---
title: "The product rule in Goodwillie calculus"
authors:
  - "Max Blans"
  - "Thomas Blom"
year: 2026
venue: "arXiv:math.AT"
arxiv: "2608.25682"
doi: null
source: "https://arxiv.org/abs/2608.25682"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1650
seed_batch: "curiosity-2026-09-09"
reviewed: "2026-09-09"
pool: "maths-foundations"
relevance_score: 10
lineage: goodwillie-calculus
cites:
  - title: "The product rule in Goodwillie calculus"
    url: "https://arxiv.org/abs/2608.25682"
    year: 2026
    arxiv: "2608.25682"
    doi: null
see:
  - "1588-the-2-monoidal-structure-of-symmetric-sequences"
---

# The product rule in Goodwillie calculus

## One-sentence takeaway

Goodwillie derivatives satisfy a product rule: for C with Sp-stabilization, the derivatives functor turns products into Day convolution of spectra sequences.

## Why it's lovely

Why you might love this: Goodwillie calculus is the Taylor tower for functors, and a product rule is the sort of thing you expect from any honest derivative — but proving it at the ∞-categorical level is non-trivial. Flagged as a Sep-3 Curiosity near-keeper; finally minted. Pure homotopy-calculus sparkle.

## Key ideas

- Differentiable ∞-categories whose stabilization is Sp.
- Derivatives functor ∂_* on finitary functors.
- Product rule: derivatives of a product become a Day-type convolution of spectra sequences.
- Closes a folklore gap in modern Goodwillie calculus.

## Caveats

Assumes Sp-stabilization; not the most general differentiable setting. Technique paper more than an application showcase.

## Links

- arXiv abs: https://arxiv.org/abs/2608.25682
- PDF: https://arxiv.org/pdf/2608.25682.pdf