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Oisín Flynn-Connolly

Publications and source records attributed to Oisín Flynn-Connolly.

8 recordsLinked to original sources
arXiv2026

Enriched coalgebras are sometimes comonadic

We introduce an enriched notion of a coalgebra over an operad P in a symmetric monoidal V-category C. When C is semicartesian and P is unital, we construct a V-endofunctor on C associated to P and give conditions under which it is a V-comonad with co-Eilenberg-Moore V-category isomorphic to the V-category of P-coalgebras in C. In many cases, this permits computation of V-categories of coalgebras. The key example is the category of pointed topological spaces with wedge product, enriched over topological spaces with Cartesian product, where this construction recovers the comonadic description of C_n-coalgebras of Moreno-Fernández, Wierstra and the present author. We further recover one direction of Fox's theorem.

math.CT↗
arXiv2026

Central Limits via Dilated Categories

The Central Limit Theorem (CLT) establishes that sufficiently large sequences of independent and identically distributed random variables converge in probability to a normal distribution. This makes the CLT a fundamental building block of statistical reasoning and, by extension, in reasoning about computing systems that are based on statistical inference such as probabilistic programing languages, programs with optimisation, and machine learning components. However, there is no general theory of CLT-like results currently, which forces practitioners to redo proofs without having a good handle on the essential ingredients of CLT-type results. In this paper, we introduce dilated seminorm-enriched category theory as a unifying framework for central limits, and we establish an abstract central limit theorem within that framework. We illustrate how a strengthened version of the classical CLT and the law of large numbers can be obtained as instances of our framework. Moreover, we derive from our framework a novel central limit theorem for symplectic manifolds, the CLT for observables, which finds applications in statistical mechanics.

cs.LO↗
arXiv2026

A recognition principle for iterated suspensions as coalgebras over the little cubes operad

Our main result is a recognition principle for iterated suspensions as coalgebras over the little disks operads. Given a topological operad, we construct a comonad in pointed topological spaces endowed with the wedge product. We then prove an approximation theorem that shows that the comonad associated to the little $n$-cubes operad is weakly equivalent to the comonad $Σ^n Ω^n$ arising from the suspension-loop space adjunction. Finally, our recognition theorem states that every little $n$-cubes coalgebra is homotopy equivalent to an $n$-fold suspension. These results are the Eckmann--Hilton dual of May's foundational results on iterated loop spaces.

math.AT↗
arXiv2025

An obstruction theory for strictly commutative algebras in positive characteristic

This is the first in a sequence of articles exploring the relationship between commutative algebras and $E_\infty$-algebras in characteristic $p$ and mixed characteristic. In this paper we lay the groundwork by defining a new class of cohomology operations over $\mathbb F_p$ called cotriple products, generalising Massey products. We compute the secondary cohomology operations for a strictly commutative dg-algebra and the obstruction theories these induce, constructing several counterexamples to characteristic 0 behaviour, one of which answers a question of Campos, Petersen, Robert-Nicoud and Wierstra. We construct some families of higher cotriple products and comment on their behaviour. Finally, we distingush a subclass of cotriple products that we call higher Steenrod operations and conclude with our main theorem, which says that $E_\infty$-algebras can be rectified if and only if the higher Steenrod operations vanish coherently.

math.AT↗
arXiv2025

A $p$-adic de Rham complex

This is the second in a sequence of three articles exploring the relationship between commutative algebras and $E_\infty$-algebras in characteristic $p$ and mixed characteristic. Given a topological space $X,$ we construct, in a manner analogous to Sullivan's $A_{PL}$-functor, a strictly commutative algebra over $\padic$ which we call the de Rham forms on $X$. We show this complex computes the singular cohomology ring of $X$. We prove that it is quasi-isomorphic as an $E_\infty$-algebra to the Berthelot-Ogus-Deligne \emph{décalage} of the singular cochains complex with respect to the $p$-adic filtration. We show that one can extract concrete invariants from our model, including Massey products which live in the torsion part of the cohomology. We show that if $X$ is formal then, except at possibly finitely many primes, the $p$-adic de Rham forms on $X$ are also formal. We conclude by showing that the $p$-adic de Rham forms provide, in a certain sense, the "best functorial strictly commutative approximation" to the singular cochains complex.

math.AT↗
arXiv2024

Higher order Massey products for algebras over algebraic operads

We introduce higher-order Massey products for algebras over algebraic operads. This extends the work of Fernando Muro on secondary ones. We study their basic properties and behavior with respect to morphisms of algebras and operads and give some connections to formality. We prove that these higher-order operations represent the differentials in a naturally associated operadic Eilenberg--Moore spectral sequence. We also study the interplay between particular choices of higher-order Massey products and quasi-isomorphic $\mathcal P_\infty$-structures on the homology of a $\mathcal P$-algebra. We focus on Koszul operads over a characteristic zero field and explain how our results generalize to the non-Koszul case.

math.AT↗
arXiv2022

Three Schur functors related to pre-Lie algebras

We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the PBW theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of Kähler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra.

math.RA↗