal subtree of T with leaves ϕ(l1),..., ϕ(ln) and root ϕ(r). Clearly we can factor ϕ as composition of two injective maps δ : S → S0 and ε : S0 → T. The map ε is a composition of elementary outer face maps, where each such outer face adds a single vertex to S0 and its input edges. Now we should see that the map δ is instead a composition of elementary inner faces. Recall that every face map sends a vertex v of S to a subtree of T. Now suppose that δ sends every vertex of S to a corolla, then δ is the inclusion of a subtree. Therefore, δ must be the identity, since S and S0 have the same leaves and root. Otherwise, suppose that there is a vertex v of S such that the subtree ϕ(v) of T has an inner edge, say e. Then ϕ factors as

0 δ 0 δe 0 S −→ ∂eS −→ S where the map δ0 still preserves the leaves and root. The proof is then completed with an induction argument on the number of inner edges in the trees ϕ(v), for v ∈ V(S). (2) It sufﬁces to argue that a composition

δe δv ∂e(∂vT) −→ ∂vT −→ T of an inner face δe followed by an outer face δv can also be factored the other way around, i.e. as an outer face followed by an inner face. Note that the inner edge e of ∂vT is also an inner edge of T, which moreover is not connected to v (otherwise it could never be inner in ∂vT). But then our claim follows immediately from the codendroidal identity (3)

∂v∂e = ∂e∂v listed in Lemma 3.15. 40 CHAPTER 3. TREES

3.3.3 Linear faces Finally, we will now discuss a very important class of face maps that can resemble the combinatorial structure of the simplex category within each branch of a tree.

Let T be a tree, for any n ≥ 1 an n-linear subtree is an embedding Ln ,→ T; moreover, we call it maximal if it satisﬁes the following property: for any another embedding Lm ,→ T for n ≤ m ﬁtting into a commutative diagram of inclusions

Ln T

Lm then necessarily m = n. One could also talk about 0-linear subtree, but we avoid this trivial case because it is just equivalent to picking an edge e via the embedding e : η → e.

In other words, a maximal linear subtree Ln ,→ T corresponds to a maximal sequence n + 1 of edges connecting n unary vertices, namely the image of such an embedding. Since morphisms of trees preserve the partial order, 0 ∈ Ln must be embedded as the closest edge to the root and n ∈ Ln as the most far away from the root.

Deﬁnition 3.19. An elementary face map δ : R → T in Ω is called linear if there exist a maximal linear subtree Ln ,→ T and another embedding Ln−1 ,→ R ﬁtting into a commutative diagram

Ln−1 R

δi δ

Ln T for some elementary face map δi : Ln−1 → Ln. In a situation as the diagram above, R is isomorphic to the tree obtained from T by contracting one edge on the maximal linear subtree corresponding to the bottom embedding. Note that the map δi : Ln−1 → Ln is uniquely deﬁned because embeddings are monomorphisms in Ω and it corresponds precisely to an elementary face map δi : [n − 1] → [n] via the functor i : ∆ → Ω. Remark 3.20. Comparing with Remark 3.16, we can now identify what are the special elementary faces that form (non-trivial) section-retraction pairs together with degeneracies: they are precisely the linear faces!

As we can give a linear order to the edges on a maximal linear subtree ι : Ln ,→ T, we can also order elementary face maps sitting on the same maximal linear part. Indeed, for such a maximal linear part there are precisely n + 1 elementary faces acting on it, corresponding to the action of ’skipping the ith edge’ in the order induced by the embedding. Therefore, we can label these (ι) (ι) elementary faces as δ0 ,..., δn . Faces acting on the same maximal linear subtree are called connected. (ι) Deﬁnition 3.21. An elementary linear face map δ : R → T in Ω is called normal if δ = δi for some (ι) 0 ≤ i < n in the induced order, and maximal linear in the remaining case δ = δn .

If the elementary face δ : R → T lives on a maximal linear part Ln ,→ T, then δ is connected to precisely n other elementary faces. Of these n + 1 faces altogether, n are normal and exactly one is not normal (the maximal one in the induced order). Linear faces are always inner faces, except for two cases: the root face for a maximal linear part embedding at the bottom of a tree, and the leaf face for a maximal linear part embedding at the top of a tree. Of these two, the ﬁrst one is a normal face while the second one is not. 3.4. DENDROIDAL ABELIAN GROUPS & DENDROIDAL CHAIN COMPLEXES 41

Remark 3.22. In the last deﬁnition we made the arbitrary choice of picking the ﬁrst n as normal and excluding the case i = n, but we could instead exclude i = 0 and all the following treatise would be still valid mutatis mutandis. This is the same as choosing among the two embeddings δ0, δn : [n − 1] → [n] in ∆, as arbitrarily done for the simplicial Dold-Kan correspondence.

Inner Faces Outer Faces

Linear Faces

Max. Faces Leaf Faces

Normal Faces

Non max. Faces Root Faces

Figure 3.1: Elementary scheme for elementary face maps.

We conclude this section by recapping all the discussion about elementary faces and ﬁtting them into the scheme pictured above. First we distinguished between inner and outer faces, corresponding respectively to ’skipping an inner edge’ and ’skipping an outer vertex and its eventual leaves’. We have seen that linear faces, the only ones forming section-retraction pairs, might be both inner and outer. Finally, among n + 1 linear faces we set up the convention of calling the ﬁrst n normal, in symmetry with the fact that there are only n degeneracies mapping in the opposite direction.

3.4 Dendroidal abelian groups & dendroidal chain complexes

As for simplicial objects, we now start the study of presheaves on the category Ω. Instead of approaching the topic in the most general case, we just discuss our case of interest: presheaves of abelian groups.

Deﬁnition 3.23. A dendroidal abelian group is a functor X : Ωop → Ab.

With natural transformations as morphisms between them, dendroidal abelian groups form a category which we denote by dAb := Fun(Ωop, Ab).

Concretely, a dendroidal abelian group is given by a family of abelian groups XT, indexed over all the trees T ∈ Ω, together with a group homomorphism

∗ α : XT → XS for every morphism α : S → T in Ω. Functoriality implies that we have (βα)∗ = α∗β∗ for α β ∗ R −→ S −→ T in Ω and idT = idXT . The elements of XT are called T-dendrices of X. Equivalently, due to the combinatorial properties of Ω, in the description of a dendroidal abelian group X = (XT)T∈Ω we can focus on the generating morphisms of the category, i.e. elementary faces and degeneracies. 42 CHAPTER 3. TREES

Hence we deﬁne three kind of maps:

∗ de := (δe) : XT → X∂eT inner face, for each inner edge e of T, ∗ dv := (δv) : XT → X∂vT outer face, for any external vertex v of T, ∗ se := (σe) : XT → XσeT degeneracy, for each edge e of T.

Not surprisingly, these structural maps satisfy the so-called dendroidal identities, dual to the codendroidal identities of the previous section, and induced from them by functoriality.

Lemma 3.24 (Dendroidal Identities). Faces and degeneracies satisfy the following relations:

(1)d wdv = dvdw if v and w are distinct outer vertices.

(2)d f de = ded f if e and f are distinct inner edges.

(3)d vde = dedv if e is an inner edge and v an external vertex not attached to e.

(4)d xde = dwdv if v is an external vertex and e is an inner edge connecting v to another vertex w, provided w is an external vertex of the tree ∂vT. Here x is the new vertex of ∂eT arising as the composition of v and w along e.

(5)s f se = ses f if e and f are distinct edges.

(6)s e1 se = se2 se if e1 and e2 are the two new edges in the tree σeT.

(7)d ei se = id for i = 1, 2 if ei is an internal edge.

(8)d xse = id if e is an outer edge and x is the new vertex connecting e1 and e2.

(9)d f se = sed f if e and f are distinct and f is an inner edge.

(10)d vse = sedv if v is an external vertex and e is not an external edge attached to v.

Moreover, the face and degeneracy operators are equivariant with respect to isomorphisms of trees, in the sense that if α : S → T is an isomorphism and e an inner edge of T, then the square

∗ X(T) α X(S)

de dα(e)

∂eα X(∂eT) X(∂α(e)S)

commutes. Of course, a similar statement applies to outer faces and degeneracies.

3.4.1 Dendroidal chain complexes Now we want to introduce the concept of dendroidal chain complex. The inspiration comes again from the simplicial context: a chain complex is a sequence of abelian groups indexed over the natural numbers connected by maps satisfying ∂2 = 0. Furthermore, after Chapter 2, we could elaborate this a bit more and say that it is a simplified version of a simplicial abelian group, with no significant loss of data, as shown by the Dold-Kan correspondence. The question we have to address now is the following: how do we generalize this concept to the context of trees? In other words, what should be the right definition of a dendroidal chain complex? The two things we want to keep in mind are that it must be a simpler version of a dendroidal abelian group, namely fewer morphisms, and it should not lose any significant data, meaning it 3.4. DENDROIDAL ABELIAN GROUPS & DENDROIDAL CHAIN COMPLEXES 43 must fit into an equivalence of categories in the flavour of the Dold-Kan correspondence. First, we will introduce an intermediate step, inspired by the concept of a semi-simplicial abelian group.

Let Ωin be the wide subcategory of Ω whose morphisms are maps of trees which induce injections on edges. In other words, Ωin is obtained from Ω by removing all the degeneracy maps. Obviously, this is analogous to what discussed in Chapter 1 relatively to ∆in ⊂ ∆. Furthermore, these inclusions are compatible with the inclusion ∆ → Ω, that is we have a commutative square

∆in ∆

i|in i

Ωin Ω

op Deﬁnition 3.25. A semi-dendroidal abelian group is a functor Ωin → Ab. We indicate with sdAb the category of semi-dendroidal abelian groups.

We have a natural forgetful functor F : dAb → sdAb forgetting about all the degeneracy maps: it is given by precomposition with the inclusion Ωin ,→ Ω, i.e. F simply restricts a functor op Ω → Ab to the subcategory Ωin. Deﬁnition 3.26. A dendroidal chain complex is a semi-dendroidal abelian group such that all the normal faces induce the zero morphism.

This deﬁnition should sound very familiar, compared with the discussion done in Chapter 1 about chain complexes as a special case of semi-simplicial abelian groups. Roughly speaking, in ( ) a dendroidal chain complex XT Ωin we just remember of one non-trivial elementary face map → d : XLn XLn−1 satisfying d2 = 0 for each linear component instead of the ones induced by the n + 1 different faces Ln−1 → Ln. Indeed, due to the simplicial identity

δn+1 ◦ δn = δn ◦ δn the composition two consecutive maximal-linear faces induces the trivial map because it can be factored through a normal face, which induces the zero morphism by deﬁnition of dendroidal chain complex.

Remark 3.27. Consider in the category Ωin the two-sided ideal generated by the normal faces

N := Ωin ◦ {Normal faces} ◦ Ωin. op Since a dendroidal chain complex is a functor X : Ωin → Ab sending the arrows in N to the zero morphism, from the universal property of the quotient category expressed in Lemma 2.5, op this corresponds uniquely to a pointed functor X : (Ωin/N) → Ab. In other words, we could redeﬁne the category of dendroidal chain complexes as

Ω op dCh := Fun0 in , Ab . N

3.4.2 A description via reduced trees We now want to give an alternative and more concrete way of looking at dendroidal chain complexes. We will mimic the process occurring to go from simplicial abelian groups to chain complexes: we simplify the maps in ∆, obtaining the poset N modulo the relation ∂2 = 0. To adapt this construction to Ω, we will introduce a special kind of trees called reduced trees. 44 CHAPTER 3. TREES

Deﬁnition 3.28. A reduced tree is a tree with no unary vertices. A weighted reduced tree (T, w) is a pair given by a reduced tree T and a weight function w : E(T) → N on the set of edges. In other words, a reduced tree is a tree without n-linear subtrees for n ≥ 1, so in particular, there are no degeneracies coming out of it because all the linear subtrees have already been contracted as much as possible. Observation 3.29. There is a bijection between trees and reduced weighted trees, given by short- ening n-linear subtrees and remembering the number n of edges contracted. More precisely, for every tree T there is a unique reduced tree T whose edges form a subset of edges of T and a unique epimorphism σ : T → T in Ω. The weight function deﬁned on E(T) keeps track of how many edges have been contracted w(e) = |σ−1(e)| − 1. Conversely, given a weighted reduced tree (T, w) we can uniquely reconstruct a tree T by ex- tending each edge e ∈ E(T) with w(e) = n > 0 to a maximal linear part Ln ,→ T embedding [0] 7→ e ∈ E(T). Using this bijection, we will often identify a weighted reduced tree and the corresponding tree.

0 1 1 2 2 0 1

Remark 3.30. Notice that a morphism φ : T → S between reduced trees cannot be a degeneracy, otherwise T would not be reduced, nor a linear face map, otherwise S would not be reduced. Therefore, we are only allowed to have outer face maps or inner face maps, both of them corresponding to ’skipping’ edges connecting non-unary vertices. Comparing this with the scheme from Figure 3.1, we shall also notice that these two kind of elementary morphisms in Ω are exactly the non-linear faces, i.e. the only injective maps that do not have retractions.

We will now unpack the deﬁnition of a dendroidal chain complexes describing them as a structures graded over weighted reduced trees. First, we must introduce two notions related to weight functions. • If w : T → N is a weight function on a reduced tree T and e is an edge of T such that w(e) > 0 we deﬁne we : T → N to be the weight function obtained by reducing by 1 the weight at edge e: ( w( f ) if f 6= e we( f ) = w(e) − 1 if f = e

(T, we) is the tree obtained by contracting one edge on the linear subtree corresponding to the edge e of the tree (T, w).

f • For an elementary face map δf : ∂ f T → T between reduced trees, we denote by w the f restriction to ∂ f T of the weight function w : T → N, i.e. w = w ◦ δf . In other words, f (∂ f T, w ) has the same weights as (T, w) on the edges in im(δf ) ⊂ E(T), which means that T does not have any linear extensions of the edges of ∂ f T, and in fact this would contradict T being reduced. 3.4. DENDROIDAL ABELIAN GROUPS & DENDROIDAL CHAIN COMPLEXES 45

Now we can equivalently describe a dendroidal chain complex C = (C•) as a structure consisting the following data:

• for any weighted reduced tree (T, w) an abelian group C(T,w) together with a family of isomorphisms C(T,w) → C(T,w) one for each automorphism of the tree (T, w), ∗ • a morphism of abelian groups δ : C → C f for every elementary face map f (T,w) (∂ f T,w ) δf : ∂ f T → T between reduced trees, • → ( ) > differential maps de : C(T,w) C(T,we) for each edge of T such that w e 0 and satisfying

dede = 0 and ded f = d f de.

Moreover, these structural maps must commute in composition with each other according to edge-independence rule expressed by the dendroidal identities. A morphism of dendroidal chain complexes f : C• → D• is a collection of group homomorphisms

f(T,w) : C(T,w) → D(T,w) one for each weighted reduced tree, such that they commute with all the structural maps. Den- droidal chain complexes and morphisms between them form the category of dendroidal chain complexes indicated with dCh.

Remark 3.31. Comparing this with the previous remark, we can ensure that this description is exhaustive because there are only three kinds of morphisms in Ωin: isomorphisms, non-linear face maps (i.e. faces corresponding to maps between reduced trees) and linear face maps; in particular, the latter kind is the only one which needs to satisfy the extra condition d2 = 0.

As done for standard chain complexes, now we would like to give an even simpler characteriza- tion of dendroidal chain complexes as abelian presheaves on a quotient category which already encodes the trivialization of consecutive linear maps.

We deﬁne the reduced category of trees ΩR:

• The objects of ΩR are weighted reduced trees, equivalently seen just as trees in Ω via the 1:1 bijection T ←−→ (T, w).

• There are three kind of morphisms:

– Isomoprhisms: for any tree T and for any automorphism in Aut(T) there is a corresponding arrow T → T. – Non-linear arrows: there is exactly one arrow (T/ f , w f ) → (T, w) for any reduced tree T and any elementary face map δf : T/ f → T between reduced trees.

– Linear arrows: there is exactly one arrow (T, we) → (T, w) for any reduced tree T and any edge e ∈ E(T) with w(e) > 0.

Furthermore, these maps are imposed to commute with each other in any possible composition.

This ad hoc category ΩR is non-full subcategory of Ωin. The objects are the same, simply trees or weighted reduced trees, and the only difference is that, between two trees that differ just by a linear extension of a branch, the hom-set in ΩR is just a singleton. In other words, the simpliﬁcation we have operated to go from Ω to ΩR forgets about the different maps that we can have between two trees differing only by the length of a linear subtree: we only have one map Ln−1 → Ln instead of the different n + 1 face maps and n degeneracies in the other direction. 46 CHAPTER 3. TREES

The same simpliﬁcation on the category ∆ brings to the poset category N. In fact, ΩR admits the embedding of the poset category N into it: ∼ N ,→ ΩR n 7→ (η, n) = Ln mapping each natural numbers n to the linear tree Ln = (η, n), where the weight function assigns to the unique edge η the weight n. The last thing left now is to impose the condition ∂2 = 0 for linear arrows by selecting the right ideal of arrows to trivialize. Deﬁne the ideal of arrows generated by the composition of two or more linear arrows

0 0 S := ΩR ◦ { f : (T, w) → (T, w ) | ∃ e ∈ E(T) such that w (e) − w(e) ≥ 2} ◦ ΩR.

Considering now the pointed category obtained as quotient ΩR/S, it is just a matter of convinc- ing ourselves that this is just another equivalent description of the category Ωin/N deﬁned in Remark 3.27. Consequently, a dendroidal chain complex is the same as a pointed functor

Ω op R → Ab. S Chapter 4

The Dendroidal Dold-Kan Correspondence

In this chapter we are going to establish the equivalence of categories between dendroidal abelian groups and dendroidal chain complexes. A special case of this generalized correspondence has already been established in the paper [GLW11] adopting a planar structure on trees and mimick- ing the constructions and the proof of the simplicial one. By equipping the category Ω with an opportune DK-triple, we will show that a more general correspondence holds without assuming the planar structure hypothesis; this will be called the dendroidal Dold-Kan correspondence.

4.1 The planar case

In this section we are going to synthesize the main ideas presented in the article [GLW11] where it is proved a version of the dendroidal Dold-Kan correspondence working on the category of planar rooted trees. Although it is a little different from the correspondence we will get to, especially for a rather more complicated deﬁnition of dendroidal chain complex, it is still very interesting because it gave an important inspiration about the structures to be considered towards the proof of our ﬁnal result.

Deﬁnition 4.1. A planar rooted tree is a rooted tree such that for every vertex v its set of input edges in(v) is endowed with a linear ordering.

More precisely, let P : Ωop → Set be the presheaf on Ω that sends each tree to its set of planar structures. Then P(T) has a natural action of Aut(T): for an automorphism σ ∈ Aut(T) and a planar structure x ∈ P(T), we have that σ · x is the planar structure on T given by reordering the input edges according to the shufﬂe given by σ; concretely, if {e1 ≤ · · · ≤ e1} are the input edges of a vertex v ordered by x, then the order σ · x will be {σ(e1) ≤ · · · ≤ σ(en)}. Notice that this action is free, i.e. any non-trivial automorphism changes the planar structure.

The category Ωp is then equivalent to the category whose objects are pairs (T, x) with T ∈ Ω, x ∈ P(T) and a morphism between two objects (T, x) → (S, y) is given by a morphism of trees f : T → S in Ω such that P( f )(y) = x. Therefore, we see that morphisms in Ωp must preserve the planar structure, so in particular, every planar tree has a trivial automorphism group. In other words, as general tree T ∈ Ω generated a canonical symmetric operad Ω(T), here a planar tree S ∈ Ωp generates a canonical non-symmetric operad Ωp(S) (i.e. without the action of the symmetric group on the operations inputs), freely generated by the vertices of S.

Deﬁnition 4.2. The category of planar trees Ωp is the category whose objects are planar rooted trees and morphisms are maps between the associated non-symmetric operads.

47 48 CHAPTER 4. THE DENDROIDAL DOLD-KAN CORRESPONDENCE

As a consequence of having only trivial automorphism groups, the Epi-Iso-Mono factorization from Lemma 3.17 is simpliﬁed in the case of planar trees.

Lemma 4.3 (Lemma 3.2 from [GLW11]). Every map f : R → T in Ωp is either the identity or it decomposes uniquely as f = d ◦ s, where d is a face map and s is a degeneracy.

Using the planar structure, it is possible to deﬁne a linear order on the set of elementary face maps of any chosen planar tree. Let T be a tree in Ωp such that |V(T)| ≥ 2, we assign to each elementary face of T a natural number respecting the following rules:

(i) If the vertex above the root r is outer then assign 0 to δr. (ii) Starting from the root vertex, walk through all the edges and vertices of T going always ﬁrst to the left and upwards and when this is not possible, turn back to the closest already visited vertex and choose the next not yet covered edge left upwards.

(iii) Whenever an inner edge or an outer vertex is visited assign the smallest not yet used natural number to the corresponding elementary face of T. If T has n elementary face maps, the process described above deﬁnes a bijection

1:1 φ : {δ | δ is an elementary face map of T} ←−−−→{0, 1, ··· , n − 1} and hence induces a linear order on the elementary face maps: we deﬁne the i-th elementary face −1 of T by setting δi := φ (i). In case T is the n-corolla, the process of traversing T from left to right induces a linear order on the set of elementary faces of T as well. After renaming these faces accordingly, one will observe that δ0 is the inclusion of η into the root, δ1 is the inclusion of η in the leftmost leaf and so on. Notice that this ordering on face maps is the natural extension of the ordering that we have on the simplex category: if T = Ln then δi deﬁned above coincides with δi : [n − 1] → [n].

In a similar way, we can assign a sign ±1 to any elementary face map δ : T → R in Ωp. We start by numbering the vertices of R following as before the upward-left rule and get a bijection

1:1 ψ : V(R) ←−−−→{0, ··· , n}.

The sign of an elementary face map δ is computed using the following rules: (i) If δ is an inner face induced by an edge e and v is its upper vertex, then sgn(δ) = (−1)ψ(v).

(ii) If δ is an outer face map induced by the root-vertex r, then sgn(δ) = (−1)ψ(r) = 1.

(iii) If δ is any other outer face map induced by a vertex v, then sgn(δ) = (−1)ψ(v)+1.

For corollas we use again a different convention: the face map δ : η → Cn has sgn(δ) = 1 if it includes ηas root, and has sgn(δ) = −1 if it includes η as a leaf.

Now we define the planar versions of the characters involved in the dendroidal Dold-Kan correspondence, namely planar dendroidal abelian groups and planar dendroidal chain complexes. Let Ωp−in be the subcategory of Ωp containing only morphisms which edge-wise are given by injective maps. op Definition 4.4. A planar dendroidal abelian group is a functor X : Ωp → Ab. The category of op planar dendroidal abelian groups is defined to be presheaf-category dAbp := Fun(Ωp , Ab). op A planar dendroidal chain complex is a functor X : Ωp−in → Ab such that for any normal face δ : R → T the induced morphism X(δ) = 0 : XT → XR is the zero map. The category of planar den- op droidal chain complexes dChp is defined as the subcategory of Fun(Ωp−in, Ab) of functors satisfying this condition. 4.1. THE PLANAR CASE 49

Comparing this deﬁnition with the one given in general case over Ω, it is clear that we are dealing with the exact same objects: the only difference is really just a matter of assuming a planar structure on each tree.

Remark 4.5. Actually, the deﬁnition of planar dendroidal chain complex given in [GLW11] is different from what has been given here. They say that an abelian group A is Ωp-graded if M A = AT. Then a planar dendroidal abelian group is a Ωp-graded abelian group together T∈Ωp # with structural maps given by group homomorphisms δ : AT → AR for every elementary face map δ : R → T satisfying the following two conditions:

• If δ is a normal face, then δ# = 0.

• For any commutative diagram involving elementary faces (left)

∂# ∂1 1 S R AS AR # # δ1 ∂2 δ1 ∂2 δ# 0 δ2 2 R T AR0 AT # # # # the associated diagram (right) anticommutes, i.e. δ1δ2 = −∂1∂2.

A map ϕ : A → B of dendroidal chain complexes is a collection of maps ϕT : AT → BT for every T ∈ Ωp compatible with the structure homomorphisms. The first part of this definition is nothing strange, in fact it is perfectly aligned with the above op given definition of planar dendroidal chain complexes as functors Ωp−in → Ab trivializing normal faces. Nevertheless, the anticommutativity condition looks quite strange and unmotivated. By definition, a functor should always send commutative squares to commutative squares, so imposing this requirement seems wrong as it goes against functoriality. It turns out that this sign- twist, which uses the sign convention established via the planar case, is an additional feature that we can artificially impose on every occurring commutative square. In other words, it does not actually change the structures involved, but it is introduced as it appears to be a natural feature of structures used by the authors in their proof.

Now we will brieﬂy illustrate the proof presented in [GLW11]. It is rather evident that it proceeds by following the same steps of the proof of the Dold-Kan correspondence done in Chapter 1.

Let A ∈ dAbp, we are going to define three different planar dendroidal chain complexes. First, we define the Moore complex CA• by assigning CAT = AT for every T ∈ Ωp and for every # elementary face δ : R → T in Ωp, the structure map δ is defined as follows: (i) If δ is a normal face, then δ# = 0.

(ii) If δ is not a normal face and it is not connected to any normal face then δ# = sgn(δ) · δ∗.

(ι) (iii) In the remaining case δ = δn for a maximal linear part ι : Ln T in the induced order. n # ∗ Define in this case δ = ∑ sgn(δi) · δi . i=0 One must check that this is actually a well-defined planar dendroidal chain complex. Then we construct two subcomplexes of CA•. T ˜∗ ˜ The normalized complex NA• is defined as NAT = δ˜ ker(δ ) ⊆ AT, with δ running through all normal faces with codomain T. We restrict the structure maps of CA to get the ones for NA. ∗ Finally, the degenerate complex DA• is defined as (DA)T = ∑σ:T→S σ (AS) ⊆ AT where σ runs through all the elementary degeneracies with source T. Again, the structural map for DA are obtained restricting the ones from CA. 50 CHAPTER 4. THE DENDROIDAL DOLD-KAN CORRESPONDENCE

It is not hard to see that these constructions give rise to functors dAbp → dChp. Not surprisingly, the key one is the second one which we will refer to as normalization functor N. As in the simplicial case, these three complexes are closely related to each other as stated in the following lemma which should seem very familiar (compare with Lemma 1.13).

Lemma 4.6. For any planar dendroidal abelian group A, the Moore complex decomposes as

CA =∼ NA ⊕ DA.

Then, we deﬁne the quasi inverse Γ of the normalization functor N, pretty much in the same way as in the simplicial case. Let C be a planar dendroidal chain complex, we set

M ΓC = (ΓC)T = CR r:TR where r runs through all epimorphisms in Ωp with domain T. We omit here the explicit deﬁni- tion of the structural map for Γ, which is done in a similar ﬂavour to the simplicial case using the unique Epi-Mono factorization for morphisms in Ωp. One can check that this construction provides an actual functor dChp → dAbp. Finally we get to the planar dendroidal Dold-Kan correspondence.

Theorem 4.7. The normalization functor N : dAbp → dChp is an equivalence of categories.

The proof relies on the calculation of the compositions of N and Γ on both sides: on one hand we have NΓ(C)T = CT for C• ∈ dChp, while on the other hand we have a natural isomorphism ∼ ΓN(A)T = AT for A• ∈ dAbp.

4.2 DK-triple on Ω

We now want to ﬁnd a DK-triple Ω = (Ω, E, E∨) such that the morphisms in the normalized pointed category N0(Ω) will be generated by the elementary face maps which are not normal. As ultimate goal, we aim to apply Corollary 2.10 and get an equivalence of categories which will op allow to see a dendroidal chain complex simply as a functor N0 (Ω) → Ab. In other words, we want to simplify two classes of morphisms in Ω: the arrows generated by the elementary degeneracies and the arrows generated by the normal faces. Therefore, the natural candidates for the Epis and dual Epis will be the wide subcategories

E = {surjective maps} and E∨ = {sections preserving linear maximal edges}.

While the ﬁrst class of maps is quite clear, we need to make some useful observations to clarify what a map in E∨ looks like. First recall that section-retraction pairs in Ω occur only in two situations: pairs of inverse isomorphisms and pairs formed by degeneracies and linear face maps. While isomorphisms always preserve maximal edges on linear subtrees, this is not true for linear faces: those with this property are exactly the ones that are composition of normal faces, analogously to what we observed on ∆. In other words, the maps in E∨ are those injective maps which are only allowed to (eventually) skip non-maximal edges on linear subtrees, and in th fact normal faces allow to skip the i edge on a linear part Ln only for i 6= n. Finally, let us notice that this structure is an extension of the DK-triple we set up on ∆: if we ∨ restrict to the ∆ embedded in Ω, i.e. the full subcategory of linear trees, E|∆ an E |∆ are precisely the two wide subcategory we deﬁned in section 2.4. 4.2. DK-TRIPLE ON Ω 51

Let us ﬁrst look at what Sing, Reg, Mono are in this context. ∨ ¯ 0 • Sing := E6' ◦ B = {· · · →}, thus a map f : R → T is singular if and only if f = f ◦ f with f 0 ∈ ArB and f¯ dual Epi non-isomorphism, hence f¯ is just a normal face skipping th the i edge in a maximal linear part Ln of T for some 0 ≤ i < n. In particular, since the th i edge on Ln is not in the image of f¯, it will also not be in the image of f . Notice that the converse also holds: if a map f skips a non-maximal edge of a linear subtree, then, swapping the morphisms in its Epi-Iso-Mono factorization using dendroidal identities, it can be expressed in a composition placing a normal face at the end, so it is singular. We conclude that f is singular if and only if there is a non-maximal edge in a linear subtree which is not in the image of f . A map is regular if it does not satisfy this condition.

0 • B ◦ E6' = {→ · · · }, thus a map g : R → T belongs to B ◦ E6' if and only if g = g ◦ g¯ with g0 ∈ ArB and g¯ surjective non-isomorphism, hence g¯ is a degeneracy map, so in particular, g is not injective. Notice that the converse also holds, namely every non-injective map in Ω can be expressed in a factorization with a degeneracy at the beginning. A map is Mono if it does not satisfy this condition, so we can conclude that M = Mono consists precisely of the injective maps in Ω.

The set M ∩ Reg consists of non-singular injective maps, i.e. injective maps which do not skip non-maximal linear edges. In other words, these are precisely the isomorphisms and the elementary face maps which are not normal: the latter are non-unary inner faces, the maximal linear inner faces and all the outer faces except the one ’skipping the root’ in the case this is normal. Composition of maps in M ∩ Reg may or may not lead to a map again inside this set. Indeed, it is not close under composition for the same reason this occurred on ∆, namely composing two maximal linear faces we get a singular map. But on the other hand, any other composition in M ∩ Reg not containing two linear faces is internal.

We now go through the axioms T1-T5 and check that they are satisﬁed. T1. Unique factorization: Any morphism ϕ : S → T in Ω admits a factorization of the form E∨ ◦ (M ∩ Reg) ◦ E. Indeed, from the Epi-Iso-Mono factorization of Lemma 3.17 we can write ϕ as σ α δ S −→ S0 −→ T0 −→ T where σ is a degeneracy, hence it belongs to E, α is an isomorphism, hence it belongs to M ∩ Reg, and δ is a face map. Furthermore, by Lemma 3.18, δ can be factored as δ = δin ◦ δout, that is is a composition of outer faces δout followed by a composition of inner faces δin. Now we will to operate some changes to this factorization by swapping the order of some maps thanks to the codendroidal identities. Indeed, keep in mind that if we have face maps involving different branches of a tree, then the dendroidal identities guarantee their independence, i.e. they commute in the composition so that we can swap them freely.

We start by working on δin. Recall that linear faces are always inner, except for two cases: the root-face, which is normal, and the leaf-faces, which are not normal. From the previous observation about the independence of edges, we can arrange the non-linear inner faces (i.e. those skipping edges connecting non-unary vertices) all at the beginnning of the factorization, leaving all the linear faces at the end: δin = δlin ◦ δnot−lin. Then, for each linear subtree, we can group together all the linear faces corresponding to the same subtree placing them consecutively in the factorization. Moreover, as proved for the Dk- triple on ∆, a composition of connected linear faces can be arranged in a way that the maximal linear map comes ﬁrst and all the normal faces at the end. Doing this for every linear subtree 52 CHAPTER 4. THE DENDROIDAL DOLD-KAN CORRESPONDENCE then, again from edge-independence, we can gather together all the maximal linear faces at the beginning of the composition leaving all the normal faces at the end:

δlin = δnor ◦ δmax−lin. Summing up, so far we transformed the factorization of δ in order to write

δ = δnor ◦ δmax−lin ◦ δnot−lin ◦ δout.

Finally, examining δout, there might be a normal face left among these, namely the root face. If this is the case, then by the same arguments, we can move it within the factorization and place it at the very end. In this way, we ensure that all the normal faces appearing in the factorization come last: δ = δNor ◦ δNot−nor with δNor being the composition of all the normal faces (in general δNor can differ from the previous δnor) and δNot−nor being the composition of all the other faces that are not normal. In ∨ particular, we have δNor ∈ E and δNot−nor ∈ M ∩ Reg. Moreover, we also have δNot−nor ◦ α ∈ M ∩ Reg so we get the desired factorization for ϕ. T2. Pairing matrix: We are now going to consider the pairing induced on the isomorphism classes of Ω, which means we will not distinguish isomorphic trees, so in this section we will sometimes consider isomorphism classes of trees, indicating with [T] the isomorphism class of T ∈ Ω. Given two morphisms α, β : R T we say that α ≤ β ⇔ α(e) ≤ β(e) for any edge e of R, considered as maps induced on the set of edges. In particular, this order relation gives a linear order on E∨(R, T) for any trees R, T ∈ Ω. Now define a map (−)∨ : E(T, R) −→ E∨(R, T) ∨ σ : T R 7→ σ : R T sending a surjection σ to its maximal section σ∨ defined as map on edges by the formula σ∨(e) := max σ−1{e}. ∨ Indeed, we can immediately verify that σ ◦ σ = idR, and by definition it is maximal among the sections of σ with respect to the linear order we defined. Moreover, the map (−)∨ is well-defined because any surjection preserves maximal linear edges, and the formula imposes the section to do it as well, so σ∨ is really a dual Epi. We claim that (−)∨ is a bijection. ∨ ∨ • Injective: given two surjections σ, τ : T R suppose that σ 6= τ , i.e. there exists e edge of R such that σ∨(e) 6= τ∨(e), meaning f := max σ−1{e} 6= max τ−1{e} := f 0. If f and f 0 are uncomparable edges of T, then we have τ( f 0) = e = σ( f ) 6= σ( f 0). If f and f 0 are comparable, without loss of generality assume f < f 0, then τ( f 0) = e = σ( f ) < σ( f 0) by definition of σ∨. Either way, we can conclude σ 6= τ.

1 • Surjective: let δ : R T be a dual Epi, then we can define a surjection σδ : T R mapping {e ∈ E(T) | e ≤ δ( f )} 7→ f ∨ and for any f ∈ E(R). Then σδ ∈ E(T, R) is well-defined because δ ∈ E and clearly ∨ ∨ (σδ) = δ. This construction is basically defining the inverse of (−) . 1Here E(T) indicates the set of edges of T, beware this notation clashes with the one for the Epis under T. 4.2. DK-TRIPLE ON Ω 53

Now, observing that we have standard decompositions2 [ [ E(T) = E(T, R) E∨(T) = E∨(R, T) TR RT and considering simultaneously all the bijective maps we have just deﬁned, we can combine them to obtain a bijection ' (−)∨ : E(T) −−−−→ E∨(T). Furthermore, this induces also a bijection on the isomorphisms classes of Ω, indicated with the same notation ∨ ' ∨ ∨ (−) : E([T]) := π0E(T) −−−−→ E ([T]) := π0E (T). Similarly, we can forget about the isomorphisms, and update the previous decomposition to one relating isomorphisms classes of trees [ [ E([T]) = E([T], [R]) E∨([T]) = E∨([R], [T]). [T][R] [R][T]

Following the explaination given for the DK-triple on ∆, we now want to order these two sets to ﬁll a table with the dual Epis in the top row and the Epis in the leftmost column, matching the entries to complete the pairing matrix with their composition. We order E∨([T]) by ordering the components of the decomposition with the rule of increasing number of edges in the source tree (even though this order turns out to be irrelevant)

E∨([T], [T]) , E∨([R], [T]) ,..., E∨([S], [T]) , E∨([T], [T]) and then we order the maps in each component decreasing according to the linear order deﬁned at the beginning of this paragraph. Using the bijection (−)∨ we get a rule to give a linear order to E([T]) as well, so that we ﬁll the leftmost column.

  idT 6' ... 6'    ..   6' .     .   6' .     .   . 6'     .   .. 6'    6' ... 6' idT We clearly have isomorphisms on the diagonal, since we compose maps which form section- retraction pairs. Considering a matching of a dual Epi with an Epi having different equivalence classes of trees respectively as source and target, then of course their composition cannot be an isomorphism since it would have as source a different isomorphism class than the target. So we are left to consider the matching of dual Epi E∨([R], [T]) with Epis E([T], [R]), corresponding precisely to square submatrices appearing as blocks aligned on the diagonal: outside these blocks we are already sure that the matrix has only non-isomorphisms. Let us consider an entry below the diagonal in such a submatrix, supposing we have σ ◦ δ = id in this entry: this means δ is a section for σ, but on the other hand, by the way we ordered the dual Epis in each block (decreasing), we have σ∨ < δ contradicting the maximality of σ∨. So we conclude that each of these blocks has non-isomorphisms below the diagonal, so in particular, the whole pairing matrix has isomorphisms on the diagonal and non-isomorphisms below it. It is precisely the same argument used to describe the pairing matrix of the DK-triple used on ∆.

2Here this notation stands for the Epis under T. 54 CHAPTER 4. THE DENDROIDAL DOLD-KAN CORRESPONDENCE

T3. E∨ ◦ E closed under composition: a map in E∨ ◦ E consists of a surjective map followed by an isomorphism and a (eventually empty) composition of normal faces. Let f , g ∈ E∨ ◦ E, composing them we get a map g ◦ f which by the axiom T1 admits a factorization of the form E∨ ◦ (M ∩ Reg) ◦ E. Since both f and g only involve normal faces in their factorization, and not other inner or outer faces, the same must hold for their composition hence the factor belonging to M ∩ Reg can only be an isomorphism. Since both E and E∨ both contain all the isomorphisms, we conclude that this factorization of g ◦ f is actually of the form E∨ ◦ E. T4. (M ∩ Reg) ◦ (M ∩ Reg) ⊂ M: the statement is automatically veriﬁed because M consists of injective maps and the composition of injective maps is again injective. T5. M ◦ Sing ⊂ Sing: let g : S → T ∈ Mono, f : R → S ∈ Sing and consider their composition g ◦ f : R → T. First, since f is singular there must be a non-maximal edge e on a linear subtree of S which does not belong to im( f ). since g is injective, g(e) is again a non-maximal edge sitting on a linear part of T, moreover g(e) does not belong to the im(g ◦ f ), so g ◦ f is singular.

4.3 The correspondence

Now what we are left to do now is understand better what actually is the normalized pointed category N0(Ω). Our claim is that we already know it: the source category in our deﬁnition of dendroidal chain complex.

∨ Proposition 4.8. The normalized pointed category N0(Ω) induced by the DK-triple Ω = (Ω, E, E ) is isomorphic to the quotient category Ωin/N. Proof. Recall the deﬁnition of the normalized pointed category M N (Ω) := 0 M ∩ Sing where M is the wide subcategory Mono ⊂ Ω, which in our case consists simply of the injective maps of Ω. In other words, M coincides with the wide subcategory Ωin. Now we just have to convince ourselves that M ∩ Sing also coincides with the ideal N = Ωin ◦ {Normal faces} ◦ Ωin. This follows from the factorization properties of Ω, and hence of Ωin, for which normal faces in a composition can be placed all at the end as explained in T1. This observation shows that N ⊂ M ∈ Sing, while the inclusion M ∩ Sing ⊆ N is obvious.

Theorem 4.9 (Dendroidal Dold-Kan correspondence). The category of dendroidal abelian groups dAb is equivalent to category of dendroidal chain complexes dCh. Proof. We have checked that the category Ω can be equipped with a DK-triple structure using the two classes of morphisms we have speciﬁed. This ensures that the hypotheses of Corollary 2.10 are satisﬁed, hence by applying it we can deduce the following equivalence of categories

op ' 0 op Fun(Ω , Ab) ←−−−→ Fun (N0(Ω) , Ab). Combining the previous proposition with the Dold-Kan equivalence, we can upgrade the previous equivalence to op ' Ω Fun(Ωop, Ab) ←−−−→ Fun0 in , Ab . N Giving proper names to each side we ﬁnally obtain the dendroidal Dold-Kan correspondence ' dAb ←−−−→ dCh. 4.4. FINAL COMMENTS 55

4.4 Final comments

It is apparent that structure of the DK-triple we set up on Ω closely follows the one constructed on ∆. The dendroidal Dold-Kan correspondence is something happening only on linear subtrees, where it occurs for the same reason as in the simplicial Dold-Kan correspondence. The simplification we operated on Ω involves indeed only linear subtrees, regarding specifically the symmetry encoded between degeneracies and normal faces: in fact, we could ‘embed’ ∆, seen as sequence of linear trees, into each linear extension of any branch of a tree, and then apply the simplicial correspondence on these linear parts. In the end, the dendroidal Dold-Kan correspondence can be simply seen as a combination of this visual intuition about ∆ ‘hiding’ into each linear subtree together with the simplicial Dold-Kan correspondence. Recalling our slogan from Section 3.4.1, the category of dendroidal chain complex we defined was expected to have two features: having as objects, simpler versions of dendroidal abelian groups and fitting into a Dold-Kan correspondence. The first one was already clear from the definition, while the second one is expressed in the very last theorem. Our dendroidal chain complexes behave exactly as they are expected to do, or in other words, we can say that we have given the right definition. Nevertheless, the process through which we arrive at this point is not mathematically and logically standard. Usually, in mathematics, a new concept is introduced for various reasons (e.g. chain complexes and simplicial abelian groups) in the development of a certain theory and, only later, a connection between them can be highlighted (e.g. the Dold-Kan correspondence). In the dendroidal case, instead, the concept of dendroidal chain complex does not arise naturally as a tool, but is rather defined to be the missing piece into the equivalence we want to establish. In this sense, meta-mathematically speaking, the reasoning described in this thesis goes in the opposite direction to the common sense. Personally, for this reason, I appreciated the development of this work even more. 56 CHAPTER 4. THE DENDROIDAL DOLD-KAN CORRESPONDENCE Popular Summary

Mathematicians, especially pure mathematicians working in algebra and geometry, always try to work on results assuming as few hypotheses as possible: this makes a theorem stronger, meaning applicable to the most number of cases. This search for more and more generality reached a remarkable peak with the introduction of the concept of categories in the 1940’s. Samuel Eilen- berg and Saunders Mac Lane, while working respectively in the ﬁeld of algebraic topology and abstract algebra, realized an unexpected connection between their independent work. Unrav- eling the puzzle, they found out that a more general framework for both would have helped to develop a mathematical common ground to work with more agility between different ﬁelds. Soon, categories were born and their language quickly became the new mathematical dictionary for pure mathematics.

Roughly speaking, a category is a collection of objects with maps between them, intuitively vi- sualized as a bunch of points connected by some arrows. This very simple idea can allow to group together all objects of the same kind: we have then the category Set of all sets and functions between them, the category Ab of abelian groups and homomorphisms between them, the category Top of topological spaces and continuous maps between them, etc. With an extremely wide range of applicability, the language of categories revealed to be useful as a unifying theory in mathematics. The key power of this approach consisted in shifting the focus from a single object, to the class of similar objects, highlighting the relations encoded by the maps.

‘Functions’ between categories are called functors: they allow to coherently connect different categories preserving the respective structures. From a more intuitive point of view, a functor enables one to translate the network of a certain category into another. Moreover, functors are also used to compare categories, and in particular two categories are said to be equivalent when there are functors in both directions behaving almost like inverse maps of each other. Equivalent categories present somehow the same core structure, even though they may look quite different at a ﬁrst sight. Therefore, ﬁnding an equivalence between two categories that a priori do not look the same is always a great result, because it tells us that properties true in one category, must hold also in the other and vice versa.

Some particular categories can be artiﬁcially built to have a particular combinatorial structure, resembling just a graph which pictures a speciﬁc diagram. A functor having as a source such a category, reproduces the shape of the diagram into another category by assigning to each vertex and arrow an item picked from the target category. Examples of these kinds of combinatorial categories are for instance the simplex category ∆ and the category of trees Ω. The category ∆ is constructed with a linear order between the objects, modelled to resemble the geometric hierarchy between 0-dimensional points, 1-dimensional segments, 2-dimensional triangles and so on. A functor having as a source ∆, translates this structure into another category, abstractly realizing the geometric complex. The category of trees is an extension of the simplex category intuitively obtained by enlarging the linear order to a certain partial order; consequently, more maps are added, enriching the variety of interactions.

57 58 CHAPTER 4. THE DENDROIDAL DOLD-KAN CORRESPONDENCE

Given two categories C and D, we can consider the category of functors between these two Fun(C, D), and this is often a very interesting object of study, especially if the category C has peculiar structure. The Dold-Kan correspondence is an equivalence between two particular categories of functors over ∆: the category of simplicial abelian groups and the category of chain complexes. In fact, using the theory of DK-triples, it is possible to generalize the Dold- Kan correspondence and prove more general equivalences of the same ﬂavour, provided the source category C presents a combinatorial structure with a symmetry between two sub-classes of arrows. These two sub-classes can be cancelled out in the shape of C to obtain a simpler category NC, and the categories of functors Fun(C, Ab) and Fun(NC, Ab) are proven to be equivalent. Analyzing the category of trees Ω, we can show that it admits a DK-triple structure, which hence induces an equivalence between the category of dendroidal abelian groups and the category of dendroidal chain complexes: this is the so-called dendroidal Dold-Kan correspondence. Bibliography

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[Baš15b] Matija Baši´c. Stable homotopy theory of dendroidal sets. PhD thesis, [Sl: sn], 2015.

[GJ99] Paul G. Goerss and John F. Jardine. Simplicial homotopy theory. vol. 174. Progress in Mathematics. Basel: Birkhäuser Verlag, page 10, 1999.

[GLW11] Javier J. Gutiérrez, Andor Lukacs, and Ittay Weiss. Dold-Kan correspondence for dendroidal abelian groups. Journal of Pure and Applied Algebra, 215(7):1669–1687, 2011.

[HM18] Gijs Heuts and Ieke Moerdijk. Trees in algebra and topology. Preprint, pages 1–102, 2018.

[ML13] Saunders Mac Lane. Categories for the working mathematician, volume 5. Springer Science & Business Media, 2013.

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[Rie11] Emily Riehl. A leisurely introduction to simplicial sets. Unpublished expository article available online at http: // www. math. jhu. edu/ ~eriehl/ , 2011.

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