The excision theorems in Hochschild and cyclic homologies

2014 ◽  
Vol 144 (2) ◽  
pp. 305-317
Author(s):  
Guram Donadze ◽  
Manuel Ladra
Keyword(s):  
Hochschild Homology ◽  
Excision Property ◽  
Crossed Modules ◽  
Simplicial Algebras

We study the excision property for Hochschild and cyclic homologies in the category of simplicial algebras. We extend Wodzicki's notion of H-unital algebras to simplicial algebras and then show that a simplicial algebra I* satisfies excision in Hochschild and cyclic homologies if and only if it is H-unital. We use this result in the category of crossed modules of algebras and provide an answer to the question posed in the recent paper by Donadze et al. We also give (based on work by Guccione and Guccione) the excision theorem in Hochschild homology with coefficients.

scholarly journals ON 2-HOLONOMY

2019 ◽  
pp. 1-30
Author(s):  
HOSSEIN ABBASPOUR ◽  
FRIEDRICH WAGEMANN
Keyword(s):  
Lie Algebra ◽  
Hochschild Homology ◽  
Crossed Module ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Crossed Modules ◽  
Crucial Ingredient

We construct a cycle in higher Hochschild homology associated to the two-dimensional torus which represents 2-holonomy of a nonabelian gerbe in the same way as the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez–Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module $\unicode[STIX]{x1D707}:\mathfrak{h}\rightarrow \mathfrak{g}$ of the principal 2-bundle, the Lie algebra $\mathfrak{h}$ is abelian, up to equivalence of crossed modules.

scholarly journals Abelian extensions and crossed modules of Hom-Lie algebras

2020 ◽  
Vol 224 (3) ◽  
pp. 987-1008
Author(s):  
José Manuel Casas ◽  
Xabier García-Martínez
Keyword(s):  
Lie Algebras ◽  
Abelian Extensions ◽  
Crossed Modules

Crossed Semimodules and Schreier Internal Categories in the Category of Monoids

1998 ◽  
Vol 5 (6) ◽  
pp. 575-581 ◽  
Author(s):  
A. Patchkoria
Keyword(s):  
Internal Category ◽  
Equivalence Of Categories ◽  
Crossed Modules

Abstract We introduce the notion of a Schreier internal category in the category of monoids and prove that the category of Schreier internal categories in the category of monoids is equivalent to the category of crossed semimodules. This extends a well-known equivalence of categories between the category of internal categories in the category of groups and the category of crossed modules.

scholarly journals Computation and homotopical applications of induced crossed modules

2003 ◽  
Vol 35 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Ronald Brown ◽  
Christopher D. Wensley
Keyword(s):  
Crossed Modules

scholarly journals DETECTING TORSION IN SKEIN MODULES USING HOCHSCHILD HOMOLOGY

2006 ◽  
Vol 15 (02) ◽  
pp. 259-277 ◽  
Author(s):  
MICHAEL McLENDON
Keyword(s):  
Tensor Product ◽  
Spectral Sequence ◽  
Boundary Surface ◽  
Hochschild Homology ◽  
Heegaard Splitting ◽  
Common Boundary ◽  
Skein Modules ◽  
Skein Module ◽  
Hochschild Complex

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

scholarly journals Infinitesimal 2-braidings and differential crossed modules

2015 ◽  
Vol 277 ◽  
pp. 426-491 ◽  
Author(s):  
Lucio Simone Cirio ◽  
João Faria Martins
Keyword(s):  
Crossed Modules

scholarly journals Hochschild homology and global dimension

2009 ◽  
Vol 41 (3) ◽  
pp. 473-482 ◽  
Author(s):  
Petter Andreas Bergh ◽  
Dag Madsen
Keyword(s):  
Global Dimension ◽  
Hochschild Homology

Fibrations of 2-crossed modules

2018 ◽  
Vol 42 (16) ◽  
pp. 5293-5304
Author(s):  
Ummahan Ege Arslan ◽  
İbrahim İlker Akça ◽  
Gülümsen Onarlı Irmak ◽  
Osman Avcıoğlu
Keyword(s):  
Crossed Modules

scholarly journals Properties of the Secondary Hochschild Homology

2018 ◽  
Vol 25 (02) ◽  
pp. 225-242
Author(s):  
Jacob Laubacher
Keyword(s):  
Exact Sequence ◽  
Morita Equivalence ◽  
Hochschild Homology ◽  
Low Dimension

In this paper we study properties of the secondary Hochschild homology of the triple (A, B, ε) with coefficients in M. We establish a type of Morita equivalence between two triples and show that H•((A, B, ε); M) is invariant under this equivalence. We also prove the existence of an exact sequence which connects the usual and the secondary Hochschild homologies in low dimension, allowing one to perform easy computations. The functoriality of H•((A, B, ε); M) is also discussed.

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