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

TL;DR

This paper introduces a symmetric promonoidal kernel on the simplex category, leading to a Hadamard natural transformation that provides a classical simplicial contraction of $ abla^n$ to a point, with novel promonoidal and Hadamard structures.

Contribution

It presents a new promonoidal kernel on $ riangle$ and a Hadamard map that induce a classical contraction of simplices, not previously documented.

Findings

Established a symmetric promonoidal kernel on $ riangle$.

Derived a Hadamard natural transformation for representable functors.

Provided a simplicial contraction of $ abla^n$ to a point.

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.