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Volume 2013 (2013), Article ID 306978, 7 pages
Some Results on Strict Graded Categorical Groups
1Department of Mathematics, Hanoi National University of Education, Hanoi 123105, Vietnam
2Department of Mathematics and Applications, Saigon University, Ho Chi Minh City 700900, Vietnam
Received 21 March 2013; Accepted 15 July 2013
Academic Editor: Adolfo Ballester-Bolinches
Copyright © 2013 Nguyen Tien Quang and Che Thi Kim Phung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present some applications of strict graded categorical groups to the construction of the obstruction of an equivariant kernel and to the classification of equivariant group extensions which are central ones. The composition of a graded categorical group and an equivariant group homomorphism is also determined.
The group extension problem has an important significance in the development of modern algebra. Some notions of this problem such as crossed product, factor set, and obstruction (see ) are not only applied to rings or to algebraic types but also are raised to a categorical level.
The theory of graded categorical groups studied by Cegarra et al.  can be viewed as a generalization of both the categorical group theory of Sinh  and the graded category theory of Fröhlich and Wall . The equivariant group extension problem is one of applications of this theory.
Strict graded categorical groups, with their simple structures compared to the general case, are more likely to give a lot of interesting applications. In  we presented an application of this notion to the classification of equivariant crossed modules. In this paper we continue to introduce some other applications. Firstly, we show that if is the third invariant of the strict graded categorical group and is an equivariant kernel; then Secondly, we classify equivariant group extensions of by which are central extensions by graded monoidal autofunctors of the strict graded categorical group . Finally, we construct the composition of a -graded categorical group with a -homomorphism, which is analogous to the composition of a group extension with a group homomorphism (see [1, Chapter 3]).
2.1. Graded Categorical Groups
We recall briefly some basic notions about graded categorical groups in .
We regard the group as a category with one object, say , where the morphisms are elements of and the composition is the group operation. A category is -graded if there is a functor . The grading is said to be stable if for any object and any there exists an isomorphism in with domain and . A -graded monoidal category consists of(1)a stable -graded category , -graded functors and ,(2)natural isomorphisms of grade 1 , and such that, for all , the following two coherence conditions hold:
A graded categorical group is a graded monoidal category in which every object is invertible and every morphism is an isomorphism. In this case, the subcategory consisting of all objects of and all morphisms of grade 1 in is a categorical group.
If , are -monoidal categories, then a graded monoidal functor consists of a -graded functor , natural isomorphisms of grade 1 , and an isomorphism of grade 1 , such that, for all , the following coherence conditions hold: Let , be two -graded monoidal functors. A graded monoidal natural equivalence is a natural equivalence of functors such that all isomorphisms are of grade 1, and for all , the following coherence conditions hold:
2.2. Reduced Graded Categorical Groups
The authors of  showed that any -graded categorical group determines a triple , where(1)the set of 1-isomorphism classes of the objects in is a -group,(2)the set of 1-automorphisms of the unit object is a -equivariant -module,(3)the third invariant is an equivariant cohomology class .
Based on these data, they constructed a -graded categorical group, denoted by , which is graded monoidally equivalent to . Below, we briefly recall this construction.
Objects of are elements and its morphisms are pairs consisting of an element and such that .
The composition of two morphisms is given by
The graded tensor product is given by
The unit constrains are strict in the sense that . The associativity isomorphisms are
The stable -grading is .
The unit graded functor is given by We call the -graded categorical group a reduction of the -graded categorical group , simply denoted by .
3. Strict Graded Categorical Groups
3.1. Definitions and Examples
It is well known that each crossed module of groups can be seen as a strict categorical group (see , Remark 3.1 in ). Crossed modules of groups can be enriched in some ways to become, for example, crossed bimodules over rings or equivariant crossed modules. In the former case, each crossed bimodule can be seen as a strict Ann-category . In the later case, each crossed module of -groups can be identified with a strict -graded categorical group  whose definition is recalled as below.
Firstly, if is a monoidal functor between categorical groups, then the isomorphism can be deduced from and , so we can omit when not necessary. A monoidal functor between two categorical groups is termed regular whenever for all , . A factor set on with coefficients in a categorical group (see ) is regular if , and is a regular monoidal functor, for all .
Definition 1 (see ). A graded categorical group is said to be strict if(i) is a strict categorical group,(ii) induces a regular factor set on with coefficients in a categorical group .
Equivalently, a graded categorical group is strict if it is a -graded extension of a strict categorical group by a regular factor set.
Note 1. In this paper we denote by + for the operation of the group .
Example 2 (the strict graded categorical group ). The discrete -graded categorical group defined by a -group has the elements of as objects, and its morphisms are the elements with . Composition is multiplication in , and the grading is the obvious map . The graded tensor product is given by and the graded unit by the associativity and unit isomorphisms are identities.
Example 3 (the strict graded categorical group of a -group ). Firstly, observe that if is a -group, then the group of automorphisms of is also a -group with the action
Then, the homomorphism , ( is the inner automorphism of given by conjugation with ) is a homomorphism of -groups. Indeed, for all , one has
For each -group , we can construct a strict graded categorical group, denoted by (see ), whose objects are elements of the -group . A -morphism is a pair , where with . Composition of two morphisms is given by
The graded tensor product is and the graded unit is defined by The associativity and unit isomorphisms are identities.
Example 4 (strict graded categorical groups associated with an equivariant crossed module). The notion of -crossed module is a generalization of that of crossed module of groups introduced by Whitehead .
Definition 5 (see ). Let , be -groups. A -crossed module is a quadruple , where , are -homomorphisms satisfying the following conditions: where , , , and is the inner automorphism given by conjugation with .
A -crossed module is also called an equivariant crossed module by Noohi .
From the definition of -crossed module, it is easy to deduce the following properties: (i) is a -subgroup in ; (ii) is both a normal subgroup in and a -group; (iii) is a left -equivariant -module under the actions
The strict graded categorical group associated with the -crossed module is constructed as follows.
Objects of are the elements of the group , and a -morphism is a pair , where , with . Composition of two morphisms is defined by
The tensor product on objects is given by the multiplication in the group , and for two morphisms , , then
The associativity and unit constraints of the tensor product are strict.
The graded functor is given by and the graded unit functor by
3.2. Equivariant Kernel and Strict -Categorical Group
The notion of equivariant kernel was introduced in . It is a triple , where , are -groups and is a homomorphism of -groups. Then is -equivariant -module under the action . The obstruction of is an element , defined as follows (see [2, page 996]). For each , choose an automorphism of , , in particular . Since there exist elements such that where is the inner automorphism of -group induced by . The pair therefore induces an element defined by the following relations. The associative law for a product leads to The relations imply According to , . The cohomology class is called the obstruction of , denoted by .
The following theorem describes the invariants of the graded categorical group and shows a relation between third invariant of and .
Theorem 6. Let be an equivariant kernel. Then the invariants of the strict graded categorical group are(i), ,(ii) with .
Proof. (i) It is obvious.
(ii) According to Section 3 in , there is a -monoidal equivalence Let us observe that each morphism in is written in the form Let
Since , for , (see [12, Proposition 13]), the -monoidal functor defines a function by Set , and the pair determines a function by relations (25)–(28) which can be written in the form , even though does not take values in .
The compatibility of with the associativity constraints leads to . Thus, for , we obtain
Next, we show that .
Since is a -functor, the following diagram commutes: (35) where , .
Due to the relations (5)–(15) of the product and the composition in and , one has
Let us note that Therefore, This shows that .
Finally, we prove that based on the functorial property of . Indeed, consider the composition On one hand, On the other hand, Again, , the functorial property of , implies or . Clearly,
3.3. Classification of Central Extensions
Denote by the set of all equivalence classes of equivariant extensions , where . One can classify these extensions by the autofunctors of -graded categorical group .
Theorem 7 (the schreier theory for central extensions of equivariant groups). Let be a -group, and let be a -equivariant -module. There is a bijection where is the set of homotopy classes of monoidal -functors from to itself satisfying
Proof. Let . Then defines a function
with . The compatibilities of with the constraints imply and
The naturalness of and the functority of give, respectively,
for all , . With the function , we can construct an equivariant crossed product extension in which the operation of is
The relation (48) and the normality of show that is a group. By the relations (49), (50), is a -group under the -action
Then we obtain an exact sequence
where , . Further, it is easy to see that .
Conversely, suppose that is an equivariant extension with . For each , choose a representative , . Then the system induces a function taking values in such that The rules imply the relations (48)–(50), respectively. Thus, we can define a -monoidal functor as follows:
Now, two extensions in are equivalent if and only if two corresponding graded monoidal functors are homotopic. Indeed, let be two homotopic -graded monoidal functors in which the homotopy is given by where . From the definition of a homotopy and from the relations (5), (6), it follows that where , . Then, the corresponding is an equivalence between two central extensions if and only if equalities (59) hold.
Conversely, if is an isomorphism, then where is a function with . Thus, it follows that is a homotopy of and , as we see by retracting our steps.
Finally, each central extension is equivalent to one via the isomorphism .
3.4. Composition of a -Categorical Group with a -Homomorphism
It is well known that for a given extension where is Abelian, and a given homomorphism , there is an extension of by with . The extension is just the pull pack of the pair . We state an analogous result on strict -categorical groups in the following proposition.
Proposition 8. Let be a strict graded categorical group with three invariants , , , and let be an equivariant homomorphism. Then there exists a strict graded categorical group which is equivalent to the graded categorical group , where is viewed as a -module under the actions , , , and .
Proof. The strict graded categorical group is constructed as follows:
A -morphism is a triple , where is a -morphism in . The composition of two morphisms is given by
The tensor products on objects and on morphisms in are given by
For each morphism in , one has The graded functor is given by and the graded unit functor by
The unit object of is , where is the unit object of . The associativity and unit constraints are identities.
Suppose that the graded categorical group induces a regular factor set . Then, induces a regular factor set by
Thus, is a strict graded categorical group.
We define a pair of maps by Then, is an isomorphism of -groups and is an isomorphism of -equivariant -module, where the -module structure of is .
The graded monoidal functor given by induces one , where . Then, where is a morphism in . Thus, is a functor of type .
Suppose that . According to Theorem 3.2 in , the obstruction of pair vanishes in ; that is, Now, we denote , and then the pair , where , , is a graded monoidal functor from to . The composition is an equivalence from to .
Finally, we prove that is in the same cohomology class with . Let ; then together with is a graded monoidal functor, and the composition is a graded monoidal functor making the following diagram commute: (76) Since is a graded monoidal functor of type , its obstruction vanishes. This implies that , and that means that .
If , we obtain Proposition 14 in .
The graded categorical group is called a composition of with , denoted by . It follows that can be seen as a contravariant functor in terms of the invariant with fixed. Indeed,
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