Simplicial Galois Theory Of A Free Simplicial Groups With Given Cw-Bases

2 (2), 2008, 166-176

©BEYKENT UNIVERSITY

SIMPLICIAL GALOIS THEORY OF A FREE

SIMPLICIAL GROUPS WITH GIVEN CW-BASES

Ali MUTLU, Berrin MUTLU, Emel ÜNVER, Emine USLU

1

1 Celal Bayar University, Faculty of Science and Arts, Department of Mathematics

Muradiye Campus, 45030, Manisa/TURKEY e-mail: ali.mutlu@bayar.edu.tr

Received: 20 June 2007, Accepted: 09 May 2008

ABSTRACT

We introduce a corresponding theory in dimension two for a free simplicial group with given CW-bases in [12, 13] as a consequence of a generalised simplicial Galois theory. This verifies an equivalence between a simplicial category of a free simplicial group with given CW-bases F and a category of actions on simplicial group of a certain double group constructed from F .

Keywords: Homological Algebra, Free Simplicial Groups CW-bases.

2000 Math. Subj. Class.: 55U10; 18G55

CW-TABANLARIYLA VERİLEN SERBEST

SİMPLİŞIL GRUPLARIN SİMPLİŞIL GALOİS

TEORİSİ

ÖZET

Simplişıl Galois teorisinin bir genelleştirmesinin bir sonucu olarak [12, 13]'deki CW-tabanlarıyla verilen bir simplişıl grup için ikinci boyuta karşılık gelen bir teoriyi takdim ederiz. Bu bir F CW tabanlarıyla verilen bir serbest simplişıl grubun bir simplişıl kategorisi ve F 'den oluşturulan bir çift grubun simplişıl grup üzerindeki etkilerinin kategorisi arasında bir denkliği üretir.

0. INTRODUCTION

There is a classical definition of covering maps of a good space F in terms of actions of the fundamental group n1( F , f ) of F . This definition can instead be used as a description of the fundamental group, where F be a free simplicial group with given CW-bases with respect to [12, 13].

There are second order analogues of the fundamental group. These include not just the second homotopy group but also the crossed module formed by the second relative homotopy group and the fundamental group, as considered by Mac Lane and Whitehead [14] and by Whitehead [15]. Several closely related structures were proposed by Quillen (the crossed module of a fibration), Brown and Higgins (the double groupoid of a pair [3], crossed modules over groupoids [4]), Loday (the fundamental cat1-group of a map [6,11]), and others and however there has been corresponding theory of second order covering maps in [5], but in [5], Brown and Janelidze did not never use category of a free simplicial groups with given CW-bases for this issue.

The aim of this paper is to evolve such a theory for a free simplicial groups with given CW-bases, as a special case of simplicial Galois theory in categories of free simplicial groups with given CW-bases in [12,13] with respect to [8]. The second order notion of a free simplicial group with given CW-bases ensuring here as the simplicial Galois group of a fibration is slightly different from the above notions but it generates the same notion of the second relative homotopy group, considered as a crossed module.

The paper consists of three sections. The first section recollects an appropriate simplified version of simplicial Galois theory in categories of free simplicial groups with given CW-bases in [12,13] with respect to an adjunction [8]. The second section introduces the connection with the classical simplicial theory of coverings. The third section presents the simplicial Galois theory with respect to the adjunction between free simplicial groups and groupoids with given CW-bases in the sense of [12,13].

Acknowledgements: We would like to thank University of Celal Bayar for its supported during our study.

1. THEORY OF SIMPLICIAL GALOIS IN CATEGORIES

OF FREE SIMPLICIAL GROUPS WITH GIVEN

CW-BASES

Let

FrSimpGrps

be a category of a free simplicial groups with given CW-bases in [12,13] with pullbacks and F a class of morphisms in

FrSimpGrps

containing all isomorphisms, closed under compositions and pullback stable: F can be studied as a pseudo functor

F : FrSimpGrps

op

— Catgrp

described as follows: given an object

F

in

FrSimpGrps,

the objects of F ( F ) are all pairs

(G,a),

where

a: G — F is a morphism in F and morphisms are all commutative triangles in

FrSimpGrps

We write

F ( F )

=

(FrSimpGrps

l

F).

Note that for any morphism p : E a F in F the pullback functor

F(p) = p* = (FrSimpGrps l F) >(FrSimpGrps l E)

(G,a)

a

(EXF

G, pr

x

)

has a left adjoint

p*

=

(FrSimpGrps

l

E) »(FrSimpGrps

l

F)

which is the composition with p, i.e. for a given object ( D , 8) in

(FrSimpGrps

l

E).

We have

p*(D,

8) =

(D, p8).

If in addition p* is monadic, then we say that p : E a F is an effective F -descent morphism. Let

FrSim

p

Gr

p

s Sim

p

Gr

p

s

, n

: 1FrSimpGrps ^ H I , £ ' IH ^ lSimpGrps

be an adjunction between categories

FrSimpGrps

and

SimpGrps

with pullbacks and let

F

and

F '

be classes of morphisms in

FrSimpGrps

and

FrSimpGrps

respectively satisfying the conditions above and where SimpGprs is a category of simplicial groups. If I(F) c F ' a n d

H (F')

c

F , then for any object

F

e

FrSimpGrps

we get an induced adjunction

iF

(FrSimpGrps l F) <

F

>(SimpGrps 11(F)),

H

n : 1(FrSimpGrpslF) ^ H 1 , ^ ' 1 H ^ 1(SimpGrpsll (F))

in which IF (G, a) = (I (G), I (a)); HF (SG, 0) = (F Xm (F) H (SG), pr1) via

for any (SG,@) in (SimpGrps ^ I(F));

VFoA)

=<

a,rİG

>:

G

^

F *HI(F) HI ( G );

^(SG.D) ~^SGI ( p r2)'

that is it is the composition

I (F X

H!(F)

HI(SG))

I(pr2 )

> IH (SG)

%G

> SG

where SG is a simplicial group.

Let

r =

(FrSimpGrps,SimpGrps,I,H,tf,£,F,F')

be the data above; as in [8] we say that r is a free simplicial Galois structure.

Let p : E a F be an effective F -descent morphism, that is ( E , p ) is a monadic extension in the sense of [8, Definition 6.7] and let

FSGal

I

(E, p) = I(EX

F

EX

F

E) > I(EX

F

E) < I(E)

be its simplicial Galois pregroup in the sense of [8]. The fundamental theorem of free simplicial Galois Theory [8, Theorem 6.8] establishes a certain category equivalence

Simpl

r

(E, p) • Cosimpl

r

(FSGalj (E, p), SimpGrps) (*

x

)

between a full simplicial subcategory

Simpl

r

(E, p)

of

(FrSimpGrps

l

F)

whose objects can be defined as simplicial coverings split over ( E , p ) and a certain category

Cosimpl

r

(FSGalj (E, p), SimpGrps) of cosimplicial-split

FSGalj (E, p)

-actions in

SimpGrps

. In this article we consider merely a particular case where

Simpl

r

(E, p) = {(G, a) E (FrSimpGrps lF) I nE<

F

G,

pri)

is an isomorphism} (*

2

)

and

Cosimpl

r

(FSGal

I

(E,p),SimpGrps) = SimpGrps

FSGdI

^

p)

n(SimpGrps 11(E)) (*

3

)

see [8] for details.

As stated by the consequences of [8], we obtain a conclusion as following. Corollary 1.1. The morphisms £E , £E X f E and £E X f E X f E are isomorphisms.

2. SIMPLICIAL COVERINGS OF FREE SIMPLICIAL

FAMILIES WITH GIVEN CW-BASES

Let A be a category of the skeleton of a free simplicial group with given CW-bases in [12, 13] and let

FrSimpGrps

=

Fam(A)

be the category of families of objects in

A

; a morphism

( f ' , a ) : (G

A

)

AeA

a

(SG

A

)

A

-

eA

'

consists of a map f ' : A a A ' and morphisms a: GA a G' ^ for all

Ae A. For example

FrSimpGrps

=

Fam(I),

where

FrSimpGrps

is the usual category of free simplicial groups with given CW-bases and I is the category of free simplicial groups with given CW-bases within completely a morphism.

If A has pullbacks, then so also does FrSimpGrps, but the converse is not true; therefore, we accept merely that

FrSimpGrps

has pullbacks.

We examine the following simplicial Galois structure

r = (FrSimpGrps,SimpGrps,I,H,tf,£,F,F'):

FrSimpGrps

=

Fam(A)

as above, presuming that

A

has a terminal object t, and

FrSimpGrps

has pullbacks;

SimpGrps;

I: (GAeA a A H : SG a Z t = (G

g

.)

g

,

sSG

(SG e SimpGrps)

g'eSG 6 6

where Gg' = t for all g' e SG , with obvious T] and £ ; F and F ' are the

classes of all morphisms in

FrSimpGrps

and

SimpGrps,

respectively and both are the same.

If

p : E

a

F

is an effective descent morphism in

FrSimpGrps

with E, i.e. with

E

in

A ,

then

Simpl

r

(E, p)

consists of those

(G,a) e

(FrSimpGrps

l

F)

for which there exists an isomorphism in

(FrSimpGrps l F) of the form

( * 4 )

where Z E is a (possibly infinite) coproduct of copies of E with the canonical morphism to E, in fact this coproduct is just a family each member of which is E .

Obviously (*4) agrees with the ordinary notion of covering space. Additionally, under an appropriate select of

FrSimpGrps = Fam(A)

and p : E a F , the category equivalence (*1) gives the classical equivalence

SimpCov(F) • SimpGrps

n{F) (*5) between the category

SimpCov(F)

of simplicial covering spaces over a good topological space F , and the category SimpGrpsn1( F) of its fundamental group actions. In fact (*5) is a

special case of the simplicial covering theory in a molecular topos see [1], which itself is a special case of the situation considered here as explained in detail in [9].

Observe that the category

FrSimpGrps

of free simplicial groups with given CW-bases can also be used as

FrSimpGrps

(any category of the form

FrSimpGrpsD, where D is a small category of free simplicial groups with

given CW-bases, is a molecular topos and in particular is the category of families of its connected objects), which again give (*5) as a special case of (*1).

3. THE SIMPLICIAL GALOIS STRUCTURE FOR

SECOND ORDER SIMPLICIAL COVERINGS

We consider the following simplicial Galois structure

r = (FrSimpGrps, SimpGrps, I, H ,tf,£, F, F ' )

FrSimpGrps = SimpGrps

A is the category of free simplicial groups with given CW-bases here and we use as far as possible the terminology below and notation of Gabriel and Zisman [7] for of free simplicial groups with given CW-bases ; SimpGrps is the category of simplicial groups.

H : SimpGrps

a

FrSimpGrps

is the canonical inclusion, often called the nerve functor, and written as DI in [7].

I = n

1

: FrSimpGrps

a

SimpGrps

(written as

n :

A E a

Gr

in [7]) is the left adjoint of the canonical inclusion

H : SimpGrps

a

FrSimpGrps

with obvious

rj

and £.

F is the class of fibration in the sense of Kan [7, p. 65] and so F ' = F n SimpGrps is the class of fibrations of free simplicial groups with given CW-bases in the sense of [2] therefore H (F') c F by the description, and openly also I (F) c F ' .

A free simplicial groups with given CW-bases F is a Kan complex if and merely if the unique map F a I is a fibration in the sense of [7. p.65]. Proposition 3.1. If

F

is a Kan complex, then

£

F

: I

F

H

F a

1f

SSimpGrps

i

I

(

F

))

is an isomorphism.

Proof: As

£

: IH

a

1

SimpGrps is an isomorphism, we have to indicate that for any fibration of the form ( : SG a I(F) in SimpGrps the morphism

I(pr

2

): I(F

XHI(F)

H (SG))

a

IH(SG)

is an isomorphism of groups.

Furthermore, since the functors I and H do not change vertices of free simplicial groups with given CW-bases, it adequate to prove that for any vertex ( f , g') of F XHI (F) H (SG) the homomorphism is an isomorphism.

n

(F

xH1 (F)

H (SG), ( f , g'))

a n

(H (SG), g) ( * )

is an isomorphism. The pullback diagram

presents a commutative diagram with exact rows

where F = H ( ( )(tfF ( f ) ) , and the homomorphism (*6) is an isomorphism by the standard (non-abelian) five-lemma (see, e.g., [2]) since we know that (i) all arrows not involving n0 are free simplicial group homorphisms;

(ii) n2( H I ( F ) , 7]F ( f ) ) = 0 since HI(F) is a group considered as a free simplicial group;

(iii) the projection ( { f } x F , ( f , g')) a ( F , g') is an isomorphism and hence so also are the induced maps on n1 and n0.

(iv) n ( F , f ) a n1(HI(F), nF ( f )) is an isomorphism by the definition

of n .

The exact sequence for a fibration used above is described in [7, p. 117] we may use it here since F is a Kan complex and H (SG) a HI(F) and

pr1 : F XHI(F) H (SG) a F are fibrations.

Proposition 3.2. Let p : E a F be a surjective fibration of Kan complex. Then

p

is an effective descent morphism in

FrSimpGrps

satisfying the Corollary 1.1.

Proof: Because

p

is surjective and

FrSimpGrps

is a topos,

p

is an effective global-descent morphism and so as to clarify that p is an effective

F descent morphism we need merely appear that if

is a pullback diagram in which pr1 : E XF G a E is a fibration, then so also

is a: G a F (see [10] for details). We consult a commutative diagram

( * 7 )

in which i is the inclusion of the k -horn of A " . We have to show that there exists a completion f ' : A" a G such t h a t # f ' = f", f ' i = g'.

We elect f1 : A" a E such that pf1 = f " . This is possible because p is

pr2

y y

f 1

E

— j r

As pr1 is a fibration, the lift f exists, and then pr2 f is the required

completion.

We can use Proposition 3.1 to complete the proof once we know that p satisfies the Corollary 1.1. For this it suffices to show that

E, EXF E, EXF EXF E are Kan complexes. This follows from the

assumptions that E is Kan and p : E a F is a fibration, since then E XF E XF E a E XF E a E are also fibrations. •

From this and the conclusions of [8] defined in the first section we have. Corollary 3.3. Let p : E a F is a surjective fibration of Kan complexes, then

there is a category equivalence

Simpl

T

(E, p) • SimpGrps

FSGal

'

(E,p)

n (SimpGrps l I(E)) in which

Simpl

r

(E, p)

is full subcategory of

(FrSimpGrps

l

F)

with objects those pairs (G, (X) for which the diagram

( * 8 )

is a pullback. Furthermore we get.

Proposition 3.4. Let p : E a F and p : E' a F be surjective fibrations of Kan complexes such that each connected component of E is contractible.

Then Simpl

r

(E\ p ) c Simpl

r

(E, p).

Proof: We need to prove merely that there exists a morphism f " : E' a E with p ' fm = p . This is a standard lifting argument on each component of

Proposition 3.5. Let p : E a F be a surjective fibration of Kan complex

then the free simplicial Galois group

FSGalj (E, p)

is an internal group on

simplicial groups i.e. is a double group.

Proof: Note that in [8. 5.5c] it enough to prove that the canonical morphisms

I((EXF E)XE (EXF E)) a I ( E X F E ) \E ) I(EXF E)

I((EXF E)XE (EXF E)XE (EXF

E)

a

I(Ex

p

E)x^

(Ei I(EXF E)X (E)

I

(EXF

E)

are isomorphisms. However, this follows from the more general known statement (which can be easily showed using again standard arguments

involving the exact sequence of a fibration) The functor I = 7T1 preserves all

pullbacks

in which L, K are Kan complexes and f is both a fibration and a split surjection. •

Obviously

FSGal

I

(E, p)

contains as an object

group Loday's catl-group

^(ExFE,*)— ' )7LX(E,*) of the

fibration p: E a F . This cat1-group is known to be equivalent to other

similar structures, for example the crossed module

7T

1

(F,

*)

>ft

1

(E,

*),

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