(Co)homology of crossed modules

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(Co)homology of crossed modules

Sebastian Thomas

Diplomarbeit September 2007

Rheinisch-Westfälisch Technische Hochschule Aachen Lehrstuhl D für Mathematik

Prof. Dr. Gerhard Hiß

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This is a revised version from June 11, 2008.

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Introduction

In the 1940s,Eilenberg andMacLanedeveloped a homology theory for groups (see [10], [11] for example).

By definition, the homology of a group G is the (singular) homology of a connected CW-space T, called its classifying space, whose fundamental groupπ₁(T)is isomorphic to the given groupGand whose higher homotopy groups π_n(T)forn≥2 are all trivial. Previously, they andHurewicz (cf. [14] resp. [18]) had independently recognised that the homotopy type and hence the homology of such a topological space is uniquely determined by its fundamental group. In fact, the achievement of Eilenbergand MacLane was their purely algebraic approach to the homology of groups, circumventing topological spaces. To this end, they implicitly used a combinatorial model for the topological space in question, namely the classifying simplicial set BG of the group G. The calculation of its homology leads to the homological algebra description of the homology of G via a projective resolution.

So groups determine connected homotopy types T with only π1(T) as non-vanishing homotopy group. The homology of T is algebraically calculable starting from this group. Later, in 1949, Whitehead introduced crossed modules (cf. [30]), which determine connected homotopy types T with onlyπ1(T)and π2(T) as non- vanishing homotopy groups (cf. [25]), also known as2-types. Examples of crossed modules comprise inclusions of normal subgroups in a group, the inner automorphism homomorphism from a group to its automorphism group, or any surjection from a central extension of a group to this group.

In this work, we want to calculate the homology of T algebraically. Therefore we can now proceed as follows.

Given a crossed moduleV corresponding toT, we attach a classifying simplicial setBV toV, combinatorially modellingT. The homology ofBV is given in algebraic terms and calculates the homology ofT. In analogy to the definition of the homology of groups, we may now define the homology ofV as the homology ofT, or, what amounts to the same, ofBV.

The construction of the classifying simplicial setBV is done in two steps. First, we associate toV a simplicial group, its coskeleton CoskV. Second, we construct a classifying simplical set BG for a general simplicial groupG. Thereafter we may defineBV := B CoskV.

To motivate this construction, firstly, we can mention that to a simplicial groupG, we can associate a crossed module TruncG. If only the first two homotopy groups of G are nontrivial, then there is a weak homotopy equivalence Cosk TruncG'G, so that each such simplicial group is modelled by a crossed module via Cosk.

Secondly, simplicial groups model all connected homotopy types viaB.

To construct the classifying simplicial setBGof a simplicial groupG, there are two possibilities.

First, Kan introduced in [20] the Kan classifying functor W. This functor is the right adjoint and actually, the homotopy inverse to the Kan loop group functor, which is a combinatorial analogon to the topological loop space functor. This justifies callingWGa classifying simplicial set ofG.

Second, we can invoke bisimplicial sets in the following way. As mentioned in the beginning, we can attach a classifying simplicial set to a group, which in this context shall be called its nerve. A simplicial group is a sequence of groups, together with so-called face and degeneracy morphisms between them. So we can apply this nerve functor to the groups occuring in this sequence. We end up with two simplicial directions, one from the simplicial group and one from the nerve construction; i.e. we end up with a bisimplicial setB⁽²⁾G. Reading off the diagonal simplicial set of this bisimplicial set, we obtain the second variant for the classifying simplicial set ofG.

The second variant is more common; the first, however, yields smaller objects in a certain sense, which is more convenient for direct calculations.

v

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simplicial sets

associated

complex // of abelian groups^complexes

^homology // ^abelian_groups

crossed modules

coskeleton// ^simplicial_groups

classifyingKan functor

99

nerve

%%

bisimplicial sets

diagonal simplicial

set

OO

total simplicial

set

OO

associated double

complex // _complexes^double

total complex

OO

associated spectral

sequence // _sequences^spectral

approximation

OO

It is well-known that these two variants for the classifying simplicial set of Gare indeed homotopy equivalent after topological realisation (¹). Better still, the Kan classifying functor W can be obtained as the composite of the nerve functor with a so-called total simplicial set functor introduced byArtin andMazur[1] (²); and CegarraandRemedios[7] showed that already the total simplicial set functor and the diagonal functor yield homotopy equivalent results after topological realisation (³).

Here, we will give an algebraic proof by constructing a simplicial homotopy equivalence between both simplicial sets. In other words, we show that the triangle in the diagram above commutes up to simplicial homotopy equivalence. This confirms in an algebraic way that both variants for the classifying simplicial set ofGessentially coincide. As far as the author is aware, so far, this has only been known in a topological way.

The homology of a crossed module V is defined to be the homology of its classifying simplicial set BV. More generally, to calculate the homology of a simplicial set, one associates a complex to it and takes its homology.

Similarly, we can attach a double complex to a bisimplicial set. To a double complex in turn, we can attach its total complex. The generalised Eilenberg-Zilber theorem (due toDold, Puppeand Cartier [9]) states that the total complex of the double complex associated to a bisimplicial set is homotopy equivalent to the complex associated to its diagonal simplicial set; i.e. the quadrangle in the middle of the diagram above commutes up to homotopy equivalence of complexes.

We solve the exercise of constructing an explicit homotopy equivalence to prove Eilenberg-Zilber, adapting the arguments of EilenbergandMac Lanein [12] and [13].

The homology of the total complex of a double complex can be approximated by means of a spectral sequence.

Its starting terms are the horizontally taken homology groups of the vertically taken homology; it converges to the homology of the total complex. In our case of the classifying bisimplicial setB⁽²⁾Gof a simplicial groupG, this yields the Jardine spectral sequence [19], whose starting terms involve ordinary group homology, and which converges to the homology ofG. So the second variant of the classifying simplicial set of Genables us to use a spectral sequence. In particular, takingG= CoskV for a crossed moduleV, this yields a spectral sequence converging to the homology ofV.

To obtain results in cohomology instead of homology, we have to apply the duality functor _Z(−,Z) to the associated complex resp. to the associated double complex in the procedure described above.

Finally, we show by an example that the Jardine spectral sequence does not degenerate in the case of crossed modules.

InEllis’ approach to the (co)homology of crossed modules via quadratic modules, he develops a (co)homology theory for crossed modules that yields the (co)homology groups of its classifying space in dimensions less or equal than4[15]. Moreover,Carrasco,CegarraandGrandjéanin [6] develop still another (co)homology theory of crossed modules, andGrandjean,LadraandPirashviliestablished a long exact sequence relating this homology theory with the homology of crossed modules via classifying sets as considered here. Moreover, this alternative (co)homology theory was extended by Paoliin [27] to the case, where the coefficients are in a π1-module. These alternative (co)homology theories will not be dealt with here.

1Addendum (December 19, 2011): The fact thatWGandDiag NGare weakly homotopy equivalent has been shown byZis- man [31, sec. 3.3.4, cf. sec. 1.3.3, rem. 1]. He shows that a morphism Diag NG →WG, which is essentially the same as the morphismDGwe consider in chapter IV, §5, induces an isomorphism on the fundamental groups as well as isomorphisms on the homology groups of their universal coverings.

2This is not the total simplicial set as used byBousfieldandFriedlander[2, appendix B, p. 118].

3Addendum (December 19, 2011): To this end, Cegarra and Remediosconsider a morphism DiagX → TotX, which is essentially the same as the morphismφX we consider in proposition (3.15).

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Acknowledgements

First and foremost, I would like to thank Dr. Matthias Künzer. He taught me homological algebra and cohomology of groups, two subjects that have been the indispensable basis for me while I wrote this diploma thesis. Moreover, I would like to thank him for the interesting topic, for his comments and for the criticisms he gave to me, the numerous hours, in which we talked about mathematics, his helpfulness, supporting me in writing this diploma thesis, and for his patience.

I would like to thankProf. Dr. Gerhard Hißfor his willingness to supervise my diploma thesis. Moreover, I thank him for his lectures on algebraic topology and for the stimulating work environment at the Lehrstuhl D für Mathematik.

Furthermore, I would like to thank all professors I have met during my studies of mathematics at the RWTH Aachen University for the excellent education I had there. In particular, I would like to mentionProf. Dr.

Volker Enß, Prof. Dr. Gabriele Nebe, Prof. Dr. Ulrich Schoenwaelder and Prof. Dr. Eva Zerz.

I would like to thank the Studentenwerk Aachen for supporting my studies financially with BAföG.

Moreover, I would like to thank my family for giving me their love and trust, and especially my parents for their continuing support.

Last but not least I would like to thank my girlfriend Désirée for supporting me in each area of my life, giving me love and help everywhere and everytime.

Aachen, September 26, 2007 Sebastian Thomas

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Conventions and notations

We use the following conventions and notations.

• The composite of morphismsX −→^f Y andY −→^g Z is denoted byX −→^{f g} Z. The composite of functors C−→ D^F andD−→ E^G is denoted byC−−−→ E^G◦F .

• Isomorphy of objectsX andY in any category is denoted byX∼=Y.

• If C is a category and X, Y ∈ ObC are objects in C, we write _C(X, Y) = Mor_C(X, Y) for the set of morphisms between X andY. In particular, we write _Cat(C,D)for the set of functors between (small) categories C andD. To distinguish this notation from the functor category of functors betweenC andD as objects and natural transformations between functors as morphisms, we write(((C,D)))in the latter case.

• We suppose given categories C and D. A functor C −→ D^F is said to be an isofunctor if there exists a functorD−→ C^G such thatG◦F = id_C andF◦G= id_D. The categoriesCandDare said to be isomorphic, written C ∼=D, if an isofunctorC−→ D^F exists.

A functor C −→ D^F is said to be an (category) equivalence if there exists a functor D −→ C^G such that G◦F ∼= idC and F◦G∼= idD. The categories C and D are said to be equivalent, writtenC ' D, if a category equivalenceC−→ D^F exists.

• Given a functorI−→ C, we sometimes denote the image of a morphism^X i−→^θ j inI byX_i−−→^X^θ X_j. This applies in particular ifI=∆ôp orI=∆ôp×∆ôp.

• In certain standard categories like Set, Grp, Top, etc., we also use the common notation for the set of morphisms between two objects, for example, we write Map(X, Y) for the set of maps between sets X andY, we writeHom(G, H)for the set of group homomorphisms between groupsGandH, and we write C(T, U)for the set of continuous maps between topological spaces T andU.

• The category associated to a posetP is denoted byCat(P). Similarly, given a groupG, we writeCat(G) for the associated category with one object.

• Products of objectsX1 andX2in arbitrary categories are denoted asX1 Π X2. Pullbacks of morphisms X1 −→ Y, X2 −→ Y are denoted asX1ϕ₁Π_ϕ₂X2 = X1ϕ₁Π^Y_ϕ₂ X2. The diagonal morphism is written X −→^∆ X Π X.

• Given an index setI and a family of groups(Gi)_i∈I, we denote the direct product by

×

i∈IGi. Similarly for morphisms.

• Projections are denoted aspr, embeddings asemb.

• A subobject B of an objectAin an abelian category is denoted asBA.

• Given an additive categoryA, the additive category of complexes resp. double complexes inAis denoted by C(A) resp. C²(A). The full subcategory of C²(A) with objects C such that C_p,q ∼= 0for p < 0 or q <0is denoted byC²_q(A)

• If we have a complex C in an additive category A such that Cn ∼= 0 for n < 0, we usually omit to denote these zero objects. Similarly for morphisms, complex homotopies, etc. and for the dual situation ifCⁿ=C_−n∼= 0forn <0.

ix

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• In any complexCwith differentials∂, we writeZ_nC:= Ker(C_n−→^∂ C_n−1)andB_nC:= Im(C_n+1−→^∂ C_n).

• Homotopy equivalence of complexesC andDin an additive categoryAis denoted byC'D.

• We use the notationsN={1,2,3, . . .}andN0=N∪ {0}.

• The Kronecker delta is defined by δx,y =

(1 forx=y, 0 forx6=y,

where xandy are elements of some set.

• Given a mapf:X →Y and subsetsX⁰⊆X,Y⁰⊆Y withX⁰f ⊆Y⁰, we letf|^Y_X⁰0 the mapX⁰→Y⁰, x⁰7→

x⁰f. In the special cases, whereY⁰=Y resp.X⁰ =X, we also writef|X⁰ :=f|^Y_X0 resp.f|^Y⁰ :=f|^Y_X⁰.

• Given integersa, b, c∈Z, we write[a, b] :={z∈Z|a≤z≤b}for the set of integers lying betweenaand b. Furthermore, we write[a, b]∧c:= [a, b]\ {c} to omit elements in the interval.

Sometimes, we need some specified orientation, then we write da, be := (z ∈ Z | a ≤ z ≤ b) for the ascending interval and ba, bc= (z ∈Z|a ≥z ≥b) for the descending interval. Likewise da, be ∧c, etc.

Whereas we formally deal with tuples, we use the element notation, for example we write Y

i∈d1,3e

g_i=g₁g₂g₃and Y

i∈b3,1c

g_i=g₃g₂g₁ or

(gi)_i∈b3,1c= (g3, g2, g1) for group elements g₁,g₂,g₃.

• If we have tuples(x_j)_j∈A and(x_j)_j∈B with disjoint index setsAandB, then we write(x_j)_j∈A∪(x_j)_j∈B for their concatenation.

• A composite of zero morphisms is stipulated to be an identity. For instance,f₁. . . f_k= id ifk= 0.

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Chapter I

Simplicial objects

In this chapter, we recall the standard facts about simplicial sets or, more generally, simplicial objects in an arbitrary category. For further information, the reader is referred for example to [17], [23], [26], [29, §8].

§1 The category of simplex types

Before we can introduce simplicial sets, we have to study the following category.

(1.1) Definition(category of simplex types).

(a) Forn∈N0we let[n] :=Cat([0, n])be the category with objects[0, n]and exactly one morphismi−→j fori, j∈[0, n]if and only ifi≤j.

(b) The full subcategory ∆ in Cat with objects Ob∆ := {[n] | n ∈ N0} is called the category of simplex types.

Hence, if we disregard the category aspect of an object[n], the category∆ consists of linearly ordered sets[n]

as objects and monotonically increasing maps as morphisms.

(1.2) Example(embedding of∆inTop). For everyn∈N0we define thetopological standardn-simplex |∆ⁿ| to be

|∆ⁿ|:=

(x₀, . . . , x_n)∈Rⁿ⁺¹

X

j∈[0,n]

x_j = 1andx_j≥0for allj∈[0, n]

,

equipped with the relative topology. We consider for any morphism [m]−→^θ [n]the induced map θ∗:|∆^m| →

|∆ⁿ|defined by

(xi)_i∈[0,m]θ_∗:= ( X

i∈[0,m]

iθ=j

xi)_j∈[0,n] for allx= (xi)_i∈[0,m]∈ |∆^m|.

Since |∆^m| and |∆ⁿ| carry the relative topologies of R^m+1 resp.Rⁿ⁺¹, the map θ_∗ is continuous. If we have morphisms[m]−→^θ [n]and[n]−→^ρ [p]in ∆, this yields

(x_i)_i∈[0,m]θ_∗ρ_∗= ( X

i∈[0,m]

iθ=j

x_i)_j∈[0,n]ρ_∗= ( X

j∈[0,n]

jρ=k

X

i∈[0,m]

iθ=j

x_i)_k∈[0,p]= ( X

i∈[0,m]

i(θρ)=k

x_i)_k∈[0,p]= (x_i)_i∈[0,m](θρ)_∗

and

(xi)_i∈[0,m](id_[m])_∗= ( X

i∈[0,m]

iid_[m]=j

xi)_j∈[0,m]= (xj)_j∈[0,m]= (xi)_i∈[0,m]id_|∆m|

1

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for all(xi)_i∈[0,m]∈ |∆^m|. Hence

∆ ^|∆

−|

−−−→Top,([m]−→^θ [n])7→(|∆^m|−→ |∆^θ^∗ ⁿ|)

is well defined as a functor from the category of simplex types∆to the categoryTopof topological spaces.

In order to prove that |∆⁻| is faithful, we let [m] −→^θ [n] be an arbitrary morphism in ∆. Further we let {ei |i∈[0, m]}resp.{ej|j∈[0, n]}denote the standard basis ofR^m+1 resp.Rⁿ⁺¹. Then we get

eiθ_∗= ( X

i⁰∈[0,m]

i⁰θ=j

(ei)i⁰)_j∈[0,n] = ( X

i⁰∈[0,m]

i⁰θ=j

δi,i⁰)_j∈[0,n]= (δiθ,j)_j∈[0,n]=eiθ

for everyi∈[0, m]. Thus if we have morphisms[m]−→^θ [n]and[m]−→^ρ [n]in∆withθ_∗=ρ_∗, then in particular we haveeiθ=eiθ_∗=eiρ_∗=eiρ and thereforeiθ=iρfor alli∈[0, m]. Henceθ=ρ, and consequently|∆⁻|is a faithful functor.

We aim to distinguish generators for∆, which we will define now.

(1.3) Definition(cofaces and codegeneracies).

(a) Forn∈N,k∈[0, n], the morphism[n−1] ^δ

k

−→[n] defined by iδ^k :=

(i fori∈[0, k−1], i+ 1 fori∈[k, n−1]

is called thek-th coface of[n].

(b) Forn∈N0,k∈[0, n], the morphism[n+ 1] ^σ

k

−→[n]defined by iσ^k:=

(i fori∈[0, k], i−1 fori∈[k+ 1, n+ 1]

is called thek-th codegeneracy of[n].

(1.4) Proposition(cosimplicial identities). We letn∈Nbe a natural number. For the cofaces and codegeneracies the following identities hold:

δ^kδ^l=δ^l−1δ^k for0≤k < l≤n+ 1as morphisms[n−1]−→[n+ 1], σ^kσ^l=σ^l+1σ^k for0≤k≤l≤n−1as morphisms[n+ 1]−→[n−1], δ^kσ^l=







σ^l−1δ^k fork < l,

id_[n−1] forl≤k≤l+ 1, σ^lδ^k−1 fork > l+ 1







as morphisms [n−1]−→[n−1], wherek∈[0, n], l∈[0, n−1].

Proof.

(a) Ifk < l, then

iδ^kδ^l=

(iδ^l fori∈[0, k−1], (i+ 1)δ^l fori∈[k, n−1]

)

=







i fori∈[0, k−1], i+ 1 fori∈[k, l−2], i+ 2 fori∈[l−1, n−1]







=

(iδ^k fori∈[0, l−2], (i+ 1)δ^k fori∈[l−1, n−1]

)

=iδ^l−1δ^k for alli∈[0, n−1].

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(b) Fork≤l we calculate

iσ^kσ^l=

(iσ^l fori∈[0, k], (i−1)σ^l fori∈[k+ 1, n+ 1]

)

=







i fori∈[0, k], i−1 fori∈[k+ 1, l+ 1], i−2 fori∈[l+ 2, n+ 1]







=

(iσ^k fori∈[0, l+ 1], (i−1)σ^k fori∈[l+ 2, n+ 1]

)

=iσ^l+1σ^k for every i∈[0, n+ 1].

(c) Furthermore:

iδ^kσ^l=

(iσ^l fori∈[0, k−1], (i+ 1)σ^l fori∈[k, n−1]

)

=











i fori∈[0, k−1], k≤l+ 1, i+ 1 fori∈[k, l−1], k≤l+ 1, i fori∈[l, n−1], k≤l+ 1, i fori∈[0, l], k > l+ 1,

i−1 fori∈[l+ 1, k−1], k > l+ 1, i fori∈[k, n−1], k > l+ 1











=











iδ^k fori∈[0, l−1], k < l, (i−1)δ^k fori∈[l, n−1], k < l, i forl≤k≤l+ 1, iδ^k−1 fori∈[0, l], k > l+ 1, (i−1)δ^k−1 fori∈[l+ 1, n−1], k > l+ 1











=







iσ^l−1δ^k fork < l, i forl≤k≤l+ 1, iσ^lδ^k−1 fork > l+ 1.

Our next aim is to show that, in some sense, the cofaces and codegeneracies generate the category of simplex types∆and that the cosimplicial identities of the preceding proposition yield a set of relations defining∆.

(1.5) Notation. Givenm, n∈N0and0≤m1<· · ·< mt< mand0≤n1<· · ·< nu≤nfor somet, u∈N0, we write

σ^m^bt,1c:=σ^m^t· · ·σ^m¹ as morphism[m]−→[m−t]

and

δⁿ^d1,ue :=δⁿ¹· · ·δⁿ^u as morphism[n−u]−→[n].

(1.6) Remark. We let[m]−→^θ [n]in ∆be defined by θ:=σ^m^bt,1cδⁿ^d1,ue,

where 0 ≤m₁ <· · · < m_t < m and 0 ≤n₁ < · · · < n_u ≤n, and where t, u ∈N0 such that m−t =n−u.

Furthermore, we let k ∈ [0, t] and l ∈ [0, u] be the unique elements such that i ∈ [m_k+ 1, m_k+1] and iθ ∈ [n_l, n_l+1−1], wherem₀:=−1,m_t+1:=m,n₀:= 0and n_u+1:=n+ 1. Then we have

iθ=i−k+l for everyi∈[0, m].

Proof. By induction ont, the caset= 0being trivial, we have iσ^m^bt,1c =

(iσ^m^bt−1,1c fori∈[0, m_t],i.e. k∈[0, t−1], (i−1)σ^m^bt−1,1c fori∈[mt+ 1, m],i.e. k=t

)

=

(i−k fork∈[0, t−1], (i−1)−(t−1) fork=t

)

=i−k

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for alli∈[0, m]withi∈[mk+ 1, mk+1],k∈[0, t]. Furthermore, by induction onu, the caseu= 0being trivial, we have

iσ^m^bt,1cδⁿ^d1,ue =

(iσ^m^bt,1cδⁿ^d1,u−1e foriσ^m^bt,1cδⁿ^d1,u−1e ∈[0, nu−1],i.e. l∈[0, u−1], iσ^m^bt,1cδⁿ^d1,u−1e+ 1 foriσ^m^bt,1cδⁿ^d1,u−1e ∈[n_u, n−1],i.e. l=u

)

=

(iσ^m^bt,1c+l forl∈[0, u−1], iσ^m^bt,1c+ (u−1) + 1 forl=u

)

=iσ^m^bt,1c+l for alli∈[0, m]. Finally, we getiθ=iσ^m^bt,1cδⁿ^d1,ue = (i−k) +l.

(1.7) Theorem. Every morphism[m]−→^θ [n]in ∆can uniquely be written as θ=σ^m^bt,1cδⁿ^d1,ue,

where0≤m1<· · ·< mt< mand0≤n1<· · ·< nu≤n, and where t, u∈N0.

Proof. We begin by showing the existence of a factorisation. We letm₁ <· · ·< m_t be the elements of[0, m]

such that m_kθ= (m_k+ 1)θ for everyk∈[1, t]and we let n₁<· · ·< n_u be the elements of [0, n], that do not lie in[0, m]θ. Settingp:=m−t=n−uas well as σ:=σ^m^bt,1c and δ:=δⁿ^d1,ue, we have to show that we get the factorisationθ=σδ.

[m] ^θ //

σ

[n]

[p]

δ

??

Thereto we proceed by induction oni∈[0, m].

Ifi= 0andl:= 0θ∈[0, n], then, due to the monotony ofθ, we have[0, l−1]∩[0, m]θ=∅. Sincei∈[0, m1], remark (1.6) yields0σδ= 0−0 +l=l= 0θ.

Ifi∈[1, m], we choosek∈[0, t]andl∈[0, u]such thati∈[mk+ 1, mk+1]andiθ∈[nl, nl+1−1]. We distinguish the following two cases: Ifiθ= (i−1)θ, then by the choice ofm1, . . . , mtwe geti−1 =mk ∈[mk−1+ 1, mk].

Using the induction hypothesis and remark (1.6), this yields

iθ= (i−1)θ= (i−1)σδ= ((i−1)−(k−1))δ= (i−k)δ=iσδ.

Otherwise,iθ >(i−1)θandi−1∈[m_k+ 1, m_k+1]. We letl⁰∈[0, u]be such that(i−1)θ∈[n_l⁰, n_l⁰₊₁−1]. If l⁰=l, then, by the induction hypothesis and remark (1.6),

iθ= (i−1)θ+ 1 = (i−1)σδ+ 1 = ((i−1)−k+l) + 1 =i−k+l=iσδ.

Ifl⁰ < l, we must have(i−1)θ=nl⁰+1−1andiθ=nl+ 1. Further we havenl−nl⁰+1=l−(l⁰+ 1) =l−l⁰−1 since[nl⁰+1, nl]⊆[0, n]\([0, m]θ). By induction hypothesis and remark (1.6), we obtain

iθ= (i−1)θ+ (iθ−(i−1)θ) = (i−1)σδ+ ((nl+ 1)−(nl⁰+1−1))

= ((i−1)−k+l⁰) + (nl−nl⁰+1+ 2) =i−1−k+l⁰+l−l⁰−1 + 2 =i−k+l=iσδ.

Thus we haveθ=σδ.

Now, we show the uniqueness of the factorisation. We supposeθ=σδwithσ=σ^m^bt,1c andδ=δⁿ^d1,ue, where 0≤m1<· · ·< mt< mand0≤n1<· · ·< nu≤nwitht, u∈N0.

We claim thatm₁, . . . , m_t consists of exactly those elements i∈[0, m] withiθ = (i+ 1)θ. To this end, we let k ∈ [1, t] be such that i ∈ [m_k−1+ 1, mk]. Then the injectivity of δ yields the equivalence of (i+ 1)θ = iθ and (i+ 1)σ=iσ. But, by remark (1.6), this is equivalent to (i+ 1)σ=i−(k−1) = (i+ 1)−k, that is, to i+ 1∈[mk+ 1, mk+1]. Sincei∈[m_k−1+ 1, mk], saying i+ 1∈[mk+ 1, mk+1] is the same as sayingi=mk. This proves the claim.

Further, surjectivity ofσandδ=δⁿ¹· · ·δⁿ^u show that[0, n]\([0, m]θ) ={n1, . . . , nu}. Therefore, the morphism θdetermines the numbersm1, . . . , mt andn1, . . . , nu. This shows the uniqueness of the representation.

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§2 Simplicial objects in arbitrary categories

(1.8) Definition(simplicial objects and their morphisms).

(a) We let C be an arbitrary category. The category of simplicial objects in C is defined to be the functor category

sC:= (((∆^op,C))).

An object insCis called asimplicial object inC, a morphism insCis calledmorphism of simplicial objects in C or asimplicial morphism inC.

(i) A simplicial object inSetis called asimplicial set, a morphism is called asimplicial map.

(ii) A simplicial object in Grp is called a simplicial group, a morphism is called a simplicial group homomorphism.

(iii) A simplicial object inAbGrpis called a simplicial abelian group, a morphism is called a simplicial homomorphism of abelian groups.

(iv) We let R be a ring. A simplicial object in R-Mod is called asimplicial R-module, a morphism is called asimplicialR-module homomorphism.

(v) A simplicial object in Topis called asimplicial topological space, a morphism is called asimplicial continuous map.

(b) Dually, we define for every categoryC thecategory of cosimplicial objects inC by csC:= (((∆,C))).

(1.9) Example (constant simplicial object). We letC be a category andX ∈ObC an object in C. Then the constant functor

∆^{op Const}−−−−−→ C^X

with(ConstX)_[n] =X forn∈N0and(ConstX)_θ= id_X forθ∈Mor∆is a simplicial object inC, theconstant simplicial object.

This yields a functorC−−−−→^Const sC by letting(Constf)_[n]:=f forn∈N⁰,f ∈_C(X, Y),X, Y ∈ObC.

(1.10) Example (singular simplicial set).

(a) We letn∈Nbe a natural number. Concerning example (1.2), the topological standard simplex functor

|∆⁻|is a (covariant) functor ∆−→Top, that is, a cosimplicial topological space.

(b) For an arbitrary topological spaceT, we let∆^{op S}−−→^T Setbe the contravariant functor given by ST := C(|∆⁻|, T).

This is a simplicial set, which is called the singular simplicial set to the topological spaceT. In fact, we have a functor

Top−→^S sSet

given byS(=) = C(|∆⁻|,=).

(1.11) Example. For any commutative ringR we let Set−−→^R− R-Modbe the functor that assigns to every setM the freeR-left-moduleRM on the setM and to every mapf:M →N for setsM andN theR-module homomorphism Rf: RM → RN, which is defined by the operation of f on the basis M. Since sSet and sR-Modare functor categories, this functorR−lifts to a functorsSet−−→^R− sR-Mod. If we have an arbitrary simplicial set X, then RX is per definitionem the simplicialR-module with(RX)[n] =RX[n], that is, the set ofn-simplices(RX)[n] ofRX is a freeR-module on the setX[n].

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(1.12) Definition (faces and degeneracies). For a simplicial objectX in a categoryC, we define morphisms X_[n]−→^d^k X_[n−1]

bydk := d^X_k :=X_δk fork∈[0, n],n∈N, called faces, and morphisms X_[n]−→^s^k X_[n+1]

bysk:= s^X_k :=X_σk fork∈[0, n],n∈N0, calleddegeneracies.

(1.13) Notation. We letXbe a simplicial object in a categoryC. Givenm, n∈N0and0≤m1<· · ·< mt< m and0≤n1<· · ·< nu≤nfor somet, u∈N0, we write

sm_d1,te:= sm₁· · ·sm_t as morphismX_[m−t] −→X_[m]

and

d_n_bu,1c := d_n_u· · ·d_n₁ as morphismX_[n]−→X_[n−u]. Furthermore, we use the interval notations

s_dk−t+1,ke:= s_k−t+1· · ·sk as morphismX_m−t−→Xm

and

d_bl,l−u+1c= dl· · ·dl−u+1 as morphismsXn −→Xn−u

fork∈[t−1, m−1],l∈[u−1, n],t∈[0, m−1],u∈[0, n].

(1.14) Proposition (simplicial identities). We let X be a simplicial object in a category C. The faces and degeneracies satisfy the following identities:

d_ld_k= d_kd_l−1 for0≤k < l≤n+ 1as morphisms X_[n+1]−→X_[n−1], slsk= sksl+1 for0≤k≤l≤nas morphismsX[n+1]−→X_[n−1], sldk =







dks_l−1, fork < l, idX_[n−1], forl≤k≤l+ 1, d_k−1sl, fork > l+ 1







as morphismsX_[n−1]−→X_[n−1], where k∈[0, n], l∈[0, n−1].

In particular, every face is a retraction and every degeneracy is a coretraction.

Proof. The required identities result from proposition (1.4).

The identities in the previous proposition are even characterising a simplicial object, as we will see now.

(1.15) Theorem (classical definition of a simplicial object). We let (Xn)_n∈N₀ be a sequence of objects in a categoryC and we suppose given morphisms

Xn d_k

−→X_n−1 fork∈[0, n], n∈N, and

Xn s_k

−→Xn+1 fürk∈[0, n], n∈N0, which satisfy the simplicial identities

d_ld_k=d_kd_l−1 for0≤k < l≤n+ 1as morphismsX_[n+1]−→X_[n−1], slsk =sksl+1 for0≤k≤l≤nas morphismsX[n+1]−→X_[n−1], sldk=







dks_l−1, fork < l, idX_[n−1], forl≤k≤l+ 1, d_k−1sl, fork > l+ 1







as morphismsX_[n−1]−→X_[n−1], wherek∈[0, n], l∈[0, n−1].

Then there exists a simplicial objectX in C withX_[n] =Xn for alln∈N0 andd^X_k =dk fork∈[0, n], n∈N, as well ass^X_k =sk fork∈[0, n],n∈N0.

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Proof. We define X on the objects of ∆ by X_[n] := Xn for n ∈ N⁰. On the morphisms in the category of simplex types ∆, we define X as follows: Given a morphism [m] −→^θ [n] in ∆, then according to theorem (1.7), there is a unique representation ofθas a composite of codegeneracies and cofaces,θ=σ^m^bt,1cδⁿ^d1,ue with 0≤m1<· · ·< mt< mand 0≤n1<· · · < nu ≤n. We letXθ :=dn_bu,1csm_d1,te :=dn_u· · ·dn₁sm₁· · ·sm_t. In particular, we have X_δk =dk fork∈[0, n],n∈N, andX_σk =sk fork∈[0, n], n∈N0. Since the morphisms dk for k∈[0, n],n∈N, andsk fork∈[0, n], n∈N0, satisfy the simplicial identities, while the corresponding cofaces and codegeneracies in ∆ satisfy the cosimplicial identities, X is compatible with the composition of morphisms and therefore a well defined functor

∆^op−→ C,^X

that is, a simplicial object inC.

(1.16) Proposition (classical definition of a simplicial morphism). We letX and Y be simplicial objects in a category C and we suppose given morphisms X_n −→^fⁿ Y_n forn∈N0. If these morphisms commute with the faces and degeneracies ofX andY, that is, if

f_nd_k = d_kf_n−1 fork∈[0, n], n∈N, and

f_ns_k= s_kf_n+1 fork∈[0, n], n∈N0,

then there exists a simplicial morphismX −→^f Y withf[n] =fn for alln∈N0. Proof. Follows from theorem (1.7).

At the end of this section, we want to fix some notions.

(1.17) Definition (n-simplices). We let X, Y be simplicial objects in a categoryC andX −→^f Y a simplicial morphism. We setX_n:=X_[n] andf_n:=f_[n] for alln∈N0. IfX_n is a set or has a set as underlying structure, then the elements ofX_n are called n-simplices. The0-simplices are also called verticesand the1-simplices are also callededges ofX. Then-simplices of the formx_n−1s_k forx_n−1∈X_n−1,k∈[0, n−1],n∈N, are said to bedegenerate.

(1.18) Definition (reduced simplicial set). A simplicial set X is called reduced, if it has exactly one vertex, i.e. if|X0|= 1. The full subcategory of reduced simplicial sets insSetis denoted by sSet0.

(1.19) Definition (cartesian product of simplicial sets).

(a) Given simplicial setsX, Y, we define their (cartesian)product X×Y by(X ×Y)_n :=X_n×Y_n for all n∈N0 and(X×Y)θ:=Xθ×Yθfor all morphisms [m]−→^θ [n]in the category of simplex types∆.

(b) Given simplicial setsX,Y,X⁰,Y⁰and simplicial mapsX −→^f X⁰,Y −→^g Y⁰, the simplicial mapX×Y −^f×g−−→ X⁰×Y⁰ is defined by(f×g)_n :=f_n×g_n for every n∈N0.

§3 The standard n-simplex

We consider a standard example of a family of simplicial sets which will be needed later.

(1.20) Definition (standard n-simplex). We let n ∈N0 be a non-negative integer. The standardn-simplex

∆ⁿ in the categorysSetis defined by

∆ⁿ:=∆(•,[n]),

that is, ∆ⁿ is the functor ∆^op −→ Set represented by [n]. For the set of m-simplices of ∆ⁿ, we write

∆ⁿ_m:= (∆ⁿ)m=∆([m],[n]).

(1.21) Lemma. The standardn-simplices ∆ⁿ withn∈N0 form a cosimplicial object ∆⁻ in the category of simplicial setssSet, that is,∆⁻∈Obcs(sSet).

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Proof. The homfunctor∆(•,−) = ∆⁻ is a functor with two arguments, contravariant in the first argument and covariant in the second one.

(1.22) Lemma. We letX be a simplicial set andn∈N0. Then we have a bijective correspondence between then-simplices inX and the simplicial maps∆ⁿ−→X. It is given by

X_n→sSet(∆ⁿ, X), x_n7→(θ7→x_nX_θfor every θ∈∆ⁿ_m) with inverse

sSet(∆ⁿ, X)→X_n, f 7→(id_[n])f_n.

Proof. This is a consequence from the Yoneda lemma.

The next corollary justifies the namecategory of simplex types for the category∆.

(1.23) Corollary. For all non-negative integersm, n∈N0, we have

sSet(∆^m,∆ⁿ)∼=∆([m],[n]).

Proof. Lemma (1.22) impliessSet(∆^m,∆ⁿ)∼= ∆ⁿ_m=∆([m],[n])for allm, n∈N0.

§4 The nerve

In this section, we study an example of a simplicial set, which is going to be the most important one for our purposes - the nerveNC of a given categoryC. Intuitively explained, the nerveNC of a given categoryC has the objects of C as vertices, while the morphisms are the edges. Furthermore, the 2-simplices are exactly the pairs of composable morphisms, the3-simplices are the triples, and so on. Thekth face is given by "deleting"

the object number k, that is, the two morphisms that end resp. start with this object are composed, and the kth degeneracy inserts an identity morphism for the object number k.

Since any group can be regarded as a category with a single object, we also obtain a nerve functor for groups.

We need the notion of a nerve for a category object and a group object in an arbitrary category (under certain technical conditions).

Throughout this section, we assume given a category C, in which pullbacks and a terminal object exist. A terminal object inCis denoted byT and the unique morphism from an objectX∈ObCis written asX −→^∗ T.

Examples of algebraic structures within arbitrary categories

An introduction to category objects and group objects can be found in [16].

(1.24) Definition (category objects and functors).

(a) Acategory object (orinternal category) inC consists of objects O, M ∈ObC and morphisms M −→^s O, M −→^t O, O−→^e M andM_tΠ_sM −→^c M, whereM _tΠ_sM is a pullback of the morphismst ands, such that the following four diagrams commute.

(STC) Source and target of the composition morphism:

M

t

M_tΠ_sM

pr₂

oo ^pr¹ //

c

M

s

O oo ^t M ^s //O

(STI) Source and target of the identity morphism:

Ooo îdÔ O îdÔ //

e

O

M

t

``

s

>>

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(AC) Associativity of the composition morphism:

M _tΠ_sM_tΠ_sM ^c^t^Π^s^id^M //

id_{M t}Πsc

M _tΠ_sM

c

M _tΠ_sM ^c //M

(CI) Composition of the identity morphism:

O_id_OΠ_sM ^e^Π^id^M //

pr₂

((

M_tΠ_sM

c

M_tΠ_id_OO

idMΠe

oo

pr₁

vvM

We call O the object of objects and M the object of morphisms in the category object, the morphisms s, t, e and c are called source (morphism), target (morphism), identity (morphism) and composition (morphism), respectively.

Given a category object C in C with object of objects O, object of morphisms M, source s, target t, identity eand compositionc, we write ObC:=O, MorC :=M,s := s^C:=s,t := t^C :=t,e := e^C :=e andc := c^C:=c.

(b) We letC,D be category objects inC. A functor fromC toD in C consists of two morphismsObC−→^o ObD andMorC−→^m MorD, that are compatible with the categorical structure morphisms, that is,

s^Co=ms^D,t^Co=mt^D,e^Cm=oe^D andc^Cm= (mΠm)c^D.

We call othemorphism on the objects andmthemorphism on the morphisms of the functor.

Given a functorffromCtoDconsisting of a morphism on the objectsoand a morphism on the morphisms m, we writeObf :=o,Morf :=mandC−→^f D.

Composition of functors is defined by the composition on the objects and on the morphisms.

(c) Thecategory of category objects inC, where the objects are the category objects inC and the morphisms are the functors inC, is denoted byCat(C).

(1.25) Example (category objects inSet). A category object inSet is just an arbitrary (small) category.

(1.26) Definition (group objects and group homomorphisms).

(a) Agroup(object) inC consists of an objectGin Cand morphismsG Π G−→^m G,T −→ⁿ GandG−→ⁱ G, such that the following diagrams commute.

(AM) Associativity of the multiplication:

G Π G Π G ^id^G^Π^m //

mΠidG

G Π G

m

G Π G ^m //G

(MN) Multiplication with identity:

T Π G ⁿ^Π^id^G //

pr₂

((

G Π G

m

G Π T

idGΠn

oo

pr₁

vvG

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(MI) Multiplication with inverse:

G ⁽ⁱ^id^G⁾ //

∗

G Π G

m

T ⁿ //G

G ^{( id}^Gⁱ⁾ //

∗

OO

G Π G

m

OO

We callm,nandithemultiplication(operation),identityorneutral(operation) andinversion(operation), respectively.

Given a group objectGinCwith multiplicationm, identitynand inversioni, then we writem := m^G :=m, n := n^G:=nandi := i^G:=i.

(b) We letG,H be group objects inC. Agroup homomorphism fromGto H inC is a morphismG−→^ϕ H, that is compatible with multiplication, neutral operation and inversion, that is

m^Gϕ= (ϕ Π ϕ)m^H,n^Gϕ= n^H andi^Gϕ=ϕi^H.

Composition of group homomorphisms inC is given just by the ordinary composition inC.

(c) Thecategory of group objects in C, where the objects are the group objects inC and the morphisms are the group homomorphisms inC, is denoted byGrp(C).

(1.27) Example (group objects inSet, TopandsSet).

(a) The group objects inSetare just ordinary groups.

(b) In the category of topological spaces Top, the group objects are the topological groups. These are topological spaces whose underlying sets are endowed with a group structure such that the multiplication map and the inversion map are continuous.

(c) The group objects in the category of simplicial sets sSet are the simplicial objects in Grp and hence simplicial groups (more precisely, there is an equivalenceGrp(sSet)−→sGrp).

(1.28) Lemma.

(a) We suppose given a group object Gin C. Then_C(X, G)is a group for every X ∈ObC with m^C^(X,G)=

C(X,m^G),n^C^(X,G)=_C(X,n^G)andi^C^(X,G)=_C(X,i^G).

(b) We suppose given a group homomorphismG−→^ϕ H in C, where G, H ∈ ObGrp(C). Then C(X, ϕ) is a group homomorphism for every X∈ObC.

Proof. Follows from definition (1.26) and the fact that the hom functor_C(X,−)commutes with products.

As an application, we show by an example how results ordinary group theory (proven by calculations with elements) can be used to obtain results for category objects inC.

(1.29) Proposition. We suppose given group objectsGandH in C and a morphismG−→^ϕ H. Thenϕis a group homomorphism inC if and only ifm^Gϕ= (ϕ Π ϕ)m^H.

Proof. Ifϕis a group homomorphism inC, then in particularm^Gϕ= (ϕ Π ϕ)m^H.

So let us conversely assume thatϕis a morphism withm^Gϕ= (ϕ Π ϕ)m^H. Then we have

m^C^(X,G)_C(X, ϕ) =_C(X,m^G)_C(X, ϕ) =_C(X,m^Gϕ) =_C(X,(ϕ Π ϕ)m^H) =_C(X, ϕ Π ϕ)_C(X,m^H)

= (C(X, ϕ)×C(X, ϕ))m^C^(X,H)

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for everyX ∈ObC. Hence_C(X, ϕ)is a semigroup homomorphism and thus, by ordinary group theory, already a group homomorphism, that is, we have n^C^(X,G)_C(X, ϕ) = n^C^(X,H⁾ and i^C^(X,G)_C(X, ϕ) = _C(X, ϕ)i^C^(X,H) for every X∈ObC. In particular, we obtain

n^Gϕ= n^GC(T, ϕ) = (idT)C(T,n^G)C(T, ϕ) = (idT)n^C^{(T ,G)}C(T, ϕ) = (idT)n^C^{(T ,H)}= (idT)C(T,n^H) = n^H and

i^Gϕ= i^G_C(G, ϕ) = (idG)_C(G,i^G)_C(G, ϕ) = (idG)i^C^(G,G)_C(G, ϕ) = (idG)_C(G, ϕ)i^C^(G,H)

= (idG)C(G, ϕ)C(G,i^H) =ϕC(G,i^H) =ϕi^H Henceϕis a group homomorphism inC.

The nerve of a category object

Since we need the notion of a nerve for a category object in an arbitrary category, existing in every category with pullbacks, we have to introduce some notation.

(1.30) Definition. We let C, Dbe category objects inC andC−→^f D be a functor. We set

(MorC)^t^Π^sⁿ:=







ObC ifn= 0,

MorC ifn= 1,

MorC_tΠ^ObC_s (MorC)^t^Π^s⁽ⁿ⁻¹⁾ ifn >1, and analogously

(Morf)^Πⁿ:=







Obf ifn= 0,

Morf ifn= 1,

MorfΠ(Morf)^Π⁽ⁿ⁻¹⁾ ifn >1.

A morphismX −→^f (MorC)^t^Π^sⁿ can be denoted as the tuple(fpr_j)_{j∈bn−1,0c}. Furthermore, we define morphisms(MorC)^t^Π^sⁿ−→^t^j ObC and(MorC)^t^Π^sⁿ

c_bj

1,j0c

−−−−−→MorC by t_j :=

(pr_jt ifj < n, pr_n−1s ifj =n

forj∈[0, n],n∈N, resp.t₀:= idObC forn= 0, and c_bj

1,j₀c:=

(t_j

0e ifj₁=j₀,

(pr_j₁₋₁,c_bj

1−1,j0c)c ifj1> j0

forj0, j1∈[0, n]withj1≥j0,n∈N⁰. Thinking in elements, the notationc_bj

1,j₀c should simply express that the morphisms which start with object numberj1and end with the object numberj0 are composed. Similarly, the morphismt_j picks the object with numberj.

(1.31) Remark. We let C be a category object inC. There is a simplicial objectNC inC given by NnC:= (NC)n:= (MorC)^t^Π^Ob^s ^Cⁿ forn∈N0

and

NθC:= (NC)θ:=

(t_0θ ifm= 0, (cb(i+1)θ,iθc)_{i∈bm−1,0c} ifm >0

)

as morphismsNnC−→NmC,

for all morphisms[m]−→^θ [n]in ∆,m, n∈N0.

(Co)homology of crossed modules