Any simplicial Hopf algebra involves $2n$ different projections between the Hopf algebras $H_n,H_{n-1}$ for each $n \geq 1$. The word projection, here meaning a tuple $\partial \colon H_{n} \to H_{n-1}$ and $i \colon H_{n-1} \to H_{n}$ of Hopf algebra morphisms, such that $\partial \, i = \mathrm{id}$. Given a Hopf algebra projection $(\partial \colon I \to H,i)$ in a braided monoidal category $\mathfrak{C}$, one can obtain a new Hopf algebra structure living in the category of Yetter-Drinfeld modules over $H$, due to Radford's theorem. The underlying set of this Hopf algebra is obtained by an equalizer which only defines a sub-algebra (not a sub-coalgebra) of $I$ in $\mathfrak{C}$. In fact, this is a braided Hopf algebra since the category of Yetter-Drinfeld modules over a Hopf algebra with an invertible antipode is braided monoidal. To apply Radford's theorem in a simplicial Hopf algebra successively, we require some extra functorial properties of Yetter-Drinfeld modules. Furthermore, this allows us to model Majid's braided Hopf crossed module notion from the perspective of a simplicial structure.
For a finite braided tensor category we introduce its Picard crossed module consisting of the group of invertible module categories and the group of braided tensor autoequivalences. We describe the Picard crossed module in terms of braided autoequivalences of the Drinfeld center of the braided tensor category. As an illustration, we compute the Picard crossed module of a braided pointed fusion category.
We introduce, for a symmetric fusion category $\mathcal{A}$ with Drinfeld centre $\mathcal{Z}(\mathcal{A})$, the notion of $\mathcal{Z}(\mathcal{A})$-crossed braided tensor category. These are categories that are enriched over $\mathcal{Z}(\mathcal{A})$ equipped with a symmetric tensor product, while being braided monoidal with respect to the usual tensor product on $\mathcal{Z}(\mathcal{A})$. In the Tannakian case where $\mathcal{A}=\mathbf{Rep}(G)$ for a finite group $G$, the 2-category of $\mathcal{Z}(\mathcal{A})$-crossed braided categories is shown to be equivalent to the 2-category of $G…
If $Γ$ is a group, then braided $Γ$-crossed modules are classified by braided strict $Γ$-graded categorial groups. The Schreier theory obtained for $Γ$-module extensions of the type of an abelian $Γ$-crossed module is a generalization of the theory of $Γ$-module extensions.
We show that the author's notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed G-categories, recently introduced for the purposes of 3-manifold topology [31]. The Galois extensions C \rtimes S are studied in detail, and we determine for which g in G non-trivial objects of grade g exist in C \rtimes S.
In this paper we study the categories of braided categorical associative algebras and braided crossed modules of associative algebras and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras.
In this paper we study the category of braided categorical Leibniz algebras and braided crossed modules of Leibniz algebras and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras using the Loday-Pirashvili category.
Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the {\it fundamental crossed complex} of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with so…
We show how the framework of crossed simplicial groups may be used to provide a classification of topological field theories on open cobordism categories defined by reductions of the structure group to a planar Lie group. Such theories are equivalent to algebras equipped with a group action and a non-degenerate trace satisfying certain invariance requirements which generalize the notion of a frobenius algebra.
We address the (pointed) homotopy of crossed module morphisms in modified categories of interest; which generalizes the groups and various algebraic structures. We prove that, the homotopy relation gives rise to an equivalence relation; furthermore a groupoid structure, without any restriction on neither domain nor co-domain of the crossed module morphism. Additionally, we consider the particular cases such as associative algebras, Leibniz algebras, Lie algebras and dialgebras of crossed modules of this generalized homotopy definition. Then as the main part of the paper, we prove that the func…
The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel'd modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. In the present paper these three sources of solutions are unified inside the framework of Yetter-Drinfe' d modules over a braided system. A systematic construction of braiding structures on such modules is provided. Some general categorical methods of obtaining such generalized Yetter-Drinfel'd (=GYD) modules are described. Among the b…
A crossed module is (A,H,d,\la) where d:A\to H is a homomorphism of groups and H acts on A, with conditions leading to a groupoid A\lcross H{\to\atop \to}H as an example of a strict 2-group. We give the corresponding notion of a quantum 2-group where we replace the above by Hopf algebras and introduce a new version of quantum groupoid. The work also suggests a natural notion of braided crossed module where A a braided-Hopf algebra in the braided category Z({}_H\CM) of crossed H-modules, although without the full groupoid picture in this more general case.
We study the Moore complex of a simplicial cocommutative Hopf algebra through Hopf kernels. The most striking result to emerge from this construction is the coherent definition of 2-crossed modules of cocommutative Hopf algebras. This unifies the 2-crossed module theory of groups and of Lie algebras when we take the group-like and primitive functors into consideration.
This is the last part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans - crossed modules of monoids relative to this class of spans - simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this …
Thomason's Homotopy Colimit Theorem has been extended to bicategories and this extension can be adapted, through the delooping principle, to a corresponding theorem for diagrams of monoidal categories. In this version, we show that the homotopy type of the diagram can be also represented by a genuine simplicial set nerve associated with it. This suggests the study of a homotopy colimit theorem, for diagrams $\b$ of braided monoidal categories, by means of a simplicial set {\em nerve of the diagram}. We prove that it is weak homotopy equivalent to the homotopy colimit of the diagram, of simplic…
We extend the notion of the nerve of a category for a small class of crossed simplicial groups, explicitly describing them using generators and relations. We do this by first considering a generalised bar construction of a group before looking at twisted versions of some of these nerves. As an application we show how we can use the twisted nerves to give equivariant versions of certain derived stacks.
We show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category GLoc A of twisted representations. This category is a braided crossed G-category in the sense of Turaev. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT -> modular ca…
Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure containing the prescribed internal structure is studied. Following the work of Lack, these universal objects must be constructed from simplicial objects arising from our monad-theoretic framework, as certain 2-categorical colimits called codescent objects. We isolate the extra structure present on these simplicial objects which enable their codescent object…
In this paper we define 3-crossed modules for commutative (Lie) algebras and investigate the relation between this construction and the simplicial algebras. Also we define the projective 3-crossed resolution for investigate a higher dimensional homological information and show the existence of this resolution for an arbitrary $\mathbf{k}$-algebra.
We introduce the notion of 3-crossed module, which extends the notions of 1-crossed module (Whitehead) and 2-crossed module (Conduché). We show that the category of 3-crossed modules is equivalent to the category of simplicial groups having a Moore complex of length 3. We make explicit the relationship with the cat$^{3}$-groups (Loday) and the 3-hyper-complexes (Cegarra-Carrasco), which also model algebraically homotopy 4-types.
We compare three different ways of defining group cohomology with coefficients in a crossed-module: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where the crossed-module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed-modules and also prove the "long" exact cohomology sequence associated to a short exact sequence of crossed-modules and weak morphisms.
This is the second part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans - crossed modules of monoids relative to this class of spans - simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In thi…
In this work, we explore the close relationship between an ideal map structure S --> End(R) on a homomorphism of commutative k-algebras R --> S and an ideal simplicial algebra structure on the associated bar construction Bar(S, R).
We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy relation, and prove that it yields an equivalence relation in very unrestricted cases (freeness up to order one of the domain 2-crossed module). This latter condition strictly includes the case when the domain is cofibrant. Furthermore, we prove that this notion of homotopy yields a groupoid with objects being the 2-crossed module maps between two fixed 2-crosse…
We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.
This is the first part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans - crossed modules of monoids relative to this class of spans - simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this…
In this work, it is shown that the category $\mathsf{BXMod/R}$ of braided crossed modules over a fixed commutative algebra $R$ is an exact category in the sense of Barr.
In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs, modular operads, twisted (modular) operads, properads, hyperoperads, their colored versions, as well as algebras over operads and an abundance of other related structures, such as crossed simplicial groups, the augmented simplicial category or FI--modules. The usefulness of this approach is that it allows us to handle all the classical as well as more esoteric …
