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

Florian Lengyel

TL;DR

This work constructs a Cartesian promonoidal framework on the simplex category $Δ$ and shows that Day convolution on presheaves is the levelwise product. By introducing a Hadamard product on monotone maps, it defines a natural transformation $δ_{p,q}$ that induces a Day-convolution map $Θ_{p,q}: Δ^p * Δ^q → Δ^{pq}$, and demonstrates the (non-invertible) correspondence with the Hadamard map $H_{p,q}$. Specializing to $q=1$ yields a simplicial homotopy $H^n: Δ^n × Δ^1 → Δ^n$ contracting each $Δ^n$ to its $0$-vertex, which, upon geometric realization, coincides with the standard PL contraction to the $0$-vertex. The paper also provides a detailed realization-theoretic analysis, including prism-cell decompositions and affine behavior on cells, tying promonoidal/Day-convolution structures to explicit topological contractions.

Abstract

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

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

TL;DR

This work constructs a Cartesian promonoidal framework on the simplex category and shows that Day convolution on presheaves is the levelwise product. By introducing a Hadamard product on monotone maps, it defines a natural transformation that induces a Day-convolution map , and demonstrates the (non-invertible) correspondence with the Hadamard map . Specializing to yields a simplicial homotopy contracting each to its -vertex, which, upon geometric realization, coincides with the standard PL contraction to the -vertex. The paper also provides a detailed realization-theoretic analysis, including prism-cell decompositions and affine behavior on cells, tying promonoidal/Day-convolution structures to explicit topological contractions.

Abstract

We exhibit a symmetric promonoidal kernel on the simplex category with Cartesian unit, yielding on representable functors a Hadamard natural transformation based on pointwise multiplication of nondecreasing maps. Specializing to yields a simplicial homotopy contracting to its -vertex. The contraction is classical; the promonoidal presentation and induced Hadamard map appear not to be recorded.
Paper Structure (17 sections, 5 theorems, 32 equations)

This paper contains 17 sections, 5 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Promonoidal Categories and Day Convolution
  3. The Cartesian Promonoidal Structure on $\Delta$
  4. The Kernel $P$ and Unit $J$
  5. Identification and Coherence
  6. The Hadamard Map
  7. The Hadamard Product of Morphisms
  8. The Hadamard Map as a Natural Transformation
  9. The Induced Map under Day Convolution
  10. Contraction of $\Delta^n$
  11. Geometric Realization
  12. Prism cells determined by $k$.
  13. Affineness of $|H^n|$ on $S_k$.
  14. The standard PL contraction to the $0$-vertex.
  15. Verification of the Cartesian Product via Day Convolution
  16. ...and 2 more sections

Key Result

Proposition 1

The kernel $P$ is naturally isomorphic to the Cartesian product profunctor:

Theorems & Definitions (17)

  • Definition 1: Promonoidal category
  • Definition 2: Day convolution
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Definition 3: Hadamard Product
  • Definition 4
  • Proposition 2
  • ...and 7 more