From simplicial homotopy to crossed module homotopy in modified categories of interest

We address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence.

Simplicial homotopy theory, link homology and Khovanov homology

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

The Verdier Hypercovering Theorem

This note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms [X, Y] in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where Y is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object X to be locally fibrant.

Simplicial homotopy in semi-abelian categories

We study Quillen's model category structure for homotopy of simplicial objects in the context of Janelidze, Márki and Tholen's semi-abelian categories. This model structure exists as soon as is regular Mal'tsev and has enough regular projectives; then the fibrations are the Kan fibrations of S. When, moreover, is semi-abelian, weak equivalences and homology isomorphisms coincide.