(Injective) facet-complexity between simplicial complexes

Abstract

We present the notion of facet-complexity, $\text{C}(\mathsf{L};\mathsf{K})$, for two simplicial complexes $\mathsf{L}$ and $\mathsf{K}$, along with basic results for this numerical invariant. This invariant $\text{C}(\mathsf{L};\mathsf{K})$ quantifies the \aspas{complexity} of the following question: When does there exist a facet simplicial map $\mathsf{L}\to \mathsf{K}$? A facet simplicial map is a simplicial map that preserves non-unitary facets. Likewise, we introduce the notion of injective facet-complexity, $\text{IC}(\mathsf{L};\mathsf{K})$. These invariants generalize the notion of (injective) hom-complexity between graphs, recently introduced by Zapata et al. We demonstrate a triangular inequality for (injective) facet-complexity and show that it is a simplicial complex invariant. Additionally, these invariants provide an obstruction to the existence of facet simplicial maps. We explore the sub-additivity of (injective) facet-complexity and we present a lower bound in terms of the chromatic number. Moreover, we provide an upper bound for $\mathrm{C}(\mathsf{L};\mathsf{H})$ in terms of the number of facets of $L$. Finally, we establish a formula for $\mathrm{IC}(\mathsf{L};\mathsf{K})$ when $\mathsf{L}$ is a pure simplicial complex and $K$ is a complete simplicial complex.