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Point-set models for homotopy coherent coalgebras

This paper establishes an equivalence between the $\infty$ -category of homotopy coherent coalgebras over a cofibrant operad and the localized $\infty$ -category of differential graded coalgebras, thereby providing explicit point-set models for $\mathbb{E}_n$ and $E_\infty$ -coalgebras that enable an algebraic description of nilpotent $p$ -adic homotopy types.

Original authors: Dan Petersen, Victor Roca i Lucio, Sinan Yalin

Published 2026-04-21

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CC BY 4.0

Original authors: Dan Petersen, Victor Roca i Lucio, Sinan Yalin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe a complex, wobbly, 3D sculpture made of clay. In the world of mathematics, this sculpture represents a "homotopy coherent coalgebra." It's a structure that holds together not just by rigid rules, but by a flexible, shifting set of relationships that change slightly depending on how you look at them.

Mathematicians have two ways of looking at these sculptures:

The "Infinity" View: A high-level, abstract perspective where the wobbles and flexibilities are the main feature. This is the "true" shape of the object.
The "Point-Set" View: A rigid, concrete model where everything is fixed in place, like a blueprint or a Lego set. This is easier to build and measure, but it's hard to know if your rigid Lego model actually captures the true, wobbly nature of the original clay sculpture.

For a long time, mathematicians had a perfect way to turn rigid "algebra" sculptures (like building blocks that snap together) into their flexible "infinity" versions. But when they tried to do the same for "coalgebra" sculptures (which are like building blocks that split apart), they hit a wall. The rigid models they built didn't seem to match the flexible infinity versions at all. It was like trying to build a rigid Lego version of a water balloon; the moment you tried to snap the pieces together, the balloon popped or changed shape entirely.

The Problem: The "Splitting" Problem

The authors of this paper, Dan Petersen, Victor Roca i Lucio, and Sinan Yalin, tackled this specific problem. They asked: Can we build a rigid, concrete model (a point-set model) that perfectly represents these wobbly, splitting coalgebra structures?

The difficulty was that coalgebras are tricky. In the rigid world, if you try to force a structure to split in a specific way, it often forces it to be "commutative" (order doesn't matter) or "cocommutative" in a way that kills all the interesting, wobbly features. It's like trying to force a fluid to flow through a rigid pipe; the fluid loses its ability to swirl and twist.

The Solution: The "Cell-by-Cell" Construction

The authors didn't try to build the whole sculpture at once. Instead, they used a clever construction strategy:

Start Simple: They started with the simplest possible "free" structures (like a single, unconnected block). They proved that for these simple cases, the rigid model and the flexible infinity model are actually the same thing.
Build Up (Cell Attachments): They showed that if you have a good rigid model for a shape, and you add a new piece to it (like attaching a new Lego brick), the new, slightly more complex shape still has a good rigid model.
The Induction: Since any complex coalgebra can be built up by adding these "cells" one by one, they proved that if the simple ones work, the complex ones must work too.

They essentially proved that you can build a rigid, concrete Lego model for these wobbly coalgebras, provided you build them using a specific type of "cofibrant" (very well-behaved) blueprint.

The Big Payoff: Mapping the Universe

Why does this matter? The paper connects this abstract math to something very real: Topology (the study of shapes and spaces).

There is a famous theorem by Mandell that says: If you take a shape (like a donut or a sphere) and look at its "cochains" (a way of measuring its holes), you can perfectly reconstruct the shape's "p-adic" version (a specific way of zooming in on the shape using prime numbers).

However, Mandell's theorem had a catch: it only worked for shapes that were "finite type" (relatively simple shapes). A recent breakthrough by Bachmann and Burklund removed this catch, showing that the theorem works for any shape, even the infinitely complex ones. But they did it using only the abstract "Infinity" language, which is hard for many mathematicians to use for actual calculations.

This paper bridges the gap.
By proving that we can build rigid, concrete models for these coalgebras, the authors allow mathematicians to:

Take the powerful, infinite results of Bachmann and Burklund.
Translate them into the concrete, calculable language of "point-set" models (like the cellular chains functor $C_*(-; k)$ ).
Now, anyone can use standard algebraic tools to study the most complex, infinite shapes in the universe, without needing to be a wizard in abstract infinity-category theory.

The Analogy in a Nutshell

Imagine you have a magical, shape-shifting cloud (the homotopy coalgebra).

Before: Mathematicians could describe the cloud's behavior in a magical, abstract language, but they couldn't build a physical model of it to test it in a lab.
The Paper: The authors figured out how to build a physical, rigid model of the cloud using a special kind of "magnetic clay."
The Result: Now, scientists can take this physical model, run experiments on it, and use it to understand the fundamental structure of the universe (specifically, p-adic homotopy types), proving that even the most chaotic, infinite shapes can be understood through concrete, step-by-step construction.

In short, they turned a "wobbly, impossible-to-build" mathematical object into a sturdy, calculable tool, opening the door to understanding the deep structure of space itself.

1. Problem Statement

The paper addresses a fundamental gap in homotopical algebra: the lack of explicit point-set models (i.e., 1-categorical models like model categories) for homotopy coherent coalgebras over operads in the derived $\infty$ -category of chain complexes.

The Algebra vs. Coalgebra Asymmetry: While rectification theorems exist for algebras (e.g., Lurie's theorem stating that $E_1$ -algebras in a monoidal $\infty$ -category can be modeled by strictly associative algebras in a model category), analogous results for coalgebras have been elusive.
Obstacles to Rectification:
1. Non-Closedness: The standard strategy for algebras relies on the monoidal category being closed (so the free algebra functor is a left adjoint). The opposite category of a closed monoidal category is rarely closed, making the "free coalgebra" construction (dual to free algebras) extremely complex and often non-existent in a simple form.
2. Sensitivity to 1-Categorical Structure: Unlike algebras, the $\infty$ -category of coalgebras is not solely determined by the underlying $\infty$ -category; it is highly sensitive to the specific 1-categorical model (e.g., the Dold-Kan correspondence fails to induce an equivalence for coalgebras between simplicial modules and chain complexes).
3. Lack of Models: Prior to this work, there was no way to compute limits, colimits, or derived functors in the $\infty$ -category of coalgebras (e.g., $E_\infty$ -coalgebras) using classical tools like model categories.

Goal: To prove a rectification theorem for coalgebras over cofibrant dg operads over a field $k$ . Specifically, to show that the $\infty$ -category of coalgebras in the derived category $D(k)$ is equivalent to the localization of a specific 1-category of strict point-set coalgebras.

2. Methodology

The authors employ a multi-step strategy combining $\infty$ -categorical definitions with inductive arguments on cell attachments.

A. Defining Coalgebras over Enriched $\infty$ -Operads

Since the standard definition of coalgebras (as algebras over the opposite operad) fails in the enriched setting (because the opposite of a presentable category is not presentable), the authors adopt a novel definition inspired by Heuts:

Dual Schur Functor: They consider the dual Schur functor $\bar{S}c$ , which maps symmetric sequences to endofunctors. Unlike the Schur functor for algebras, $\bar{S}c$ is only lax monoidal, preventing it from directly defining a comonad.
Pro-Objects Extension: To overcome the laxity, they extend the dual Schur functor to the $\infty$ -category of pro-objects ( $\text{pro}(D)$ ). They construct a strong monoidal extension $\bar{S}c_{\text{pro}}$ on $\text{pro}(D)$ .
Pullback Definition: The $\infty$ -category of $P$ -coalgebras in $D$ is defined as the pullback of the category of coalgebras over the comonad $\bar{S}c_{\text{pro}}(P)$ in $\text{pro}(D)$ along the inclusion $D \hookrightarrow \text{pro}(D)$ .
$\text{Coalg}_P(D) \simeq D \times_{\text{pro}(D)} \text{Coalg}_{\bar{S}c_{\text{pro}}(P)}(\text{pro}(D))$

B. The Rectification Strategy (Dévissage)

The core proof relies on the fact that the functor sending an operad to its category of coalgebras converts colimits of operads into limits of categories.

Free Operads: They first establish the result for free operads $T(M)$ . Coalgebras over a free operad are equivalent to coalgebras over the underlying endofunctor. They prove that for an endofunctor $F$ preserving quasi-isomorphisms, the localization of strict $F$ -coalgebras is equivalent to the $\infty$ -category of coalgebras over the induced endofunctor in $D(k)$ .
Cell Attachments: They use the fact that any cofibrant dg operad is a retract of a cell cofibrant operad (built by iterated pushouts of generating cofibrations).
Inductive Step: They show that if a pushout of operads induces a pullback of categories, and if the "base" operads in the pushout are rectifiable, then the resulting operad is also rectifiable. This involves proving that the restriction functors between categories of coalgebras are fibrations of quasi-categories (using properties of isofibrations and fullness).
Retracts: Finally, they extend the result from cell cofibrant operads to all cofibrant operads using the fact that cofibrant operads are retracts of cell cofibrant ones.

3. Key Contributions

Definition of Enriched $\infty$ -Coalgebras: The paper provides the first robust, general definition of coalgebras over an enriched $\infty$ -operad in a stable symmetric monoidal $\infty$ -category, resolving the issue of non-closedness in the opposite category.
Rectification Theorem (Theorem C): The main result establishes an equivalence of $\infty$ -categories:
$P\text{-coalg}[Q.\text{iso}^{-1}] \simeq \text{Coalg}_P(D(k))$
where $P$ is a cofibrant dg operad over a field $k$ . This means homotopy coherent coalgebras can be modeled by strict coalgebras in the 1-category of chain complexes with the transferred model structure.
Explicit Point-Set Models: The authors construct explicit models for:
- $E_\infty$ -coalgebras (via the Barratt-Eccles operad $\Omega BE$ ).
- The cellular chains functor $C_*(-; k)$ with its $E_\infty$ -coalgebra structure.
Comparison with Lurie's Definition: They prove that their definition of coalgebras over an enriched operad agrees with Lurie's definition (via the opposite category) when the operad is non-enriched (i.e., comes from spaces).

4. Key Results

Theorem A: For a field $k$ , the $\infty$ -category of $A_\infty$ -coalgebras in the derived category $D(k)$ is equivalent to the localization of the 1-category of strict $A_\infty$ -coalgebras (with the transferred model structure).
Theorem B: The authors' definition of coalgebras over an enriched $\infty$ -operad is equivalent to Lurie's definition of coalgebras over a classical $\infty$ -operad when the enrichment is trivial.
Theorem D: The functor $C_*(-; k): \text{sSet} \to \Omega BE\text{-coalg}$ is a point-set model for the $\infty$ -categorical singular chains functor equipped with its $E_\infty$ -coalgebra structure.
Corollary 1.15 (Application to Homotopy Types): For a separably closed field $k$ of characteristic $p > 0$ , the derived unit of the adjunction between pointed simplicial sets and absolute partition $L_\infty$ -algebras is an equivalence in homology with $\mathbb{F}_p$ coefficients. This removes the "finite type" and "connected" assumptions previously required in related work (e.g., by Riou).

5. Significance and Applications

Computational Power: The result allows researchers to use classical model category techniques (Quillen adjunctions, derived functors, explicit resolutions) to study homotopy coherent coalgebras, which were previously only accessible via abstract $\infty$ -categorical machinery.
p-Adic Homotopy Theory: The paper provides a point-set algebraic model for nilpotent $p$ -adic homotopy types.
- It bridges the gap between the $\infty$ -categorical result of Bachmann–Burklund (which fully faithfully encodes $p$ -adic homotopy types via chains) and classical algebraic models.
- It enables the removal of the "finite type" hypothesis in Mandell's theorem and subsequent applications to mapping spaces and Lie models, extending these results to a broader class of spaces.
Unified Framework: The authors note that these models are essential for their forthcoming work on unifying various bar-cobar adjunctions (both 1-categorical and $\infty$ -categorical) in the literature, including generalizations to $E_n$ -coalgebras.

In summary, this paper resolves a long-standing obstruction in homotopical algebra by constructing a rigorous bridge between the abstract world of $\infty$ -coalgebras and the concrete world of differential graded coalgebras, enabling new computations in $p$ -adic homotopy theory.