Philip T. Hackney

collaborators

7 papers

math.AT2026

Coskeletality and the higher Segal conditions

Philip Hackney

A simplicial set is $d$ -Segal if and only if it is $(d + 1)$ -coskeletal and satisfies the $d$ -Segal condition in the two lowest relevant simplicial dimensions.

math.GR2026

Embeddable partial groups

Philip Hackney, Justin Lynd, Edoardo Salati

We record a folklore theorem that says a partial group embeds in a group if and only if each word has at most one possible multiplication, regardless of choice of parenthesization.…

math.GR2025

Higher Segal spaces and partial groups

Philip Hackney, Justin Lynd

The d-Segal conditions of Dyckerhoff and Kapranov are exactness properties for simplicial objects based on the geometry of cyclic polytopes in d-dimensional Euclidean space. 2-Sega…

math.AT2025

Models for cyclic infinity operads

Brandon Doherty, Philip Hackney

We construct model structures on cyclic dendroidal sets and cyclic dendroidal spaces for cyclic quasi-operads and complete cyclic dendroidal Segal spaces, respectively. We show the…

math.GR2025

Dimension and partial groups

A partial group with $n + 1$ elements is, when regarded as a symmetric simplicial set, of dimension at most $n$ . This dimension is $n$ if and only if the partial group is a group. As…

math.AT2025

The decomposition space perspective

Philip Hackney

This paper provides an introduction to decomposition spaces and 2-Segal spaces, unifying the two perspectives. We begin by defining decomposition spaces using the active-inert fact…