Pro-simplicial setFrom Wikipedia, the free encyclopedia

Contents

1 Finite group 11.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Finite abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.4 Groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Lagrange’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Sylow theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 Cayley’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.4 Burnside theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.5 Feit-Thompson theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.6 Classification of finite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Number of groups of a given order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Table of distinct groups of order n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Homotopy group 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Long exact sequence of a fibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Methods of calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 A list of methods for calculating homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Relative homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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ii CONTENTS

3 Inverse system 113.1 The category of inverse systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Direct systems/Ind-objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Pro-simplicial set 134.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Shape theory (mathematics) 145.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.1.1 Warsaw Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Simplicial set 166.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 Face and degeneracy maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.6 The standard n-simplex and the category of simplices . . . . . . . . . . . . . . . . . . . . . . . . . 186.7 Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.8 Singular set for a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.9 Homotopy theory of simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.10 Simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.11 History and uses of simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.12 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Étale homotopy type 227.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter 1

Finite group

A Cayley graph of the symmetric group S4

In abstract algebra, a finite group is a mathematical group with a finite number of elements. A group is a set ofelements together with an operation which associates, to each ordered pair of elements, an element of the set.[1] Witha finite group, the set is finite.

1

2 CHAPTER 1. FINITE GROUP

1.1 History

During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth,especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, thecomplete classification of finite simple groups was achieved, meaning that all those simple groups from which allfinite groups can be built are now known.During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased ourunderstanding of finite analogs of classical groups, and other related groups. One such family of groups is the familyof general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical orphysical objects, when those objects admit just a finite number of structure-preserving transformations. The theoryof Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associatedWeyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space. Theproperties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.

1.2 Examples

1.2.1 Permutation groups

Main article: Permutation group

The symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the nsymbols, and whose group operation is the composition of such permutations, which are treated as bijective functionsfrom the set of symbols to itself.[2] Since there are n! (n factorial) possible permutations of a set of n symbols, itfollows that the order (the number of elements) of the symmetric group Sn is n!.

1.2.2 Cyclic groups

Main article: Cyclic group

A cyclic group Zn is a group all of whose elements are powers of a particular element a where an = a0 = e, the identity.A typical realization of this group is as the complex nth roots of unity. Sending a to a primitive root of unity givesan isomorphism between the two. This can be done with any finite cyclic group.

1.2.3 Finite abelian groups

Main article: Finite abelian group

An abelian group, also called a commutative group, is a group in which the result of applying the group operationto two group elements does not depend on their order (the axiom of commutativity). They are named after NielsHenrik Abel.[3]

An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and theseorders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abeliangroup can be described directly in terms of these invariants. The theory had been first developed in the 1879 paperof Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generatedmodules over a principal ideal domain, forming an important chapter of linear algebra.

1.2.4 Groups of Lie type

Main article: Group of Lie type

1.3. MAIN THEOREMS 3

A group of Lie type is a group closely related to the group G(k) of rational points of a reductive linear algebraicgroup G with values in the field k. Finite groups of Lie type give the bulk of nonabelian finite simple groups. Specialcases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.Finite groups of Lie type were among the first groups to be considered in mathematics, after cyclic, symmetric andalternating groups, with the projective special linear groups over prime finite fields, PSL(2, p) being constructed byÉvariste Galois in the 1830s. The systematic exploration of finite groups of Lie type started with Camille Jordan'stheorem that the projective special linear group PSL(2, q) is simple for q ≠ 2, 3. This theorem generalizes to projectivegroups of higher dimensions and gives an important infinite family PSL(n, q) of finite simple groups. Other classicalgroups were studied by Leonard Dickson in the beginning of 20th century. In the 1950s Claude Chevalley realizedthat after an appropriate reformulation, many theorems about semisimple Lie groups admit analogues for algebraicgroups over an arbitrary field k, leading to construction of what are now called Chevalley groups. Moreover, as inthe case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups(Tits simplicity theorem). Although it was known since 19th century that other finite simple groups exist (for example,Mathieu groups), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriateextensions of Chevalley’s construction, together with cyclic and alternating groups. Moreover, the exceptions, thesporadic groups, share many properties with the finite groups of Lie type, and in particular, can be constructed andcharacterized based on their geometry in the sense of Tits.The belief has now become a theorem – the classification of finite simple groups. Inspection of the list of finite simplegroups shows that groups of Lie type over a finite field include all the finite simple groups other than the cyclic groups,the alternating groups, the Tits group, and the 26 sporadic simple groups.

1.3 Main theorems

1.3.1 Lagrange’s theorem

Main article: Lagrange’s theorem (group theory)

For any finite groupG, the order (number of elements) of every subgroupH ofG divides the order ofG. The theoremis named after Joseph-Louis Lagrange.

1.3.2 Sylow theorems

Main article: Sylow theorems

This provides a partial converse to Lagrange’s theorem giving information about how many subgroups of a givenorder are contained in G.

1.3.3 Cayley’s theorem

Main article: Cayley’s theorem

Cayley’s theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of thesymmetric group acting on G.[4] This can be understood as an example of the group action of G on the elements ofG.[5]

1.3.4 Burnside theorem

Main article: Burnside theorem

Burnside’s theorem in group theory states that if G is a finite group of order

4 CHAPTER 1. FINITE GROUP

paqb

where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non-Abelianfinite simple group has order divisible by at least three distinct primes.

1.3.5 Feit-Thompson theorem

The Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It wasproved by Walter Feit and John Griggs Thompson (1962, 1963)

1.3.6 Classification of finite simple groups

Main article: Classification of finite simple groups

The classification of the finite simple groups is a theorem stating that every finite simple group belongs to one offour categories described below. These groups can be seen as the basic building blocks of all finite groups, in a wayreminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Höldertheorem is a more precise way of stating this fact about finite groups. However, a significant difference with respectto the case of integer factorization is that such “building blocks” do not necessarily determine uniquely a group, sincethere might be many non-isomorphic groups with the same composition series or, put in another way, the extensionproblem does not have a unique solution.The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about100 authors, published mostly between 1955 and 2004. Gorenstein (d.1992), Lyons, and Solomon are graduallypublishing a simplified and revised version of the proof.

1.4 Number of groups of a given order

Given a positive integer n, it is not at all a routine matter to determine how many isomorphism types of groups oforder n there are. Every group of prime order is cyclic, because Lagrange’s theorem implies that the cyclic subgroupgenerated by any of its non-identity elements is the whole group. If n is the square of a prime, then there are exactlytwo possible isomorphism types of group of order n, both of which are abelian. If n is a higher power of a prime, thenresults of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphismtypes of groups of order n, and the number grows very rapidly as the power increases.Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, asa consequence, for example, of results such as the Sylow theorems. For example, every group of order pq is cyclicwhen q < p are primes with p−1 not divisible by q. For a necessary and sufficient condition, see cyclic number.If n is squarefree, then any group of order n is solvable. Burnside’s theorem, proved using group characters, statesthat every group of order n is solvable when n is divisible by fewer than three distinct primes, i.e. if n = paqb, wherep and q are prime numbers, and a and b are non-negative integers. By the Feit–Thompson theorem, which has a longand complicated proof, every group of order n is solvable when n is odd.For every positive integer n, most groups of order n are solvable. To see this for any particular order is usually notdifficult (for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60) but theproof of this for all orders uses the classification of finite simple groups. For any positive integer n there are at mosttwo simple groups of order n, and there are infinitely many positive integers n for which there are two non-isomorphicsimple groups of order n.

1.4.1 Table of distinct groups of order n

Main articles: oeis:A000001, oeis:A000688 and oeis:A060689

1.5. SEE ALSO 5

1.5 See also• Association scheme

• List of finite simple groups

• Cauchy’s theorem (group theory)

• Abelian group

• Non-abelian group

• P-group

• List of small groups

• Representation theory of finite groups

• Modular representation theory

• Monstrous moonshine

• Profinite group

• Finite ring

1.6 Notes[1] Clark, Allan (1984). Elements of abstract algebra. Dover. ISBN 0486647250.

[2] Jacobson 2009, p. 31

[3] Jacobson 2009, p. 41

[4] Jacobson 2009, p. 38

[5] Jacobson 2009, p. 72, ex. 1

[6] Humphreys, John F. (1996). A Course in Group Theory. Oxford University Press. pp. 238–242. ISBN 0198534590. Zbl0843.20001.

1.7 Further reading• Jacobson, Nathan (2009). Basic Algebra I (2nd ed.). Dover Publications. ISBN 978-0-486-47189-1.

1.8 External links• Number of groups of order n (sequence A000001 in OEIS)

• Number of Abelian groups of order n (sequence A000688 in OEIS)

• Number of non-Abelian groups of order n (sequence A060689 in OEIS)

• A classifier for groups of small order

Chapter 2

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplesthomotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopygroups record information about the basic shape, or holes, of a topological space.To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point)into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappingsare homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called then-th homotopy group, πn(X), of the given space X with base point. Topological spaces with differing homotopygroups are never equivalent (homeomorphic), but the converse is not true.The notion of homotopy of paths was introduced by Camille Jordan.[1]

2.1 Introduction

In modern mathematics it is common to study a category by associating to every object of this category a simplerobject that still retains a sufficient amount of information about the object in question. Homotopy groups are such away of associating groups to topological spaces.That link between topology and groups lets mathematicians apply insights from group theory to topology. For exam-ple, if two topological objects have different homotopy groups, they can't have the same topological structure—a factthat may be difficult to prove using only topological means. For example, the torus is different from the sphere: thetorus has a “hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with thelocal structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however,carry information about the global structure.As for the example: the first homotopy group of the torus T is

π1(T)=Z2,

because the universal cover of the torus is the complex plane C, mapping to the torus T ≅ C / Z2. Here the quotientis in the category of topological spaces, rather than groups or rings. On the other hand the sphere S2 satisfies

π1(S2)=0,

because every loop can be contracted to a constant map (see homotopy groups of spheres for this and more compli-cated examples of homotopy groups).Hence the torus is not homeomorphic to the sphere.

2.2 Definition

In the n-sphere Sn we choose a base point a. For a space X with base point b, we define πn(X) to be the set ofhomotopy classes of maps

6

2.2. DEFINITION 7

A torus

f : Sn → X

that map the base point a to the base point b. In particular, the equivalence classes are given by homotopies that areconstant on the basepoint of the sphere. Equivalently, we can define πn(X) to be the group of homotopy classes ofmaps g : [0,1]n → X from the n-cube to X that take the boundary of the n-cube to b.For n ≥ 1, the homotopy classes form a group. To define the group operation, recall that in the fundamental group,the product f ∗ g of two loops f and g is defined by setting

f ∗ g =

f(2t) ift ∈ [0, 1/2]

g(2t− 1), ift ∈ [1/2, 1]

The idea of composition in the fundamental group is that of traveling the first path and the second in succession, or,equivalently, setting their two domains together. The concept of composition that we want for the n-th homotopygroup is the same, except that now the domains that we stick together are cubes, and we must glue them along a face.We therefore define the sum of maps f, g : [0,1]n → X by the formula (f + g)(t1, t2, ... tn) = f(2t1, t2, ... tn) for t1in [0,1/2] and (f + g)(t1, t2, ... tn) = g(2t1 − 1, t2, ... tn) for t1 in [1/2,1]. For the corresponding definition in termsof spheres, define the sum f + g of maps f, g : Sn → X to be Ψ composed with h, where Ψ is the map from Sn to thewedge sum of two n-spheres that collapses the equator and h is the map from the wedge sum of two n-spheres to Xthat is defined to be f on the first sphere and g on the second.If n ≥ 2, then πn is abelian. (For a proof of this, note that in two dimensions or greater, two homotopies can be“rotated” around each other. See Eckmann–Hilton argument). Further, similar to the fundamental group, for a pathconnected space any two basepoint choices gives rise to isomorphic πn (see Allen Hatcher#Books section 4.1).It is tempting to try to simplify the definition of homotopy groups by omitting the base points, but this does notusually work for spaces that are not simply connected, even for path connected spaces. The set of homotopy classesof maps from a sphere to a path connected space is not the homotopy group, but is essentially the set of orbits of thefundamental group on the homotopy group, and in general has no natural group structure.A way out of these difficulties has been found by defining higher homotopy groupoids of filtered spaces and of n-cubes of spaces. These are related to relative homotopy groups and to n-adic homotopy groups respectively. A higherhomotopy van Kampen theorem then enables one to derive some new information on homotopy groups and even on

8 CHAPTER 2. HOMOTOPY GROUP

A sphere

homotopy types. For more background and references, see “Higher dimensional group theory” and the referencesbelow.

2.3 Long exact sequence of a fibration

Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy liftingproperty with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence ofhomotopy groups

... → πn(F) → πn(E) → πn(B) → πn₋₁(F) →... → π0(E) → 0.

Here the maps involving π0 are not group homomorphisms because the π0 are not groups, but they are exact in thesense that the image equals the kernel.Example: the Hopf fibration. Let B equal S2 and E equal S3. Let p be the Hopf fibration, which has fiber S1. Fromthe long exact sequence

⋯ → πn(S1) → πn(S3) → πn(S2) → πn₋₁(S1) → ⋯

2.4. METHODS OF CALCULATION 9

Composition in the fundamental group

and the fact that πn(S1) = 0 for n ≥ 2, we find that πn(S3) = πn(S2) for n ≥ 3. In particular, π3(S2) = π3(S3) = Z.In the case of a cover space, when the fiber is discrete, we have that πn(E) is isomorphic to πn(B) for all n greaterthan 1, that πn(E) embeds injectively into πn(B) for all positive n, and that the subgroup of π1(B) that correspondsto the embedding of π1(E) has cosets in bijection with the elements of the fiber.

2.4 Methods of calculation

Calculation of homotopy groups is in general much more difficult than some of the other homotopy invariants learnedin algebraic topology. Unlike the Seifert–van Kampen theorem for the fundamental group and the Excision theoremfor singular homology and cohomology, there is no simple known way to calculate the homotopy groups of a space bybreaking it up into smaller spaces. However, methods developed in the 1980s involving a van Kampen type theoremfor higher homotopy groupoids have allowed new calculations on homotopy types and so on homotopy groups. Seefor a sample result the 2008 paper by Ellis and Mikhailov listed below.For some spaces, such as tori, all higher homotopy groups (that is, second and higher homotopy groups) are trivial.These are the so-called aspherical spaces. However, despite intense research in calculating the homotopy groups ofspheres, even in two dimensions a complete list is not known. To calculate even the fourth homotopy group of S2 oneneeds much more advanced techniques than the definitions might suggest. In particular the Serre spectral sequencewas constructed for just this purpose.Certain Homotopy groups of n-connected spaces can be calculated by comparison with homology groups via theHurewicz theorem.

2.5 A list of methods for calculating homotopy groups

• The long exact sequence of homotopy groups of a fibration.

• Hurewicz theorem, which has several versions.

• Blakers–Massey theorem, also known as excision for homotopy groups.

10 CHAPTER 2. HOMOTOPY GROUP

• Freudenthal suspension theorem, a corollary of excision for homotopy groups.

2.6 Relative homotopy groups

There are also relative homotopy groups πn(X,A) for a pair (X,A), where A is a subspace of X. The elements of sucha group are homotopy classes of based maps Dn →X which carry the boundary Sn−1 into A. Two maps f, g are calledhomotopic relative to A if they are homotopic by a basepoint-preserving homotopy F : Dn × [0,1] → X such that, foreach p in Sn−1 and t in [0,1], the element F(p,t) is in A. The ordinary homotopy groups are the special case in whichA is the base point.These groups are abelian for n ≥ 3 but for n = 2 form the top group of a crossed module with bottom group π1(A).There is a long exact sequence of relative homotopy groups.

2.7 Related notions

The homotopy groups are fundamental to homotopy theory, which in turn stimulated the development of modelcategories. It is possible to define abstract homotopy groups for simplicial sets.

2.8 See also• Knot theory

• Homotopy class

• Homotopy groups of spheres

• Topological invariant

• Homotopy group with coefficients

• Pointed set

2.9 Notes[1] Marie Ennemond Camille Jordan

2.10 References• Hatcher, Allen (2002), Algebraic topology, Cambridge University Press, ISBN 978-0-521-79540-1

• Hazewinkel, Michiel, ed. (2001), “Homotopy group”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Ronald Brown, `Groupoids and crossed objects in algebraic topology', Homology, homotopy and applications,1 (1999) 1–78.

• G.J. Ellis and R. Mikhailov, `A colimit of classifying spaces’, arXiv:0804.3581 [math.GR]

• R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubicalhomotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011).

• H. Kamps and T.Porter, T. Abstract homotopy and simple homotopy theory., World Scientific Publishing Co.Inc., River Edge, NJ (1997).

Chapter 3

Inverse system

In mathematics, an inverse system in a category C is a functor from a small cofiltered category I to C. An inversesystem is sometimes called a pro-object in C. The dual concept is a direct system.

3.1 The category of inverse systems

Pro-objects in C form a category pro-C. The general definition was given by Alexander Grothendieck in 1959, inTDTE.[1]

Two inverse systems

F:I→ C

andG:J→ C determine a functor

Iop x J → Sets,

namely the functor

HomC(F (i), G(j))

The set of homomorphisms between F and G in pro-C is defined to be the colimit of this functor in the first variable,followed by the limit in the second variable.If C has all inverse limits, then the limit defines a functor pro-C → C. In practice, e.g. if C is a category of algebraicor topological objects, this functor is not an equivalence of categories.

3.2 Direct systems/Ind-objects

An ind-object in C is a pro-object in Cop. The category of ind-objects is written ind-C.

3.3 Examples• If C is the category of finite groups, then pro-C is equivalent to the category of profinite groups and continuoushomomorphisms between them.

• If C is the category of finitely generated groups, then ind-C is equivalent to the category of all groups.

11

12 CHAPTER 3. INVERSE SYSTEM

3.4 References• Bourbaki, Nicolas (1968), Elements of mathematics. Theory of sets, Translated from the French, Paris: Her-mann, MR 0237342.

• Hazewinkel, Michiel, ed. (2001), “S/s091930”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Segal, Jack; Mardešić, Sibe (1982), Shape theory, North-Holland Mathematical Library 26, Amsterdam:North-Holland, ISBN 978-0-444-86286-0

3.5 Notes[1] C.E. Aull; R. Lowen (31 December 2001). Handbook of the History of General Topology. Springer Science & Business

Media. p. 1147. ISBN 978-0-7923-6970-7.

Chapter 4

Pro-simplicial set

In mathematics, a pro-simplicial set is an inverse system of simplicial sets.A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets has finite homotopy groups.Pro-simplicial sets show up in shape theory, in the study of localization and completion in homotopy theory, and inthe study of homotopy properties of schemes (e.g. étale homotopy theory).

4.1 References• Edwards, David A.; Hastings, Harold M. (1980), "Čech theory: its past, present, and future”, The RockyMountain Journal of Mathematics 10 (3): 429–468, doi:10.1216/RMJ-1980-10-3-429, MR 590209.

• Edwards, David A.; Hastings, Harold M. (1976), Čech and Steenrod homotopy theories with applications togeometric topology, LectureNotes inMathematics, Vol. 542, Springer-Verlag, Berlin-NewYork,MR0428322.

13

Chapter 5

Shape theory (mathematics)

Shape theory is a branch of topology, which provides a more global view of the topological spaces than homotopytheory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates withthe Čech homology theory while homotopy theory associates with the singular homology theory.

5.1 Background

Shape theory was reinvented, further developed and promoted by the Polish mathematician Karol Borsuk in 1968.Actually, the name shape theory was coined by Borsuk.

5.1.1 Warsaw Circle

Borsuk lived and worked in Warsaw, hence the name of one of the fundamental examples of the area, the Warsawcircle. This is a compact subset of the plane produced by “closing up” a topologist’s sine curve with an arc.

The Warsaw Circle.

It has homotopy groups isomorphic to those of a point, but is not homotopy equivalent to a point—instead, theWarsaw

14

5.2. DEVELOPMENT 15

circle is shape-equivalent to a circle (one dimensional sphere). Whitehead’s theorem does not apply to Warsaw circlebecause it is not a CW complex.Remark: to be precise if a bit pedantic, a point above stands for a one point space.

5.2 Development

Borsuk’s shape theory was generalized onto arbitrary (non-metric) compact spaces, and even onto general categories,by Włodzimierz Holsztyński in year 1968/1969, and published in Fund. Math. 70 , 157-168, y.1971 (see Jean-Marc Cordier, Tim Porter, (1989) below). This was done in a continuous style, characteristic for the Čech homologyrendered by Samuel Eilenberg and Norman Steenrod in their monograph Foundations of Algebraic Topology . Dueto the circumstance, Holsztyński’s paper was hardly noticed, and instead a great popularity in the field was gained bya much less advanced (more naive) paper by Sibe Mardešić and Jack Segal, which was published a little later, Fund.Math. 72, 61-68, y.1971. Further developments are reflected by the references below, and by their contents.For some purposes, like dynamical systems, more sophisticated invariants were developed under the name strongshape. Generalizations to noncommutative geometry, e.g. the shape theory for operator algebras have been found.

5.3 References• Mardešić, Sibe (1997). “Thirty years of shape theory” (PDF). Mathematical Communications 2: 1–12.

• shape theory in nLab

• Jean-Marc Cordier, Tim Porter, (1989), Shape Theory: Categorical Methods of Approximation, Mathematicsand its Applications, Ellis Horwood. Reprinted Dover (2008)

• A. Deleanu and P.J. Hilton, On the categorical shape of a functor, Fund. Math. 97 (1977) 157 - 176.

• A. Deleanu, P.J. Hilton, Borsuk’s shape and Grothendieck categories of pro-objects, Math. Proc. Camb. Phil.Soc. 79 (1976) 473-482.

• Sibe Mardešić, Jack Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971) 41-59,

• K. Borsuk, Concerning homotopy properties of compacta, Fund Math. 62 (1968) 223-254

• K. Borsuk, Theory of Shape, Monografie Matematyczne Tom 59,Warszawa 1975.

• D.A. Edwards and H. M. Hastings, Čech Theory: its Past, Present, and Future, Rocky Mountain Journal ofMathematics, Volume 10, Number 3, Summer 1980

• D.A. Edwards andH.M.Hastings, (1976), [www.math.uga.edu/%7Edavide/Cech_and_Steenrod_Homotopy_Theories_with_Applications_to_Geometric_Topology.pdf Čech and Steenrod homotopy theories with appli-cations to geometric topology], Lecture Notes in Maths. 542, Springer-Verlag.

• Tim Porter, Čech homotopy I, II, Jour. LondonMath. Soc., 1, 6, 1973, pp. 429–436; 2, 6, 1973, pp. 667–675.

• J.T. Lisica, S. Mardešić, Coherent prohomotopy and strong shape theory, Glasnik Matematički 19(39) (1984)335–399.

• Michael Batanin, Categorical strong shape theory, Cahiers Topologie Géom. Différentielle Catég. 38 (1997),no. 1, 3–66, numdam

• Marius Dādārlat, Shape theory and asymptotic morphisms for C*-algebras, Duke Math. J., 73(3):687-711,1994.

• Marius Dādārlat, Terry A. Loring, Deformations of topological spaces predicted by E-theory, In Algebraicmethods in operator theory, p. 316-327. Birkhäuser 1994.

Chapter 6

Simplicial set

In mathematics, a simplicial set is a construction in categorical homotopy theory that is a purely algebraic model ofthe notion of a "well-behaved" topological space. Historically, this model arose from earlier work in combinatorialtopology and in particular from the notion of simplicial complexes. Simplicial sets are used to define quasi-categories,a basic notion of higher category theory.

6.1 Motivation

A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces that can be builtup (or faithfully represented up to homotopy) from simplices and their incidence relations. This is similar to theapproach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purelyalgebraic and do not carry any actual topology (this will become clear in the formal definition).To get back to actual topological spaces, there is a geometric realization functor which turns simplicial sets intocompactly generated Hausdorff spaces. Most classical results on CW complexes in homotopy theory have analogousversions for simplicial sets which generalize these results. While algebraic topologists largely continue to prefer CWcomplexes, there is a growing contingent of researchers interested in using simplicial sets for applications in algebraicgeometry where CW complexes do not naturally exist.

6.2 Intuition

Simplicial sets can be viewed as a higher-dimensional generalization of directedmultigraphs. A simplicial set containsvertices (known as “0-simplices” in this context) and arrows (“1-simplices”) between some of these vertices. Twovertices may be connected by several arrows, and directed loops that connect a vertex to itself are also allowed. Unlikedirected multigraphs, simplicial sets may also contain higher simplices. A 2-simplex, for instance, can be thought ofas a two-dimensional “triangular” shape bounded by an ordered list of three vertices A, B, C and three arrows f:A→B,g:B→C and h:A→C. In general, an n-simplex is an object made up from an ordered list of n+1 vertices (which are0-simplices) and n+1 faces (which are (n−1)-simplices). The vertices of the i-th face are the vertices of the n-simplexminus the i-th vertex. The vertices of a simplex need not be distinct and a simplex is not determined by its verticesand faces: two different simplices may share the same list of faces (and therefore the same list of vertices).Simplicial sets should not be confused with abstract simplicial complexes, which generalize simple undirected graphsrather than directed multigraphs.Formally, a simplicial set X is a collection of sets Xn, n=0,1,2,..., together with certain maps between these sets:the face maps dn,i:Xn→Xn−₁ (n=1,2,3,... and 0≤i≤n) and degeneracy maps sn,i:Xn→Xn₊₁ (n=0,1,2,... and 0≤i≤n).We think of the elements of Xn as the n-simplices of X. The map dn,i assigns to each such n-simplex its i-th face,the face “opposite to” (i.e. not containing) the i-th vertex. The map sn,i assigns to each n-simplex the degenerate(n+1)-simplex which arises from the given one by duplicating the i-th vertex. This description implicitly requirescertain consistency relations among the maps dn,i and sn,i. Rather than requiring these simplicial identities explicitlyas part of the definition, the short and elegant modern definition uses the language of category theory.

16

6.3. FORMAL DEFINITION 17

6.3 Formal definition

Let Δ denote the simplex category. The objects of Δ are nonempty linearly ordered sets of the form

[n] = 0, 1, ..., n

with n≥0. The morphisms in Δ are (non-strictly) order-preserving functions between these sets.A simplicial set X is a contravariant functor

X: Δ → Set

where Set is the category of small sets. (Alternatively and equivalently, one may define simplicial sets as covariantfunctors from the opposite category Δop to Set.) Simplicial sets are therefore nothing but presheafs on Δ.Alternatively, one can think of a simplicial set as a simplicial object (see below) in the category Set, but this is onlydifferent language for the definition just given. If we use a covariant functor X: Δ → Set instead of a contravariantone, we arrive at the definition of a cosimplicial set.Simplicial sets form a category, usually denoted sSet, whose objects are simplicial sets and whose morphisms arenatural transformations between them. There is a corresponding category for cosimplicial sets as well, denoted bycSet.

6.4 Face and degeneracy maps

The simplex category Δ is generated by two particularly important families of morphisms (maps), whose imagesunder a given simplicial set functor are called face maps and degeneracy maps of that simplicial set.The face maps of a simplicial set are the images in that simplicial set of the morphisms δ0, . . . , δn : [n− 1] → [n] ,where δi is the only injection [n−1] → [n] that “misses” i . Let us denote these face maps by d0, . . . , dn respectively.The degeneracy maps of a simplicial set are the images in that simplicial set of the morphisms σ0, . . . , σn : [n+1] →[n] , where σi is the only surjection [n + 1] → [n] that “hits” i twice. Let us denote these degeneracy maps bys0, . . . , sn respectively.The defined maps satisfy the following simplicial identities:

1. di dj = dj₋₁ di if i < j

2. di sj = sj₋₁ di if i < j

3. di sj = id if i = j or i = j + 1

4. di sj = sj di₋₁ if i > j + 1

5. si sj = sj₊₁ si if i ≤ j.

6.5 Examples

Given a partially ordered set (S,≤), we can define a simplicial set NS, the nerve of S, as follows: for every object [n]of Δ we set NS([n]) = hom ₒ- ₑ ( [n] , S), the order-preserving maps from [n] to S. Every morphism φ:[n]→[m] in Δis an order preserving map, and via composition induces a map NS(φ) : NS([m]) → NS([n]). It is straightforward tocheck that NS is a contravariant functor from Δ to Set: a simplicial set.Concretely, the n-simplices of the nerve NS, i.e. the elements of NSn=NS([n]), can be thought of as ordered length-(n+1) sequences of elements from S: (a0 ≤ a1 ≤ ... ≤ an). The face map di drops the i-th element from such a list,and the degeneracy maps si duplicates the i-th element.A similar construction can be performed for every category C, to obtain the nerve NC of C. Here, NC([n]) is the setof all functors from [n] to C, where we consider [n] as a category with objects 0,1,...,n and a single morphism from ito j whenever i≤j.

18 CHAPTER 6. SIMPLICIAL SET

Concretely, the n-simplices of the nerve NC can be thought of as sequences of n composable morphisms in C:a0→a1→...→an. (In particular, the 0-simplices are the objects of C and the 1-simplices are the morphisms of C.)The face map d0 drops the first morphism from such a list, the face map dn drops the last, and the face map difor 0<i<n composes the (i−1)st and ith morphisms. The degeneracy maps si lengthen the sequence by inserting anidentity morphism at position i.We can recover the poset S from the nerve NS and the category C from the nerve NC; in this sense simplicial setsgeneralize posets and categories.Another important class of examples of simplicial sets is given by the singular set SY of a topological space Y. HereSYn consists of all the continuous maps from the standard topological n-simplex to Y. The singular set is furtherexplained below.

6.6 The standard n-simplex and the category of simplices

The standard n-simplex, denoted Δn, is a simplicial set defined as the functor homΔ(-, [n]) where [n] denotes theordered set 0, 1, ... ,n of the first (n + 1) nonnegative integers. In many texts, it is written instead as hom([n],-)where the homset is understood to be in the opposite category Δop.[1]

The geometric realization |Δn| is just defined to be the standard topological n-simplex in general position given by

|∆n| = (x0, . . . , xn) ∈ Rn+1 : 0 ≤ xi ≤ 1,∑

xi = 1.

By the Yoneda lemma, the n-simplices of a simplicial set X are classified by natural transformations in hom(Δn,X). (Specifically, consider ∆n = ∆op(n,−) , then the Yoneda lemma gives Nat(∆op(n,−), X) ∼= X(n) ) The n-simplices of X are then collectively denoted by Xn. Furthermore, there is a category of simplices, denoted by∆ ↓ Xwhose objects are maps (i.e. natural transformations) Δn → X and whose morphisms are natural transformations Δn→ Δm over X arising from maps [n] → [m] in Δ. That is, ∆ ↓ X is a slice category of Δ over X. The followingisomorphism shows that a simplicial set X is a colimit of its simplices:[2]

X ∼= lim−→∆n→X

∆n

where the colimit is taken over the category of simplices of X.

6.7 Geometric realization

There is a functor |•|: sSet→CGHaus called the geometric realization taking a simplicial setX to its correspondingrealization in the category of compactly-generated Hausdorff topological spaces.This larger category is used as the target of the functor because, in particular, a product of simplicial sets

X × Y

is realized as a product

|X| ×Ke |Y |

of the corresponding topological spaces, where ×Ke denotes the Kelley space product. This product is the rightadjoint functor that takes X to XC as described here, applied to the ordinary topological product |X| × |Y |.To define the realization functor, we first define it on n-simplices Δn as the corresponding topological n-simplex |Δn|.The definition then naturally extends to any simplicial set X by setting

|X| = limΔn → X |Δn|

where the colimit is taken over the n-simplex category of X. The geometric realization is functorial on sSet.

6.8. SINGULAR SET FOR A SPACE 19

6.8 Singular set for a space

The singular set of a topological space Y is the simplicial set S(Y) defined by

S(Y)([n]) = homTop(|Δn|, Y) for each object [n] ∈ Δ,

with the obvious functoriality condition on the morphisms. This definition is analogous to a standard idea in singularhomology of “probing” a target topological space with standard topological n-simplices. Furthermore, the singularfunctor S is right adjoint to the geometric realization functor described above, i.e.:

homTₒ (|X|, Y) ≅ homS(X, SY)

for any simplicial set X and any topological space Y.

6.9 Homotopy theory of simplicial sets

In the category of simplicial sets one can define fibrations to be Kan fibrations. A map of simplicial sets is defined tobe a weak equivalence if its geometric realization is a weak equivalence of spaces. A map of simplicial sets is definedto be a cofibration if it is a monomorphism of simplicial sets. It is a difficult theorem of Daniel Quillen that thecategory of simplicial sets with these classes of morphisms satisfies the axioms for a proper closed simplicial modelcategory.A key turning point of the theory is that the geometric realization of a Kan fibration is a Serre fibration of spaces.With the model structure in place, a homotopy theory of simplicial sets can be developed using standard homotopicalalgebra methods. Furthermore, the geometric realization and singular functors give a Quillen equivalence of closedmodel categories inducing an equivalence of homotopy categories

|•|: Ho(sSet) ↔ Ho(Top)

between the homotopy category for simplicial sets and the usual homotopy category of CW complexes with homotopyclasses of maps between them. It is part of the general definition of a Quillen adjunction that the right adjoint functor(in this case, the singular set functor) carries fibrations (resp. trivial fibrations) to fibrations (resp. trivial fibrations).

6.10 Simplicial objects

A simplicial object X in a category C is a contravariant functor

X: Δ → C

or equivalently a covariant functor

X: Δop → C

When C is the category of sets, we are just talking about simplicial sets. Letting C be the category of groups orcategory of abelian groups, we obtain the categories sGrp of simplicial groups and sAb of simplicial abelian groups,respectively.Simplicial groups and simplicial abelian groups also carry closed model structures induced by that of the underlyingsimplicial sets.The homotopy groups of simplicial abelian groups can be computed by making use of the Dold-Kan correspondencewhich yields an equivalence of categories between simplicial abelian groups and bounded chain complexes and isgiven by functors

N: sAb→ Ch+

and

Γ: Ch+ → sAb.

20 CHAPTER 6. SIMPLICIAL SET

6.11 History and uses of simplicial sets

Simplicial sets were originally used to give precise and convenient descriptions of classifying spaces of groups. Thisidea was vastly extended by Grothendieck's idea of considering classifying spaces of categories, and in particular byQuillen's work of algebraic K-theory. In this work, which earned him a Fields Medal, Quillen developed surprisinglyefficient methods for manipulating infinite simplicial sets. Later these methods were used in other areas on the borderbetween algebraic geometry and topology. For instance, the André-Quillen homology of a ring is a “non-abelianhomology”, defined and studied in this way.Both the algebraic K-theory and the André-Quillen homology are defined using algebraic data to write down a sim-plicial set, and then taking the homotopy groups of this simplicial set. Sometimes one simply defines the algebraicK -theory as the space.Simplicial methods are often useful when one wants to prove that a space is a loop space. The basic idea is that ifG is a group with classifying space BG , then G is homotopy equivalent to the loop space ΩBG . If BG itself is agroup, we can iterate the procedure, and G is homotopy equivalent to the double loop space Ω2B(BG) . In case Gis an abelian group, we can actually iterate this infinitely many times, and obtain that G is an infinite loop space.Even if X is not an abelian group, it can happen that it has a composition which is sufficiently commutative so thatone can use the above idea to prove that X is an infinite loop space. In this way, one can prove that the algebraicK-theory of a ring, considered as a topological space, is an infinite loop space.In recent years, simplicial sets have been used in higher category theory and derived algebraic geometry. Quasi-categories can be thought of as categories in which the composition of morphisms is defined only up to homotopy, andinformation about the composition of higher homotopies is also retained. Quasi-categories are defined as simplicialsets satisfying one additional condition, the weak Kan condition.

6.12 See also• Delta set

• Dendroidal set, a generalization of simplicial set.

• Simplicial presheaf

• infinity-category

• Homotopy type theory

• Kan complex

• Dold–Kan correspondence

• Simplicial homotopy

• Simplicial sphere

6.13 Notes[1] S. Gelfand, Yu. Manin, “Methods of Homological Algebra”

[2] Goerss & Jardine, p.7

6.14 References• Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory. Progress inMathematics 174. Basel, Boston,Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.

• Gelfand, S.; Manin, Yu. Methods of homological algebra.

6.14. REFERENCES 21

• Dylan G.L. Allegretti, Simplicial Sets and van Kampen’s Theorem (An elementary introduction to simplicialsets).

• Daniel Quillen: Higher algebraic K-theory: I. In: H. Bass (ed.): Higher K-Theories. Lecture Notes in Mathe-matics, vol. 341. Springer-Verlag, Berlin 1973. ISBN 3-540-06434-6

• G.B. Segal, Categories and cohomology theories, Topology, 13, (1974), 293 - 312.

• simplicial set in nLab

Chapter 7

Étale homotopy type

In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type oftopological spaces for algebraic varieties.Roughly speaking, for a variety or scheme X, the idea is to consider étale coverings U → X and to replace eachconnected component of U and the higher “intersections”, i.e., fiber products, Un := U ×X U ×X · · · ×X U (n+1copies of U, n ≥ 0 ) by a single point. This gives a simplicial set which captures some information related to X andthe étale topology of it.Slightly more precisely, it is in general necessary to work with étale hypercovers (Un)n≥0 instead of the abovesimplicial scheme determined by a usual étale cover. Taking finer and finer hypercoverings (which is technicallyaccomplished by working with the pro-object in simplicial sets determined by taking all hypercoverings), the resultingobject is the étale homotopy type of X. Similarly to classical topology, it is able to recover much of the usual datarelated to the étale topology, in particular the étale fundamental group of the scheme and the étale cohomology oflocally constant étale sheaves.

7.1 References• Artin, Michael; Mazur, Barry (1969). Etale homotopy. Springer.

• Friedlander, Eric (1982). Étale homotopy of simplicial schemes. Annals of Mathematics Studies, PUP.

7.2 External links• http://ncatlab.org/nlab/show/étale+homotopy

22

7.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 23

7.3 Text and image sources, contributors, and licenses

7.3.1 Text• Finite group Source: https://en.wikipedia.org/wiki/Finite_group?oldid=682567149 Contributors: AxelBoldt, Zundark, Patrick, Michael

Hardy, TakuyaMurata, Silverfish, Schneelocke, Loren Rosen, Charles Matthews, Phys, Schutz, DHN, Tobias Bergemann, Giftlite, D3,Alberto da Calvairate~enwiki, Rgdboer, Andi5, Vipul, ABCD, HenryLi, Oleg Alexandrov, R.e.b., Ysangkok, Wavelength, Cullinane,RDBrown, Mhym, Dreadstar, Cydebot, Kilva, Baccyak4H, STBot, SparsityProblem, TXiKiBoT, Geometry guy, Radagast3, GirasoleDE,Messagetolove, Thehotelambush, JackSchmidt, Mr. Stradivarius, Sfan00 IMG, SilvonenBot, Good Olfactory, Addbot, Lightbot, Luckas-bot, LGB, AnomieBOT, Ciphers, JackieBot, Omnipaedista, Wpiechowski, Quondum, Wikfr, ChuispastonBot, Rezabot, KLBot2, Fraq-tive42, Luizpuodzius, ChrisGualtieri, AHusain314, Brirush, CsDix, KasparBot and Anonymous: 24

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