What question did this study set out to answer?

To establish a Koszul duality adjunction between operads and coalgebras in stable presentably symmetric monoidal infinity-categories.

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April 17, 2026Open Access

A derived Milnor-Moore theorem

Key Points

To establish a Koszul duality adjunction between operads and coalgebras in stable presentably symmetric monoidal infinity-categories.
Construct a Koszul duality adjunction between O-algebras and coalgebras.
Utilize norm maps associated with symmetric groups to analyze categorical structures.
Prove the identification of unit maps with canonical maps involving limits of operads.
Identified a Koszul duality adjunction that reveals the interplay between O-algebras and coalgebras.
Validated that under certain equivalence conditions, specific categorical maps align perfectly.
Demonstrated structural insights into the relationships between operads and their duals.

Abstract

Abstract For every stable presentably symmetric monoidal ∞ -category C C and every non-unital ∞ -operad O O in C C, where "Equation missing", we construct a Koszul duality adjunction aligned TQ_ O: Alg_ O (C) Coalg ₎^ (C): Prim_ Oaligned TQ O: Alg O (C) ⇆ Coalg O ∨ (C): Prim O between O O -algebras in C C and coalgebras over the Koszul dual ∞ -cooperad of O O. We prove that if all norm maps in C C associated to symmetric groups are equivalences, the unit of Koszul duality X Prim_ O (TQ_ O (X) ) X → Prim O (TQ O (X) ) identifies with the canonical map X X^: = ₍ ₁ ₙ (O) _ OX X → X ∧: = lim n ≥ 1

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