We define the Atiyah class for global matrix factorisations and use it to give a formula for the categorical Chern character and the boundary-bulk map for matrix factorisations, generalising the formula in the local case obtained in [12]. Our approach is based on developing the Lie algebra analogies observed by Kapranov [7] and Markarian [9].

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.

In this paper using the Quillen–Barr–Beck (co-)homology theory of universal algebras we define (co-)homology groups for commutative algebras with divided powers in prime characteristic. In particular, we determine for A a commutative 𝔽p-algebra with divided powers, the category of Beck A-modules and the group of Beck derivations. We construct the abelianization functor and we define (co-)homology. Moreover, we determine the cohomology in low dimensions and we interpret the first cohomology in terms of extensions.

We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine–Tristram signature.

For every integer $n\,>\,1$ and infinite field $F$ we construct a spectral sequence converging to the homology of $\text{G}{{\text{L}}_{n}}\left( F \right)$ relative to the group of monomial matrices $\text{G}{{\text{M}}_{n}}\left( F \right)$. Some entries in ${{E}^{2}}$-terms of these spectral sequences may be interpreted as a natural generalization of the Bloch group to higher dimensions. These groups may be characterized as homology of $\text{G}{{\text{L}}_{n}}$ relatively to $\text{G}{{\text{L}}_{n-1}}$ and $\text{G}{{\text{M}}_{n}}$. We apply the machinery developed to the investigation of stabilization maps in homology of General Linear Groups.

We present sufficient conditions for entire cyclic cohomology to reduce to ordinary cyclic cohomology. These conditions are characteristic for finite dimensional (noncommutative) spaces.