Journal of the Egyptian Mathematical Society (2011) 19, 137-141

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

www.etms-eg.org www.elsevier.com/locate/joems

SHORT COMMUNICATION

On the (co)homology theory of index category

Yasien Ghallab Gouda

Dept. of Mathematics, Faculty of Science, South Valley University, Aswan, Egypt Available online 2 February 2012

KEYWORDS

Index category; (Co)homology group

Abstract In the present work the different types of regular index categories are given. The related cohomology groups of some of these categories are studied. We give some properties of index category related with monoid and algebra over it.

© 2011 Egyptian Mathematical Society. Production and hosting by Elsevier B.V.

All rights reserved.

1. Introduction

Definition 1 6. An index category is a pair (A, i), where A is category with natural numbers as objects and i is an inclusion i : A fi A, where A is a subcategory of A. The category A is called a simplicial index category, it has objects an ordered set [n] = {0,1,...,n}, n = 0,1,... and the group morphisms Sn : [n]![n - 1], g' : [n]![n + 1], o 6 i 6 n, 0 6j 6 n, with the following identities:

= s^s-1, if i < j (1)

«+1 = «n if i 6 j

( S'n-1<-2; if i < j

r'nd'n+1 = \ Id\n], if i = j or i = j+ 1

{ S-tj if i y j + 1

E-mail address: yasiengouda@yahoo.com

1110-256X © 2011 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. All rights reserved.

Peer review under responsibility of Egyptian Mathematical Society. doi:10.1016/j.joems.2011.12.002

Among index categories, there is one, plays an important role in this field. It's called a regular index category.

Definition 2. The index category (A, i) is called regular if any morphism a 2 A(n, m) = HomA(n, m) can be uniquely written in the form a = u ° C, where u 2 A (m, n) = HomD(m, n), C 2 An = AutA(n). Clearly a regular index category is a subcategory of an index category. A regular index category plays an important role as we will show in the sequel.

Examples of regular index category: [7,9,10].

1. Simplicial index category (A). It's generated by morphisms 3i, a> (let ô'n = ô', = r).

2. Cyclic index category (C, i). It's generated by morphisms ôi, r and tn : [n] ! [n], f+1 = 1.

3. Reflexive index category (R, i). It's generated by morphisms ô!, r and rn : [n] ! [n], r2 = 1.

4. Dihedral index category (D, i). It's generated by morphisms ô', r, tn and rn.

The group of index categories (simplicial, cyclic, reflexive and dihedral) can be extended to the following class of categories [1,6].

1. Symmetry index category (S, i). It's generated by morphisms ô', r and an : [n] fi [n].

2. Bisymmetry index category (B, i). It's generated by morphisms ô', o-7, an.

3. Hyperoctahedral(Weil) index category (H, i). It's generated by morphisms S1, d and an.

We shall explain these categories in detail:

1.1. The symmetry index category (S,i) [2,3]

Consider the family S. = {Sn}nP0, where Sn is a permutation group of the set [n] = {0,1,..., n}. It's generated by the permutations 1, 2, ... , n, such that k — (k — 1, k). S is a symmetry index category with object the natural number and morphisms:

Sn : [n] ! [n — 1], rj : [n] ![n + 1], 0 6 i, j 6 n, (2)

an : [n] ! [n], an 2 Kn, n 2@, such that

an o S'n — Sfo di(an), an o ajn — o Si(an),

an o Pn — Pn o an ■

Clearly (S, is) is regular index category, where is : A fi S is inclusion functor.

Note that a cyclic index category, with object @ and group morphisms S1, d and tn = an = (n, n — 1,... ,1,0) is an example of symmetry index category.

1.2. Hyperoctahedral(Weil) index category (W,i)

Consider a group Wn = Sn • (Z/2)n + 1, where Z/2 = {—1,1}, x is a semidirect product and Sn is symmetry group. The elements of Wn is given by (a; eo,... ,sn), where a 2 Sn and Si = ±1. The multiplication in Wn defined by:

(a; So, ..., s„)-(p; Co, ..., Cn) — (a ■ b; spo ■ Co, ..., sm ■ y„):

Then, clearly that on every class there is a graded group, then the family {Wn}n>0 is a graded group.

1.3. Bisymmetry index category (T,i)

This group lies between S and W, S. c T. c W., where, Tn — Sn x@/2 c Tn — Sn x(@/2)"+', where x is a semi-direct product. The elements (a; s) of Tn through the inclusion of Tnin Wn is given by (a; s) fi (a; s0,s!,... ,sn). The family {Tn}nP0 is a graded group.

Definition 3. Following [6]:

(1) The regular index category can be considered as crossed simplicial group.

(2) There is an isomorphism between the regular index category and the category of crossed simplicial group, that is, the study of regular index category equivalence to the use of crossed simplicial group.

For more informations about crossed simplicial group and its properties see [1,4,5].

It is known that the simplical object, for an arbitrary category }, is a functor F: Aop fi } where Aop is the inverse image of A. If } is a category of sets (group, module), we can define the simplicial set (group, module) by the following relations:

F(M)= Fn, F([ô'}) = du = Sj

where the morphisms 6,, Sj are satisfies the relation (1).

Example 1. Let D[n]n = D(m, n) be the set of all morphisms in category A from m to n. The set {D[n], 6h Sj} is simplicial set, where 6,- : A [n]m fi A[n]m-i, Sj : A[n]m fi A[n]m + i.

Note that for a simplicial set X. we have the following isomorphism A(X.)m~nioXn x A(m, n)/v — Xp x AutA[m], where the equivalent relation 3 satisfies the following identities

(di, a)v(x; d' o a), (Sj, a)v(x, a' o a), a 2 A(m, n). (3)

Definition 4. Let A be an index category, for an arbitrary category }, the A-categorical object is functor F : Aop fi }, such that the following diagram is commutative:

// # Dop ! }

We can show that index categories S and W are simplicial set by defining the face and degeneracy maps.

• For the symmetry index category S:The face and degeneracy maps 0; and Sj are given by:

k + 1, i ^ k - 1, dt(k)={ Is, i = k, k + 1, k, i y k,

Sj (k) =

' k + 1, i ^ k - 1, , i y k,

k.k + 1, i — k — 1,

. k + 1 ■ k, i — k,

The face and degeneracy maps for the composition of maps are given by:

dt(a ■ b) — db(l) (a)^ db, Si(a ■ b) — Sb(l) (a)^ Sb, a, b 2 Sn, 0 6 i 6 n

• For Hyperoctahedral(Weil) index category W:The face and degeneracy maps 0i and Sj are given by:

di : Wn ! W—, Sj : Wn ! Wn+1, 0 6 i 6 n

@i(a; So, ... , Sn) (di(a), So, ... , bi, ... , Sn), Si(a; So, ... , Sn) — (s0 Si(a); S0, ... , Si Si, ... , Sn)

a 2 Sn, Si 2 f—1, +1} 2 @/2, 0 6 i 6 n

eo S-'(a) —

Such that

(a-i, ai+i) —

Si(a), Si — +1

(ai, a+i) Si (a), e,- — —1

0 ... a;... ai+1 ... n + 1 0 ... ai+1 ...ai... n + 1

2. Related subject with index category and

2.1. The K-cohomology of for K-index simplicial module

Definition 5. Let A be regular index category. For A-index simplicial module M*. The complex (Inva (M„ d^) is invariant complex, such that:

Inva(M„) = {f: Mn ! kf(m)a = sign(e—(a)-f(m), a 2 An}

f is linear and the differential dn : InvA(M.) ! Inva+1 (M.) is given by:

d"(f)(m) ^YfmSn) -(-1)', f 2 Inva(M.), m 2 Mn+,.

Theorem 1. Let k-field with char k = o and M* is A-index simplicial k — module. Then the A-(co)homology of M* is isomorphic to the (co)homology of invariant complex, i.e.

HAt(Mt)UH(InvA(Mt))

x(HA*(Inva(M„))uH(Inva (M.)). (10)

Consider the following index categories: the simplicial A, cyclic C and symmetry index category S.

Suppose that (M*) is symmetry k — module, where k is field with characteristic zero. Consider the inclusions A fi C fi S. The groups H , HC and HS are respectively simplicial, cyclic and symmetry cohomology groups of k — modules.

The aim of this part is to prove the following assertion:

Theorem 2. Suppose that M* is symmetric k — module (chark = 0), then the following isomorphism holds.

P„(f(m)) — j-1— X sign(a)f(m(a))

(n+ 1)!

, 1 ^ sSn(a)- sSnf(m)

( + )! a2Sn

(n 1)!

f (m) —f (m), f 2 InvS

From the above consideration we have the following assertions:

• T* is projective a qnd Inv'A(M,) and Inv's(M,) have the same homology.

• The morphism (¿i, o /2>) and P* are homological equivalent.

• The maps P* is also projection and we have the extension HC"(M*) = H*(M*) © L*.

• The map ¿1t : HC(M,) ! H'(M,) is projection.

• The following isomorphisms hold HC*(M*) u HC*(M*) and H*(M*) u H*(M*).

Following [ 8] from the cyclic module M* consider the module M— where M— = Mn ® Mn—2 ® Mn—4 ® .... We have the following exact sequence 0 ! M, ! M— ! M— [—2] ! 0 which induces the following isomorphism HCn(M*) u HCn — 2(M*) © Hn(M*) but HCn — 2(M*) u HC" — 4(M*) © HC" — 2(m*), then HC"(M*) u Hn(M*) © Hn — 2(M*) © Hn — 4(M*) © ... □

Example 2. Suppose M* = kS*(X) be singular simplicial k — module of topological space X (chark = 0), then H (M*) = H (X,k). In the complex kS*(X) there is a subcomplexes InvC(kS,(X)), InvS(kS,(X)). The homology of these complexes are given by:

HC-(M„)UH'-(M,)®H'-(BC, k) k

Hn(Inv*C(kSt(X)))uHn(X, k)® H"-2(X, k) Hn(InvS(kSt(X)))UHn(X, k).

Proof. From [10] BC = BSO(2) = P1 and hence it is necessary to prove the following isomorphism:

HCn(M,)uHn(M,)© H"—2(M,)® H"—4(M,)® ... (12)

consider the following sequence of complexes

InvA(M,)'^ InvC(M,) pi Inv'S(M,) (13)

where i1t, i2, are inclusion and P, : Inv*C(M,) ! Inv*S(M,) is a

projection defined by Pn(fn(m)) = aeSnsign(a)fn(m(a)).

Note that the map P* is covering of the map T*, such that T, : Inv*C(M,) ! Inv*S(M,) and the following diagram is commutative:

Inva (M.)

% ¿1, ° il, îi T. p.

Inv'C(M.) InvS (M,)

P, ° i2, — T, °(i1, ° i2,) — idIn

By considering the following sequence of injection complexes

InvC(kS,(X)) ! InvS(kS,(X)) ! Inv'A(kS,(X))

where InvA (kS.(X)) is complex of simplicial cochain of topological space X. The sequence (17) induces the following sequence:

Hn(X, k)! Hn(X, k)® h"-2(x, k)

® Hn-4(x, k)---!fl"(x, k)

where in is inclusion and Pn is projective.

Note that the same result of Theorem 8 can be got for homology group.

2.2. Monoid and algebra over monoid

In this part we study the relation between monoid and index category. The main result is Theorems 16 and 18.

Firstly we recall definition of monoid and algebra over it.

Definition 6. Monoid T in category A is the triple (T,i, o), when T: A fi A is covariant functor, i: T o T fi T and o : idA fi A are maps with the following commutative diagram.

T o T o T(X) l(T(X)) T o T T(x) T(o(x)) To T(x) o(T(x)) T(x)

T(l(X)) # T о T(X) !

where x e obA.

# l(X ,

#l(x) //

category A and by T — A/g the category of algebra over monoid.

The following theorems give some properties of index category related with monoid and algebra over it.

There is an isomorphism between the category of index category (Ind Cat) and the category of monoid in the category of simplicial set (MON (Aop set), i.e.

(19) X : IndCat ! Mon(Aop)

Example 3. For the categories A and B suppose U: A fi B, Q: B fi A are functors such that an isomorphism U : HomA(—, Q(—)) fi HomB(U(), —) is a functor from A0p x B to category of sets where HomA(—, Q(—))(x, y) = A(x, Q(y)), HomB(U(—),—)(x,y) = B(U(x),y), x 2 obA, y 2 obB, then the triple (Q o U, i, v) is monoid in category A if v(x) = u—1(idv(x)) : x fi Q o U(x),i(x) = Q(u—Hid^x)) : Q o U o Q o U(x) fi Q o U(x).

Definition 7. Algebra over monoid (T, i, v) is defined to be (X, n), where x 2 ObA and n : T(x) fi X is an isomorphism such that the following diagram commutes.

T о T(X) i(x) T(x)

T(n) # # n

x v(x) T(x)

\\ # n

(i) Suppose a category A, for every object x in A, the pair (T(X), i(x)) is algebra over monoid T and called free T-algebra generated by object x.

(ii) The pair (Q(X), n) is algebra over monoid (Q o U, i, t) if n = Qu('du(X)) : Q o U o Q(X) fi Q(X).

A morphism W : (T,i, t) fi (Tn,in, on) between monoids is given by the map W : T fi Tn such that for any object X, the following diagram commutes.

To T(x) W(x) THx) T(x) W(x) Tn(x)

l(x) #

T(x) W(x) Tn(x)

t(x) & # v\(x) (21)

Definition 8. Where A morphism W2(x) is orthogonal in the commutative diagram

T о T(x) T(W(x)) T о T\(x)

W(T(x))# # W(T\(x)) T\

о T(x) T\(W(x)) T\ о T\(x)

A morphism f: (X, n) fi (Xn, nn) between algebras over monoid (T, i, o) is given by the morphism W : X fi Xn such that the following diagram is commutative.

T(x) T(f) T(x\) T(x) W(x) T\(

n # # n\

x f x\

v(x) & # t\(x) •

In the following we denote by Ind Cat the category of index category and by Mon(A) cat the category of monoid in

Proof. Consider an index category (A, i), the monoid (T,i, o) and the functors: i* : Aopset ! Aopset, i* : Dopset ! Aopset, where i (X.) = X. о i, X. is simplicial set and i*(X.)m — П1 оXn x A(m, n)= (xs, q) ~ (x, s о q), s 2 A(m,n). From the simplicial set theory the relation between the functors i* and i* is given by the following assertion. □

Proposition 3. ([10,11]) Given the simplicial maps Ф and Ф 1, the map is identity map that is; (Ф-1 о U)(f) (x) = U_1(U(f)) (x) = Uf)([x, 1]) = (f)(x).

Proof. Consider a simplicial map Ф such that: Ф (X,Y) : Aopset(X/(Y:)) fi Aopset(i*(X), Y) where Ф (X, Y.)(F.)([X, q]) = (Fn(x))p, F. : X. fi i*( Y.), X eX„ and q 2 A (P, n). The inverse functor Ф-1 is given by:

(20) Ф^., Y.) : Aopset(i,(X), Y) ! AopSet(X., i.( Y.))

where Ф-1(X , Y.)(g.) : Xfi i*(Y), Ф-1g(x) = g([x,1]), g. : i* (X.) fi Y.. (Ф-1 о Ф)^) = Ф-1(Ф(/))(x) = Ф(/) ([x,1]) = f(x).

From the above discussion the triple (i о i», i, о) is a monoid and there is a homomorphism X between the index category (A, i) and (i о i», i, о), since i о i» : Aop — Sets fi Dop — Sets, i(x) = T о T(X) fi T(X), "X 2 Aop — Sets such that i(x)(i о i» о i о i») (X) fi i о i»(X), where (i* о i,)(X) = П10Xn x A[n]/ i([x, a,y]) = [x, a о y] 2 i о i» (X) and o(x): X fi t(x) such that o(x) = [x,1] 2 (i» о i»)(X). To get the converse proof suppose the functor X(A, i) = (i о i», i, о), then for an arbitrary monoid T in the category AopSet, the isomorphism X-1 is given by X-1(T)(n, m) = T(A(—, n))m. □

For an index category (A, i), the following isomorphism holds.

AopSet u X(A) — algebra.

Proof. For an arbitrary AopSet Y. we consider the following algebra (i*(Y), i*U(id,«(X))) which is X-algebra.

Also for every X-algebra (X , n ) we can define the action of the operator q 2 A(m,n) on the graded set by: (x)q = nm([ • ,q]), where X2Xn. □

References

[1] Y.Gh. Gouda, Index category and related crossed simplicial groups, Int. J. Algebra 4 (12) (2010) 599-608.

[2] Y.Gh. Gouda, H.N. Alla, On the trivial and nontrivial cohomology with inner symmetry groups of some classes of operator algebras, Int. J. Math. Anal. 3 (8) (2009) 377-384.

[3] R.L. Krassauskas, S.V. Lapin, Yu P. Solovev, Dihedral homology and cohomology. Basic notions and structions, Math. USSR Sbornic 133 (175) (1987) 25-48.

[4] J.-L. Loday, Cyclic Homology, second ed., Springer-Verlag, New York, 1998.

[5] S. Ault, On the Symmetric Homology of Algebras, Ph.D. Dissertation, Ohio State University, 2008.

[6] S. Ault, Z. Fiedorowicz, Symmetric Homology of Algebras, 2007. Preprint on <arXiv:0708.1575v4>{math. AT](11-5-2007).

[7] C. Berger, I. Moerdijk, On an Extension of the Notion of Ready Category, 2008. <http://www.crm.es/Publications/08/Pr826 pdf>.

[8] M. Elhamdadi, On S3-equivariant homology, Int. J. Math. Math. Sci. 26 (4) (2001) 193-197.

[9] Z. Feidorowicz, J.-L. Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991) 57-87.

[10] C. Kassel, Cyclic homology, Comodules and mixed complexes, J. Algebra 107 (1987) 195-216.

[11] J.P. May, Simplicial Object in Algebraic Topology, Van Nostrad, 1967 (reprinted by University of Chicago Press, 1982 and 1992).