A Cartesian Promonoidal Kernel on $\Delta$ and a Hadamard Contraction of $\Delta^n$

Abstract

We exhibit a symmetric promonoidal kernel on the simplex category $\Delta$ with Cartesian unit, yielding on representable functors a Hadamard natural transformation $\Delta^p\times\Delta^q\to\Delta^{pq}$ based on pointwise multiplication of nondecreasing maps. Specializing to $q=1$ yields a simplicial homotopy contracting $\Delta^n$ to its $0$-vertex. The contraction is classical; the promonoidal presentation and induced Hadamard map appear not to be recorded.