---
title: "Koszul duality and Morita categories"
authors:
  - "Max Blans"
year: 2026
venue: "arXiv:math.AT"
arxiv: "2608.27450"
doi: null
source: "https://arxiv.org/abs/2608.27450"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1616
seed_batch: "curiosity-2026-09-07"
reviewed: "2026-09-07"
pool: "maths-foundations"
relevance_score: 10
lineage: operads-infinity
cites:
  - title: "Koszul duality and Morita categories"
    url: "https://arxiv.org/abs/2608.27450"
    year: 2026
    arxiv: "2608.27450"
    doi: null
see:
  - "1588-the-2-monoidal-structure-of-symmetric-sequences"
---

# Koszul duality and Morita categories

## One-sentence takeaway

(∞,2)-Koszul duality for operads/cooperads in spectra as an equivalence of Morita categories of (co)operads and bi(co)modules.

## Why it's lovely

Why you might love this: Koszul duality upgraded to an (∞,2)-Morita story — operads and cooperads dualize by swapping bimodules for bicomodules. Same shelf energy as Dwyer–Hess / Day operads (1588), but now the duality itself is the Morita equivalence.

## Key ideas

- Morita category of operads and bimodules ↔ dual Morita category of cooperads and bicomodules.
- Works for operads/cooperads in spectra at (∞,2)-level.
- Subsumes several classical Koszul-duality forms as instances.
- Clean packaging: duality as equivalence of module categories, not just a chain-level formula.

## Caveats

Spectra / higher-categorical technical overhead; not a hands-on rewrite of classical GK Koszul duality for beginners.

## Implications for Broadside

Strengthens the operad / 2-monoidal shelf already seeded (1588). Useful mental model whenever Broadside talks about dualizing compositional structure.

## Links

- arXiv abs: https://arxiv.org/abs/2608.27450
- PDF: https://arxiv.org/pdf/2608.27450.pdf