Categorical structures of Lie–Rinehart crossed module

On Categorical Crossed Modules

The well-known notion of crossed module of groups is raised in this paper to the categorical level supported by the theory of categorical groups. We construct the cokernel of a categorical crossed module and we establish the universal property of this categorical group. We also prove a suitable 2-dimensional version of the kernelcokernel lemma for a diagram of categorical crossed modules. We th...

Cores of Countably Categorical Structures

A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure S has a core, i.e., S has an endomorphism e such that the structure induced by e(S) is a core; moreover, the core is unique up to isomorphism. We prove that every ...

Higher Categorical Structures in Geometry

A Crossed Module giving the Godbillon-Vey Cocycle

We exhibit crossed modules (i.e. certain 4 term exact sequences) corresponding to the Godbillon-Vey generator for W1, V ect(S ) and V ect1,0(Σ). Introduction There is a well known correspondence in homological algebra between (equivalence classes of) exact sequences Λ-modules, starting in M and ending in N with nmodules in between, and elements of ExtΛ(N,M). For Λ = U(g) the universal envelopin...

Yetter-drinfeld Modules for Turaev Crossed Structures

We provide an analog of the Joyal-Street center construction and of the Kassel-Turaev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of a YetterDrinfeld module over a crossed group coalgebra H and we prove that both the category of Yetter-Dri...