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.

8 pages; short note; added reference to GarnerStreet2015