In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the $\phi$-weighted coboundary operator induced by a weight function $\phi$. Our weight function $\phi$ is a generalization of Dawson's weighted boundary map. We show that our above-mentioned generalizations include new cases that are not covered by previous literature. Our definition of weighted Laplacian for weighted simplicial complexes is also applicable to weighted/unweighted graphs and digraphs.

pdf/weighted-co-homology-and-weighted-laplacian-sj7pjhpisu.pdf

Weighted (Co)homology and Weighted Laplacian

The Twisted Homology of Simplicial Set

Let $A$ be either a simplicial complex $K$ or a small category $\mathcal C$ with $V(A)$ as its set of vertices or objects. We define a twisted structure on $A$ with coefficients in a simplicial group $G$ as a function $$ \delta\colon V(A)\longrightarrow \operatorname{End}(G), \quad v\mapsto \delta_v $$ such that $\delta_v\circ \delta_w=\delta_w\circ \delta_v$ if there exists an edge in $A$ joining $v$ with $w$ or an arrow either from $v$ to $w$ or from $w$ to $v$. We give a canonical construction of twisted simplicial group as well as twisted homology for $A$ with a given twisted structure. Also we determine the homotopy type of of this simplicial group as the loop space over certain twisted smash product.

/pdf/twisted-simplicial-groups-and-twisted-homology-of-categories-49d9s34swz.pdf

Twisted Simplicial Groups and Twisted Homology of Categories

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Twisted Simplicial Groups and Twisted Homology of Categories

Twisted Simplicial Groups and Twisted Homology of Categories