We achieve a classification of n-types of simplicial presheaves in terms of (n− 1)-types of presheaves of simplicial groupoids. This can be viewed as a description of the homotopy theory of higher stacks. As a special case we obtain a good homotopy theory of (weak) higher groupoids.

Simplicial neural networks (SNNs) have recently emerged as a new direction in graph learning which expands the idea of convolutional architectures from node space to simplicial complexes on graphs. Instead predominantly assessing pairwise relations among nodes current practice, allow us describe higher-order interactions and multi-node structures. By building upon connection between convolution...

Abstract. Simplicial complexes represent useful and accurate models of complex networks and complex systems in general. We explore the properties of spectra of combinatorial Laplacian operator of simplicial complexes and show its relationship with connectivity properties of the Q-vector and with connectivities of cliques in the simplicial clique complex. We demonstrate the need for higher order...

By integrating the simplicial model of social aggregation with existing research on opinion leadership and diffusion networks, this article introduces the constructs of simplicial diffusers (mathematically defined as nodes embedded in simplexes; a simplex is a socially bonded cluster) and simplicial diffusing sets (mathematically defined as minimal covers of a simplicial complex; a simplicial c...

Abstract We associate a simplicial complex to a sequence of ring elements that generate the ring over R (or nearly so). This is motivated by a formula for the degree of R as the “volume” of the simplicial complex. Then we give a flat degeneration of the ring to one combinatorially defined from the simplicial complex – or almost; the simplicial complex sometimes has to be promoted to a simplicia...

Simplicial sets, and more generally simplicial objects in a given category, are central to modern mathematics. While I am not a mathematical historian, I thought I would describe in conceptual outline how naturally simplicial sets arise from the classical study of simplicial complexes. I suspect that something like this recapitulates the historical development. We have described simplicial comp...

Simplicial Complexes. A (finite abstract) simplicial complex is a finite set system that is closed under taking subsets, i.e., F ⊂ H ∈ X implies F ∈ X. The sets F ∈ X are called faces of X. The dimension of a face F is dim(F ) = |F | − 1. The dimension of X is the maximal dimension of any face. A k-dimensional simplicial complex will also be called a k-complex.