In this paper,I introduce a new generalization of the concept of an operad,further generalizing the concept of an opetope introduced by Baez and Dolan(1998),who used this for the definition of their version of non-str...In this paper,I introduce a new generalization of the concept of an operad,further generalizing the concept of an opetope introduced by Baez and Dolan(1998),who used this for the definition of their version of non-strict n-categories.Opetopes arise from iterating a certain construction on operads called the+-construction,starting with monoids.The first step gives rise to plain operads,i.e.,operads without symmetries.The permutation axiom in a symmetric operad,however,is an additional structure resulting from permutations of variables,independent of the structure of a monoid.Even though we can apply the+-construction to symmetric operads,there is the possibility of introducing a completely different kind of permutations on the higher levels by again permuting variables without regard to the structures on the previous levels.Defining and investigating these structures is the main purpose of this paper.The structures obtained in this way are what I call n-actads.In n-actads with n>1,the permutations on the different levels give rise to a certain special kind of n-fold category.I also explore the concept of iterated algebras over an n-actad(generalizing an algebra and a module over an operad),and various types of iterated units.I give some examples of algebras over 2-actads,and show how they can be used to construct certain new interesting homotopy types of operads.I also discuss a connection between actads and ordinal notation.展开更多
We show that the normalized cochain complex of a nonsymmetric cyclic operad with multiplication is a Quesney homotopy BV algebra;as a consequence,the cohomology groups form a Batalin-Vilkovisky algebra,which is a resu...We show that the normalized cochain complex of a nonsymmetric cyclic operad with multiplication is a Quesney homotopy BV algebra;as a consequence,the cohomology groups form a Batalin-Vilkovisky algebra,which is a result due to L.Menichi.We provide ample examples.展开更多
We present a deformation theory associated to the higher Hochschild coho-mology H*_(S)^(2)(A,A).We also study a G-algebra structure associated to this deformation theory.
基金Natural Science Foundation of Gansu Province(22JR11RA138,20JR5RA249)National Natural Science Foundation of China(12071191)Funds for Innovative Fundamental Research Group Project of Gansu Province(23JRRA684)。
基金Supported by NSFC (Nos.11771190,1861051,12071191)the Natural Science Foundation of Gansu Province (No.20JR5RA249)the Natural Science Foundation of Shandong Province (No.ZR2020MA002)。
文摘In this paper,I introduce a new generalization of the concept of an operad,further generalizing the concept of an opetope introduced by Baez and Dolan(1998),who used this for the definition of their version of non-strict n-categories.Opetopes arise from iterating a certain construction on operads called the+-construction,starting with monoids.The first step gives rise to plain operads,i.e.,operads without symmetries.The permutation axiom in a symmetric operad,however,is an additional structure resulting from permutations of variables,independent of the structure of a monoid.Even though we can apply the+-construction to symmetric operads,there is the possibility of introducing a completely different kind of permutations on the higher levels by again permuting variables without regard to the structures on the previous levels.Defining and investigating these structures is the main purpose of this paper.The structures obtained in this way are what I call n-actads.In n-actads with n>1,the permutations on the different levels give rise to a certain special kind of n-fold category.I also explore the concept of iterated algebras over an n-actad(generalizing an algebra and a module over an operad),and various types of iterated units.I give some examples of algebras over 2-actads,and show how they can be used to construct certain new interesting homotopy types of operads.I also discuss a connection between actads and ordinal notation.
文摘We show that the normalized cochain complex of a nonsymmetric cyclic operad with multiplication is a Quesney homotopy BV algebra;as a consequence,the cohomology groups form a Batalin-Vilkovisky algebra,which is a result due to L.Menichi.We provide ample examples.
文摘We present a deformation theory associated to the higher Hochschild coho-mology H*_(S)^(2)(A,A).We also study a G-algebra structure associated to this deformation theory.
