The Dendroidal Dold-Kan Correspondence

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MSc Mathematics

Master Thesis

The Dendroidal Dold-Kan

Correspondence

Author

Marco Pievani

Supervisor

dr. Gijs Heuts

Examiner

dr. Hessel Posthuma

Second examiner

dr. Raf Bockland

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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.

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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 correspon-dence, renamed planar dendroidal Dold-Kan correspondence. This is an equivalence of categories between the category of planar dendroidal abelian groups, i.e. abelian presheaves onΩopp , and the

brand-new category of dendroidal chain complexes. By constructing a variation of the normaliza-tion 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 quotientingCmodulo some subclass of arrows, and it ultimately leads to the equivalence of categories

Fun(C,A) ∼=Fun0(NC,A),

for any weakly idempotent complete categoryA. 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 corre-spondence of [GLW11], but instead working in the more flexible category of treesΩ, i.e. without assuming the choice of a planar structure, within the frame work of dendroidal sets and sym-metric operads. By equippingΩ 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.

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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 thatΩ admits a DK-triple structure and we finish by applying Walde’s result to finally get the desired dendroidal Dold-Kan correspondence.

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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 orderif 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 categoryP with set of objects Ob(P ) = P and hom-sets defined by

HomP(a, b) =

(

{∗} if 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.

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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

δi:[n−1] → [n] δi(k) =

(

k if k<i 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

σj :[n] → [n−1] σj(k) =

(

k if k≤ j 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, decode-generacies 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 σiδj =        δj−1σi if i<j−1 id if i=j−1 or i= j δjσi−1 if i>j

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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 δjand define

h0:=δ−1_j ◦h :[m] → [n−1].

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

x7→

(

g(x) if x≤y g(x) +1 if x>y

Then g0 is surjective and hence by inductive hypothesis is a composition of elementary degen-eracies, moreover g=σj◦g0 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:

f :[m]−−− [g k],−−−→ [h 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:

g :[n] [k] mapping f−1(ij) 7→j

h :[k] ,→ [m] mapping j7→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.

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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 epi-morphism.

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∆nis defined as: ∆n_:_{= {(}_t

0, . . . , tn) ∈Rn+1|t0+ · · · +tn =1 , ti ≥0∀i}

For any map α :[m] → [n]in∆ we can define the map α∗:∆m→∆n by α∗(t0, . . . , tm) = (s0, . . . , sn) with sj =

∑

i∈α−1_(j)

ti

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

(δi)∗:∆n−1→∆n

embed∆n−1into∆n as face opposite to the ithvertex.

For an elementary degeneracy σj :[n] → [n−1]the induced map

(σj)∗:∆n→∆n−1

collapses∆n onto∆n−1by projecting parallely to the line connecting the jthvertex to the j+1th. In other words, we have just described a covariant functor∆•:∆→Topsending[n] 7→∆n.

1.2 Simplicial objects

Definition 1.8. A simplicial object in a categoryCis 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 inC indicated by sC =Fun(∆op_,_C)_.

Unpacking the definition, a simplicial object X inCcan be concretely described as a sequence of objects inC 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

(αβ)∗=β∗α∗: Xn→Xk for [k] β

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1.2. SIMPLICIAL OBJECTS 5 A morphism ϕ between two simplicial objects X and Y inC is then a sequence of morphisms

ϕn: Xn →Yn inC (n≥0) compatible with every map α∗for α :[m] → [n], that is the following

square commutes for any choice of such maps: Xn Xm

Yn Ym α∗

ϕn ϕm

α∗

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 djsi =        sidj−1 if i<j−1 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 sim-plicial 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

· · · →Cn−∂→n Cn−1→ · · · →C1 ∂1

−→C0

connected by group homomorphisms, called differentials or boundary maps, satisfying ∂n◦∂n+1 =0

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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:

. . . Cn+1 Cn Cn−1 . . . . . . Dn+1 Dn Dn−1 . . . fn+1 ∂C_n+1 ∂Cn fn fn−1 ∂D_n+1 ∂Dn

In particular, the commutation of the previous diagram implies that: • ∂Cn(x) =0⇒∂Dn(fn(x)) = fn−1(∂Cn(x)) =0⇒ fn(x) ∈ker(∂nD)

• y=∂C_n+1(x) ⇒ fn(y) = fn(∂C_n+1(x)) =∂_n+1D (fn+1(x)) ⇒ fn(y) ∈im(∂D_n+1)

In other words, fn(ker(∂Cn)) ⊆ker(∂nD)and fn(im(∂C_n+1) ⊆im(∂D_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≥0to have chain complexes

as objects and chain maps as morphisms.

Remark 1.12. Chain complexes are often indexed over the integersZ, but in this thesis we will only consider non-negatively graded chain complexes, i.e. indexed over the natural numbersN. 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∆nindicates the standard topological n-simplex and δi∗is the map embedding into it the ithface 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

S(X)n:= {σ:∆n →X|σis continuous}

denotes the set of singular n-simplices in X. For any 0≤i≤n we can restrict a singular n-simplex to the ithface of∆n by precomposing with the inclusion δi∗ :∆n−1 ,→ ∆n. We hence get a map

of sets

di : S(X)n →S(X)n−1 (σ:∆n→X) 7→σ◦δi∗,

which acts by restricting a singular simplex to the ith face. Similarly, we can define the precom-position with each σ_j∗:∆n∆n−1to get maps of sets

sj : S(X)n−1→S(X)n (σ:∆n →X) 7→σ◦σ_j∗.

Let now A be an abelian group, for all n≥0 we define the group Cn(X; A):=A[Sn(X)]

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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:=

n

∑

i=0

(−1)idi : Cn(X; A) →Cn−1(X; A)

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 nthsingular homology group of X

Hn(X; A):=Hn(Cn(X; A)).

X⇒S(X)n = {σ:∆n →X} ⇒Cn = {n-chains} ⇒ComplexChain ⇒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 δi7→di σj7→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 σ7→ 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 X7→ (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.

Hn(−; A): Top Sing

−−−→sSet−−−−−→A[−] sAb ∑(−1)

i_d i

−−−−−→Ch≥0 Hn

−−−−→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

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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:

CAn:= An ∂C_n := n

∑

i=0

(−1)idi.

It is immediate to observe that this is well-defined since taking the alternating sum of the face maps di : An →An−1gives 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:

∂C_n ◦∂C_n+1= n

∑

i=0 (−1)idi n+1

∑

j=0 (−1)jdj ! = n

∑

i=0 n+1

∑

j=0 (−1)i+jdi◦dj = n

∑

i=0 i

∑

j=0 (−1)i+jdi◦dj+ n

∑

i=0 n+1

∑

j=i+1 (−1)i+jdi◦dj = n

∑

i=0 i

∑

j=0 (−1)i+jdi◦dj+ n

∑

i=0 n+1

∑

j=i+1 (−1)i+jdj−1◦di = n

∑

i=0 i

∑

j=0 (−1)i+jdi◦dj− n

∑

i=0 n

∑

j0_=i (−1)i+j0dj0◦di =0.

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

face maps di : An →An−1, except the nth:

N An:= n−1

\

i=0

ker(di) ⊆An ∂nN := (−1)ndn.

For n = 0 we have an empty intersection, so we set N A0 := A0. We must check that this is

well-defined chain complex, namely that ∂Nn(N An) ⊆ N An−1. Let x∈ N An, then for i< n−1

we have

di(∂N_n(x)) = (−1)ndi◦dn(x) = (−1)ndn−1◦di(x) =0

by the same simplicial identity as before, so we see that ∂nN(x) ∈ N An−1. Now we should also

check that ∂2=0. Let x∈ N An+1, then we have:

∂_nN◦∂N_n+1(x) = (−1)n+n+1dn◦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

(−1)n _{is actually irrelevant in this calculation, although it is needed to have ∂}N ₌ _∂C_|

N A•. In particular, the latter fact could have also been used to deduce ∂2, but more importantly we have N A•⊆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: DAn:= n−1

∑

j=0 im(sj) ⊆An ∂nD:=∂Cn|DA•.

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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

well-defined, i.e. ∂nD(DAn) ⊆DAn−1. Let x=sj(y) ∈An for some y∈ An−1and 0≤i≤n, then

we have: ∂Dn(x) = n

∑

i=0 (−1)idisj(y) = j−1

∑

i=0 (−1)idisj(y) + (−1)jdjsj(y) + (−1)j+1dj+1sj(y) + n

∑

i=j+1 (−1)idisj(y) = j−1

∑

i=0 (−1)isidj−1(y) + (−1)j( y− y) + n

∑

i=j+1 (−1)isi−1dj(y).

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

used simplicial identity, show that ∂_nD(x) ∈ DAn−1. We can extend this result to the whole

group generated by the degeneracies to conclude ∂Dn(DAn) ⊆ DAn−1. The property ∂2 = 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 equiv-alence of categories proving the Dold-Kan correspondence.

Lemma 1.13. The morphism N An⊕DAn ∼ =

−→An =CAninduced 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 N A0 = A0, so this

tautologically shows DA0⊕N A0∼= 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

NkAn := ∩k_i=0ker(di) DkAn := k

∑

i=0

im(si).

Notice that we have the following chain of inclusions:

N0An⊇ N1An ⊇ · · · ⊇Nn−1An =N An D0An⊆D1An ⊆ · · · ⊆Dn−1An=DAn.

We will prove by induction on k that NkAn⊕DkAn∼= An for every 0≤k<n and for k=n−1

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We start the induction with k= 0. In this case we have N0An = ker(d0)and D0An = im(s0).

Using the simplicial identity d0◦s0=idAn−1, the result follows from these two facts:

• N0An∩D0An = {0}: let x=s0(x0)for some x0∈ An−1and x∈ker(d0). Then

0=d0(x) =d0s0(x0) =x0so we conclude x=s0(0) =0.

• N0An+D0An = An: let x ∈ An, define x0 := s0(d0(x)) so that d0(x0) = d0(x)meaning

x−x0∈ker(d0), thus we have x= (x−x0) +x0with(x−x0) ∈N0An and x0∈D0An.

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

• NkAn∩DkAn = {0}: let x = ∑ki=0si(xi) = ∑k−1_i=0si(xi) +sk(xk) for some xi ∈ An−1 and

di(x) =0 for i≤k. By the inductive hypothesis on An−1we have the splitting

An−1∼=Nk−1An−1⊕Dk−1An−1

so we can write xk=α+βwith α∈Nk−1An−1and β∈Dk−1An−1, i.e. di(α) =0 for every

i≤k−1 and β=_∑k−1_i=0si(bi)for some bi∈ An−2. Then applying skwe get

sk(xk) =sk(α) +sk(β) =sk(α) + k−1

∑

i=0 sksi(bi) =sk(α) + k−1

∑

i=0 sisk−1(bi)

using the simplicial identity sjsi = sisj−1 for i< j; in particular, we have sk(β) ∈Dk−1An.

Now we substitute in the original expression of x gathering together all the elements in the image of each si: x= k−1

∑

i=0 si(xi+sk−1(bi)) +sk(α) =γ+sk(α) with γ := k−1

∑

i=0 si(xi+sk−1(bi)) ∈ Dk−1An.

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

di(sk(α)) =sk−1(di(α)) =0 for i<k

since α ∈ Nk−1An−1. Hence, we have sk(α) ∈ Nk−1An and x ∈ NkAn ⊆ Nk−1An, so

in particular γ ∈ Nk−1An too, hence γ = 0 by inductive hypothesis. Finally we have

x =sk(α), but applying dkwe have 0=dk(x) =dk(sk(α)) =αso we conclude x=0.

• NkAn+DkAn =An: let x∈Anand write it as x=α+βwith α∈ Nk−1Anand β∈Dk−1An

by inductive hypothesis. Defining α0 := sk(dk(α)) ∈ DkAn, then we have dk(α0) = dk(α)

and so α−α0 ∈ker dk. Using simplicial identities, for i <k we get

di(α0) =di(sk(dk(α))) =sk−1(di(dk(α))) =sk−1(dk−1(di(α))) =sk−1(di(0)) =0,

so α0 ∈ ker(di) for every i < k. Ultimately, we are able to write x = (α−α0) + (β+α0)

where(α−α0) ∈NkAnand(β+α0) ∈DkAnas desired.

Roughly speaking, we could rephrase the result from this lemma by saying that “the normalized complex N A•keeps track of non-degenerate information of CA•”. Moreover, one can prove (see for

example [GJ99]) that the inclusion N A•,→CA•is a chain homotopy equivalence. Even though

we will not need this last fact, it tells us that somehow N A• loses no homological information

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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 com-plexes, 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.

Definition 1.14. A semi-simplicial abelian group is a functor∆op_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.

sAb Ch≥0

ssAb

F

N

Last but not least, we can construct a rather special semi-simplicial abelian group(Cn)n≥0from

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

dn= (δn)∗:= (−1)n∂n : Cn→Cn−1.

One can check that this is indeed a well-defined semi-simplicial abelian group, i.e. cosimplicial identities are satisfied. In particular, from ∂2 = 0 we find dn◦dn+1 = 0 as an instance of

the simplicial identity dn◦dn+1 = dn◦dn. The sign (−1)n 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≥0since 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 → Dis 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≥0is an equivalence of categories.

To prove the result, we will construct another functorΓ : Ch≥0 → sAb that will be the

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1.4.1 Γ quasi-inverse of N

Given a chain complex C= (C•, ∂•) ∈Ch≥0we 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

ΓC([n]) =Γn(C):=

M

[n][k]

Ck

meaning that we take the direct sum of as many copies of Ckas 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) =C0corresponding to id[0]

– Γ1(C) =C1⊕C0corresponding respectively to id[1]and to[1] [0]

– Γ2(C) =C2⊕C1⊕C1⊕C0corresponding 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 Ck in degree n. Using the Epi-Mono factorization we can express τν as[m]

σ

−− [j],−→ [ι k]

as shown in the following commutative square:

[m] [n]

[j] [k]

ν

σ τ

ι

Now we define a homomorphism Ck →Cjinduced by ι as follows:

(Ck →Cj) =        idCk if j=k (−1)k∂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 compo-nent 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 functorΓC : ∆op→Ab.

First of all, it is not hard to see that id[n] induces the identity mapΓn(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 →Ckin each component, which of course

sum up to an identity in degree n.

Then, consider a composition[l]−→ [µ m]−→ [ν n]in∆, we want to see that(νµ)∗=µ∗ν∗. As before,

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[l] [m] [n]

[i] [j] [k]

µ _ν

θ ι

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

differential (with sign): in this case(ιθ)∗ = 0 = ∂2since (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 ofΓ actually provides a functor Γ : Ch≥0 →sAb,

that is a chain map f• : C• → D• defines a natural transformationΓ(f) : ΓC → ΓD. In fact,

we have componentwise maps fn : Cn →Dn, i.e. maps between each summands ofΓC and ΓD,

which commute with the chain complexes differentials; therefore, by universal property, we will get mapsΓ(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 ofΓ.

Observation 1.16. Let A= (An)n≥0be a simplicial abelian group, consider the composition of

functorsΓ◦N : sAb→sAb. For the simplicial abelian groupΓ(N A•)in degree n we have:

Γn(N A•) =

M

[n][k]

N Ak=N An⊕N An−1⊕ · · ·

For every surjection σ : [n] [k]we have a term N Ak associated to σ in the direct sum and a

structural map σ∗ : Ak → An being this a simplicial abelian group. Moreover, N Ak naturally

includes into Akby definition, so combining these together we can consider the composition

N Ak,→ Ak σ

∗

−→An

for any such term N Ak indexed by a surjection σ : [n] [k]. Then, by universal property, this

family of morphisms from each summand uniquely induces the sum map: Ψn :Γn(N A•) = M [n][k] N Ak→An mapping(xσ)σ7→

∑

σ σ∗(x).

Furthermore, considering this construction for every n ≥ 0 leads to a natural transformation Ψ= (Ψn)n≥0given 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(N A•) → An is an isomorphism for every n ≥ 0. In

particular, the family of morphismsΨ= (Ψn)n≥0provides a natural isomorphismΓ◦N ∼ =

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Proof. Let A∈ sAbbe a simplicial abelian group, we are going to prove that the natural map constructed in the previous observationΨ = (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(N A•) = N A0 =A0, so the statement is tautologically satisfied. As inductive hypothesis, let

us assume thatΨm is proven to be an isomorphism for every m<n; we now want to show that

Ψn is again an isomorphism.

• Ψnis surjective: When restrictingΨnto the component indexed by id[n]we have

N An ,→An id∗_[_n_]

−−→ An

which is just the inclusion N An ,→ An, hence we get N An ⊆ 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(N A) 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 ∼=N An⊕DAn, we conclude that An ⊆imΨn since both DAn and N Anlie in imΨn.

• Ψn is injective: suppose thatΨn maps(xσ) ∈Γn(N A)to zero, i.e.

∑

σ:[n][k]

σ∗(x) = 0. For

k<n, a surjection σ :[n] [k]has a canonical maximal section

νσ :[k] ,→ [n] νσ(i):=max{j∈ [n] |σ(j) =i}.

Given two surjections σ, σ0:[n] [k]we say

σ≤σ0 ⇔ νσ(i) ≤νσ0(i)for all i∈ [k].

In particular, σ0νσ = id[k] ⇒ σ ≤ σ0 since νσ0 is the maximal section of σ

0_{, hence ν} σ ≤

ν_σ0. If there exists a surjection τ : [n] [k] such that the component x_τ 6= 0, choose a maximal such τ (with respect to the ordering just defined). Then consider the morphism

ν_τ∗:Γn(N A) →Γk(N A)which on the τ component is defined by the diagram

[k] [n]

[k] [k]

ντ

id[k] τ

id[k]

The bottom arrow induces the identity map id : N Ak → N Ak, which then is composed

with the inclusion N Ak ,→Γk(N A)of the component indexed by id[k]. In particular, by the

simplicial structure ofΓ we see that the component of ν_τ∗(xσ) ∈Γk(N A)indexed by id[k] is

precisely xτ∈N Ak. But then by commutativity we have

(xσ) ∈kerΨn ⇒ ν_τ∗(xσ) ∈kerΨk ⇒ 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], butΨn on the

component N Anindexed by id[n]is just the inclusion N An ,→ An, so we conclude xid[n]=0 too, and hence(xσ) =0 meaningΨnis 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 withΓ: from

the canonical inclusion of normalized complex into the Moore complex for every n≥ 0 we get the maps:

Φn : N(ΓC)n,→C(ΓC)n=Γn(C) =

M

[n][k]

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Proposition 1.18. The natural inclusionsΦn : N(ΓC)n ,→ C(ΓC)n have image the factor Cn indexed

by id[n]. In particular, the family of morphismsΦ= (Φn)n≥0is a natural isomorphism N◦Γ ∼ =

−→idCh≥0.

Proof. We need to compute N(ΓC)in degree n. It consists of the elements ofΓn(C) =

M

[n][k]

Ck

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

[n]−− [σi 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

N(ΓC)n⊕D(ΓC)n ∼=Γ(C)n =

M

[n][k]

Ck=Cn⊕ · · ·

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

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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 machin-ery 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 corre-spondence 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 categoryC is called pointed if it has a zero object, i.e. if there exists an object 0∈ C

which is both initial and terminal inC. 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 isomor-phism, 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) LetC be a category, we define a pointed categoryC+

by freely adjoining a zero object toC. 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 ofC+ are the same as the objects ofCwith 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

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Given two objects x, y∈ Cwe set

HomC+(x, y):=HomC(x, y)∪ {˙ 0} where 0 denotes the unique zero morphism x→0→y.

— The composition inC+is induced by the composition inC.

The free pointed category C+ comes equipped with the canonical inclusion functor C −→ Ci +

which is obviously not full.

We can rephrase construction 2.2 into a functor

+: Cat→Cat0

acting as follows:

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

• Morphisms: for a functor F :C → PwithP a pointed category, we can define the pointed functor F+:C+→ Pwhich we call the free pointed extension of F. F+coincides with F when

restricted to the original categoryC, plus it sends 0∈ C+ to 0 ∈ P. In particular, for any

two objects x, y∈ C we have

(x→0→y)7−−→ (F+ Fx→0→Fy)

so we observe that F+ maps the zero morphisms inC+to the zero morphisms inP.

More generally, given a functor F :C → D, then+ maps F to the free pointed extension of

the compositionC −→ DF −→ Di +, i.e. we get the pointed functor(F◦i)+:C+→ D+.

Notice that even for a pointed categoryPwe haveP 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 idxand 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

+: Cat→Cat0

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 categoryP, we have a natural bijection

Fun0(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 canonical inclusionC −→ Ci +

(F :C+ → P ) 7→ (F|C :C −→ Ci + → P )

From the RHS to the LHS we have the free pointed extension taking advantage of the fact thatP

is already pointed

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2.1. CATEGORICAL PRELIMINARIES 19 where G+ is defined to agree with G onC 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

by definition G+ and G agree on the original category C, while for F ∈ Fun0(C+,P )we have

(F|C)+ = F because a pointed functor is completely characterized by its action on the original

categoryC, 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 toCand is imposed to map 0∈ C+ 7→0∈ P, i.e. we get back F itself.

2.1.2 Quotient categories

LetN be the poset category over the natural numbers as in Chapter 1, a (non-negatively graded)

chain complexin a pointed categoryP is a functorNop→ 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 mor-phism inP. Recalling that chain maps are just natural transformations between such functors, we can see the category of chain complexes inPas 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. LetCbe a category, a two-sided ideal S⊆ Cis 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 wordsC ◦S◦ C ⊆S.

In this setup we define the quotient categoryC_S to be the following pointed category: — The objects of C_S are the objects ofC plus an additional zero object 0.

— Morphisms in C_S are determined by setting HomC

S

(x, y):= HomC(x, y)

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

Similarly to the situation with quotient of algebraic structures, the category C_S comes canonically equipped with a projection functor p :C → 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 construction has the universal property expressing the fact thatC_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, whereP 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 P

C/S

p F

F

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

S(x, y)we set F f =F f ∈HomP(Fx, Fy)and without

much choice the zero morphisms must be mapped to the zero morphisms inP.

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 categoryN. 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 pointed functor _(→→)N op→ P. So we define the category

Ch≥0(P ):=Fun0 _N (→→) op ,P

of (non-negatively graded) chain complexes inP.

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 arrowsand 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.

• • •

b

• • •

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.

• We will consider the right ideal Sing := E∨_6'◦B= {−→f −→ |g 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.

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2.2. DK-TRIPLES 21 • We will consider the left ideal B◦E6' = {−→h

f

−→ |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 composi-tion:

(f , g) 7→ f◦g= • g b •f .

This induces a pairing on the isomorphism classes of the categories, denoted by:

h−,−i_b: π0E(b) ×π0E(b)∨→π0ArB.

Definition 2.7. The dataB := (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

• e

0

•−→ • ¯f e∨ •.

T2. For every object b ∈ B the pairing h−,−i_b 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 pairingh−,−ibinduces

an n×n matrix(aij)1≤i,j≤n, where aij=i◦j with j standing for the jthdual 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). T3. The set E∨◦E= {f |g 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.

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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-tripleB we construct a pointed category N0(B)called the normalized pointed category ofB which is defined as the

quotient

N0:= M

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 N0as follows:

• The pointed category N0has a zero object 0 and for each object b∈B an object ¯b∈ N.

• For every pair of objects ¯b, ¯b0 ∈N, we have the hom-set

N0(¯b, ¯b0):=M(b, b0)/(M∩Sing) = (M∩Reg)(b0, b) ∪ {¯b→0→b¯0}.

• Composition in N0is 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). LetB = (B, E, E∨) be a DK-triple with asso-ciated normalized pointed category N0 = N0(B), then for every weakly idempotent complete additive

categoryAwe have an equivalence of categories

Fun0(B+,A)←−−−→' Fun0(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)

V := N0 R0 0 B+ := M Sing Sing\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 N0or to B+is inherited from

N0and B+respectively.

– The zero morphism is the only arrow in V from an object in N0to 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}.

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2.3. THE MAIN THEOREM 23 • Composition in V is induced by the composition in N0and B+; the composition

N0(n, n0) ×R0(b, n) ×B+(b0, b) −→R0(b0, n0)

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

This construction is made in such a way that both B+ and N0embed fully faithfully into it

B+ ,→V←- N0.

Then we can consider the restriction functors along these two embeddings ResB+ : Fun

0₍_V,_{A) →}_Fun0₍_B

+,A) and ResN0 : Fun

0₍_V,_{A) →}_Fun0₍_N 0,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

Fun0(B+,A) Fun0(V,A) Fun0(N0,A) LKE ResB+ ⊥ Res_N0 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 Abopis 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

Fun(Bop, Ab)←−−−→' Fun0(N0(B)op, Ab)

Proof. By the remark, we know that Abopsatisfies the hypothesis of Theorem 2.8, so we have Fun0(B+, Abop)←−−−→' Fun0(N0(B), Abop).

On the other hand, we know from the adjunction of Lemma 2.2 that Fun(B, Abop) =Fun(B, UAbop) ∼=Fun0(B+, Abop)

and composing these two equivalences we get

Fun(B, Abop)←−−−→' Fun0(N0(B), Abop).

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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

normal-ization functor forgets about all the maps of the simplicial structure and keeps track only of the nthface 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 nthface 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 of elementary face maps, but none of them can be of the nth 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 :=E∨_6'◦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.

The Dendroidal Dold-Kan Correspondence

MSc Mathematics

Master Thesis